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Duality Principles and Numerical Procedures for a Large Class of Non-convex Models in the Calculus of Variations

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Fabio Botelho^*

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Abstract

This article develops duality principles and numerical results for a large class of non-convex variational models. The main results are based on fundamental tools of convex analysis, duality theory and calculus of variations. More specifically the approach is established for a class of non-convex functionals similar as those found in some models in phase transition. Finally, in some sections we present concerning numerical examples and the respective softwares.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

In this section we establish a dual formulation for a large class of models in non-convex optimization. It is worth highlighting the main duality principle is applied to double well models similar as those found in the phase transition theory.

Such results are based on the works of J.J. Telega and W.R. Bielski [1,2,3,4] and on a D.C. optimization approach developed in Toland [5]. About the other references, details on the Sobolev spaces involved are found in [6]. Related results on convex analysis and duality theory are addressed in [7,8,9,10,11,12,13].

Similar models on the superconductivity physics may be found in [14,15,16].

At this point we recall that the duality principles are important since the related dual variational formulations are either convex (in fact concave) or have a large region of convexity around their critical points. These features are relevant considering that, from a concerning strict convexity, the standard Newton, Newton type and similar methods are in general convergent. Moreover, the dual variational formulations are also relevant since in some situations, it is possible to assure the global optimality of some critical points which satisfy certain specific constraints theoretically established.

Among the main results here developed, we highlight the duality principles for the quasi-convex formulations in the context of the vectorial calculus of variations. An important example in non-linear elasticity is addressed along the text in details.

Also, for the applications in physics in the final sections, we believe to have found a path to connect the quantum approach with a more classical one in a unified framework.

Indeed, we have presented a path to model a great variety of chemical reactions through such a connection between the atomic and classical worlds.

Finally, in this text we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated.

In order to clarify the notation, here we introduce the definition of topological dual space.

Definition 1.1

(Topological dual spaces). Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined on U. We suppose such a dual space of U, may be represented by another Banach space

U^{*}

, through a bilinear form

{〈 \cdot, \cdot 〉}_{U} : U \times U^{*} \to R

(here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given

f : U \to R

linear and continuous, we assume the existence of a unique

u^{*} \in U^{*}

such that

\begin{matrix} f (u) = {〈 u, u^{*} 〉}_{U}, \forall u \in U . \end{matrix}

(1)

The norm of f , denoted by

{∥ f ∥}_{U^{*}}

, is defined as

\begin{matrix} {∥ f ∥}_{U^{*}} = sup_{u \in U} {| {〈 u, u^{*} 〉}_{U} {| : ∥ u ∥}_{U} \leq 1} \equiv ∥ u^{*} ∥_{U^{*}} . \end{matrix}

(2)

At this point we start to describe the primal and dual variational formulations.

2. A general duality principle non-convex optimization

In this section we present a duality principle applicable to a model in phase transition.

This case corresponds to the vectorial one in the calculus of variations.

Let

Ω \subset R^{n}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u_{1}, \dots, \nabla u_{N}) + G (u_{1}, \dots, u_{N}) - {〈 u_{i}, h_{i} 〉}_{L^{2}},

and where

F (\nabla u_{1}, \dots, \nabla u_{N}) = \int_{Ω} f (\nabla u_{1}, \dots, \nabla u_{N}) d x

f : R^{N \times n} \to R

is a three times Fréchet differentiable function not necessarily convex. Moreover,

V = {u = (u_{1}, \dots, u_{N}) \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω},

h = (h_{1}, \dots, h_{N}) \in L^{2} (Ω; R^{N}),

and

1 < p < + \infty .

We assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Furthermore, suppose G is Fréchet differentiable but not necessarily convex. A global optimum point may not be attained for J so that the problem of finding a global minimum for J may not be a solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Define

V_{0} = W_{0}^{1, 2} (Ω; R^{N})

and

V_{0} (u) = {ϕ \in V_{0} : supp ϕ \subset B (u)},

where

B (u) = {x \in Ω : f^{* *} (\nabla u (x)) < f (\nabla u (x))} .

Moreover,

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{N \times n})

Y_{2} = Y_{2}^{*} = L^{2} (Ω; R^{N \times n})

Y_{3} = Y_{3}^{*} = L^{2} (Ω; R^{N}),

so that at this point we define,

F_{1} : V \times V_{0} \to R

G_{1} : V \to R

G_{2} : V \to R

G_{3} : V_{0} \to R

and

G_{4} : V \to R,

\begin{matrix} F_{1} (u, ϕ) & = & F (\nabla u_{1} + \nabla ϕ_{1}, \dots, \nabla u_{N} + \nabla ϕ_{N}) + \frac{K}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x \\ + \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x \end{matrix}

(3)

and

G_{1} (u_{1}, \dots, u_{n}) = G (u_{1}, \dots, u_{N}) + \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x - {〈 u_{i}, f_{i} 〉}_{L^{2}},

G_{2} (\nabla u_{1}, \dots, \nabla u_{N}) = \frac{K_{1}}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x,

G_{3} (\nabla ϕ_{1}, \dots, \nabla ϕ_{N}) = \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x,

and

G_{4} (u_{1}, \dots, u_{N}) = \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x .

Define now

J_{1} : V \times V_{0} \to R

J_{1} (u, ϕ) = F (\nabla u + \nabla ϕ) + G (u) - {〈 u_{i}, h_{i} 〉}_{L^{2}} .

Observe that

\begin{matrix} J_{1} (u, ϕ) & = & F_{1} (u, ϕ) + G_{1} (u) - G_{2} (\nabla u) - G_{3} (\nabla ϕ) - G_{4} (u) \\ \leq & F_{1} (u, ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + sup_{v_{1} \in Y_{1}} {{〈 v_{1}, z_{1}^{*} 〉}_{L^{2}} - G_{2} (v_{1})} \\ + sup_{v_{2} \in Y_{2}} {{〈 v_{2}, z_{2}^{*} 〉}_{L^{2}} - G_{3} (v_{2})} \\ + sup_{u \in V} {{〈 u, z_{3}^{*} 〉}_{L^{2}} - G_{4} (u)} \\ = & F_{1} (u, ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}) \\ = & J_{1}^{*} (u, ϕ, z^{*}), \end{matrix}

(4)

\forall u \in V, ϕ \in V_{0} (u), z^{*} = (z_{1}^{*}, z_{2}^{*}, z_{3}^{*}) \in Y^{*} = Y_{1}^{*} \times Y_{2}^{*} \times Y_{3}^{*}

From the general results in [5], we may infer that

\begin{matrix} inf_{(u, ϕ) \in V \times V_{0} (u)} J (u, ϕ) & = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) . \end{matrix}

(5)

On the other hand

inf_{u \in V} J (u) \geq inf_{(u, ϕ) \in V \times V_{0} (u)} J_{1} (u, ϕ) .

From these last two results we may obtain

inf_{u \in V} J (u) \geq inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) .

Moreover, from standards results on convex analysis, we may have

\begin{matrix} inf_{u \in V} J_{1}^{*} (u, ϕ, z^{*}) & = & inf_{u \in V} {F_{1} (u, ϕ) + G_{1} (u) \\ - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})} \\ = & sup_{(v_{1}^{*}, v_{2}^{*}) \in C^{*}} {- F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})}, \end{matrix}

(6)

where

C^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in Y_{1}^{*} \times Y_{3}^{*} : - div {(v_{1}^{*})}_{i} + {(v_{2}^{*})}_{i} = 0, \forall i \in {1, \dots, N}},

F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) = sup_{u \in V} {{〈 u, - div (z_{1}^{*} + v_{1}^{*}) 〉}_{L^{2}} - F_{1} (u, ϕ)},

and

G_{1}^{*} (v_{2}^{*} + z_{2}^{*}) = sup_{u \in V} {{〈 u, v_{2}^{*} + z_{2}^{*} 〉}_{L^{2}} - G_{1} (u)} .

Thus, defining

J_{2}^{*} (ϕ, z^{*}, v^{*}) = F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}),

we have got

\begin{matrix} inf_{u \in V} J (u) & \geq & inf_{(u, ϕ) \in V \times V_{0}} J_{1} (u, ϕ) \\ = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) \\ = & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0}} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} . \end{matrix}

(7)

Finally, observe that

\begin{matrix} inf_{u \in V} J (u) \\ \geq & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0} (u)} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} \\ \geq & sup_{v^{*} \in C^{*}} \{inf_{(z^{*}, ϕ) \in Y^{*} \times V_{0} (u)} J_{2}^{*} (ϕ, z^{*}, v^{*})\} . \end{matrix}

(8)

This last variational formulation corresponds to a concave relaxed formulation in

v^{*}

concerning the original primal formulation.

3. Another duality principle for a simpler related model in phase transition with a respective numerical example

In this section we present another duality principle for a related model in phase transition.

Let

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Denoting

V_{0} = W_{0}^{1, 4} (Ω)

, at this point we define,

F : V \to R

and

F_{1} : V \times V_{0} \to R

F (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x,

and

F_{1} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x .

Observe that

F (u) \geq inf_{ϕ \in V_{0}} F_{1} (u, ϕ), \forall u \in V .

In order to restrict the action of

ϕ

on the region where the primal functional is non-convex, we redefine a not relabeled

V_{0} = \{ϕ \in W_{0}^{1, 4} (Ω) : {(ϕ^{'})}^{2} - 1 \leq 0, in Ω\}

and define also

F_{2} : V \times V_{0} \to R,

F_{3} : V \times V_{0} \to R

and

G : V \times V_{0} \to R

F_{2} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

\begin{matrix} F_{3} (u, ϕ) & = & F_{2} (u, ϕ) + \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(9)

and

\begin{matrix} G (u, ϕ) & = & \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(10)

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

G^{*} : Y^{*} \times Y^{*} \to R

G^{*} (v^{*}, v_{0}^{*}) = sup_{(u, ϕ) \in V \times V_{0}} {{〈 u, v^{*} 〉}_{L^{2}} + {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} - G (u, ϕ)} .

Observe that

inf_{u \in U} J (u) \geq inf_{((u, ϕ), (v^{*}, v_{0}^{*})) \in V \times V_{0} \times {[Y^{*}]}^{2}} {G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ)} .

With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by

J_{1}^{*} : V \times V_{0} \times {[Y^{*}]}^{2} \to R

, where

J_{1}^{*} (u, ϕ, v^{*}, v_{0}^{*}) = G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ) .

Having defined such a functional, we may obtain numerical results by solving a sequence of convex auxiliary sub-problems, through the following algorithm (in order to obtain the concerning critical points, at first we have neglected the constraint

{(ϕ^{'})}^{2} - 1 \leq 0 in Ω

1.: Set $K \approx 0.1$ and $K_{1} = 120.0$ and $0 < ε ≪ 1 .$
2.: Choose $(u_{1}, ϕ_{1}) \in V \times V_{0},$ such that $∥ u_{1} ∥_{1, \infty} < 1$ and $∥ ϕ_{1} ∥_{1, \infty} < 1 .$
3.: Set $n = 1 .$
4.: Calculate $(v_{n}^{*}, {(v_{0}^{*})}_{n})$ solution of the system of equations:

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} = 0,$

that is

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} - u_{n} = 0$

and

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} - ϕ_{n} = 0$

so that

$v_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial u}$

and

${(v_{0}^{*})}_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial ϕ}$
5.: Calculate $(u_{n + 1}, ϕ_{n + 1})$ by solving the system of equations:

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial u} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial ϕ} = 0$

that is

$- v_{n}^{*} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial u} = 0$

and

$- {(v_{0}^{*})}_{n} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial ϕ} = 0$
6.: If $max {∥ u_{n} - u_{n + 1} ∥_{\infty}, ∥ ϕ_{n + 1} - ϕ_{n} ∥_{\infty}} \leq ε$ , then stop, else set $n : = n + 1$ and go to item 4.

At this point, we present the corresponding software in MAT-LAB, in finite differences and based on the one-dimensional version of the generalized method of lines.

Here the software.

***********************

1.: clear all

m8=300;

d=1/m8;

K=0.1;

K1=120;

for i=1:m8

$u o (i, 1) = i^{2} * d / 2;$

vo(i,1)=i*d/10;

yo(i,1)=sin(i*d*pi)/2;

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4.3})$ and $(k < 230000)$

k=k+1;

for i=1:m8-1

duo(i,1)=(uo(i+1,1)-uo(i,1))/d;

dvo(i,1)=(vo(i+1,1)-vo(i,1))/d;

end;

m9=zeros(2,2);

m9(1,1)=1;

i=1;

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2;$

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2};$

m50(:,:,i)=m80(:,:,i)*inv(m12);

z(:,i)=inv(m12)*y11(:,i)* $d^{2}$ ;

for i=2:m8-1

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2$ ;

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1) + u o (i - 1, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1) + v o (i - 1, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2} - m 80 (:, :, i) * m 50 (:, :, i - 1);$

m50(:,:,i)=inv(m12)*m80(:,:,i);

$z (:, i) = i n v (m 12) * (y 11 (:, i) * d^{2} + m 80 (:, :, i) * z (:, i - 1));$

end;

U(1,m8)=1/2;

U(2,m8)=0.0;

for i=1:m8-1

U(:,m8-i)=m50(:,:,m8-i)*U(:,m8-i+1)+z(:,m8-i);

end;

for i=1:m8

u(i,1)=U(1,i);

v(i,1)=U(2,i);

end;

b12=max(abs(u-uo))

uo=u;

vo=v;

u(m8/2,1)

end;

for i=1:m8

y(i)=i*d;

end;

plot(y,uo)

**************************************

For the case in which

f (x) = 0

, we have obtained numerical results for

K = 0.1

and

K_{1} = 120

. For such a concerning solution

u_{0}

obtained, please see Figure 1. For the case in which

f (x) = sin (π x) / 2

, we have obtained numerical results also for

K = 0.1

and

K_{1} = 120

. For such a concerning solution

u_{0}

obtained, please see Figure 2.

Remark 3.1.

Observe that the solutions obtained are approximate critical points. They are not, in a classical sense, the global solutions for the related optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

3.1. A general proposal for relaxation

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}},

where

V = \{u \in W^{1, 4} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (Ω; R^{N}),

f_{1} \in L^{2} (Ω; R^{N})

G : V \to R

is convex and Fréchet differentiable, and

F (\nabla u) = \int_{Ω} f (\nabla u) d x,

where

f : R^{N \times n} \to R

is also Fréchet differentiable.

Assume there exists

\hat{N} \in N

such that

W_{h} \equiv \{y \in R^{N \times n} : f^{* *} (y) < f (y)\} = \cup_{j = 1}^{\hat{N}} W_{j}

where for each

j \in {1, \dots, \hat{N}}

W_{j} \subset R^{N \times n}

is an open connected set such that

\partial W_{j}

is regular. We also suppose

\bar{W_{j}} \cap \bar{W_{k}} = \emptyset, \forall j \neq k .

Define

{\hat{W}}_{j} = \{v_{j} \in W_{0}^{1, 4} (Ω; R^{N}); \nabla v_{j} (x) \in W_{j}, a . e . in Ω\}

and define also

W = \{v = (v_{1}, \dots, v_{\hat{N}}) : v_{j} \in {\hat{W}}_{j} \forall j \in {1, \dots, \hat{N}} and supp v_{j} \cap supp v_{k} = \emptyset, \forall j \neq k\} .

At this point we define

h_{5} (u (x), v (x)) = \{\begin{matrix} f (\nabla u (x) + \nabla v_{j} (x)), & if \nabla u (x) \in W_{j}, \\ f (\nabla u (x)), & if \nabla u (x) \notin W_{h}, \end{matrix}

(11)

and

H (u) = inf_{v \in W_{u}} \int_{Ω} h_{5} (u, v) d x,

where

W_{u} = {v \in W : \nabla u (x) + \nabla v_{j} (x) \in W_{j}, if \nabla u (x) \in W_{j}, a . e . in Ω, \forall j \in {1, \dots, \hat{N}}} .

Moreover, we propose the relaxed functional

J_{1} (u) = H (u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}} .

Observe that clearly

inf_{u \in V} J_{1} (u) \leq inf_{u \in V} J (u) .

4. A convex dual variational formulation for a third similar model

In this section we present another duality principle for a third related model in phase transition.

Let

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close to the infimum of J.

We intend to use the duality theory to solve such a global optimization problem in an appropriate sense to be specified.

At this point we define,

F : V \to R

and

G : V \to R

\begin{matrix} F (u) & = & \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x \\ = & \frac{1}{2} \int_{Ω} {(u^{'})}^{2} d x - \int_{Ω} | u^{'} | d x + 1 / 2 \\ \equiv & F_{1} (u^{'}), \end{matrix}

(12)

and

G (u) = \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

F_{1}^{*} : Y^{*} \to R

and

G^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v^{*}) & = & sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - F_{1} (v)} \\ = & \frac{1}{2} \int_{Ω} {(v^{*})}^{2} d x + \int_{Ω} | v^{*} | d x, \end{matrix}

(13)

and

\begin{matrix} G^{*} ({(v^{*})}^{'}) & = & sup_{u \in V} {- {〈 u^{'}, v^{*} 〉}_{L^{2}} - G (u)} \\ = & \frac{1}{2} \int_{Ω} {({(v^{*})}^{'} + f)}^{2} d x - \frac{1}{2} v^{*} (1) . \end{matrix}

(14)

Observe this is the scalar case of the calculus of variations, so that from the standard results on convex analysis, we have

inf_{u \in V} J (u) = max_{v^{*} \in Y^{*}} {- F_{1}^{*} (v^{*}) - G^{*} (- {(v^{*})}^{'})} .

Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some

{\hat{v}}^{*} \in Y^{*}

Moreover, the corresponding solution

u_{0} \in V

is obtained from the equation

u_{0} = \frac{\partial G ({({\hat{v}}^{*})}^{'})}{\partial {(v^{*})}^{'}} = {({\hat{v}}^{*})}^{'} + f .

Finally, the Euler-Lagrange equations for the dual problem stands for

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) + f (0) = 0, {(v^{*})}^{'} (1) + f (1) = 1 / 2, \end{matrix}

(15)

where

sign (v^{*} (x)) = 1

v^{*} (x) > 0,

sign (v^{*} (x)) = - 1,

v^{*} (x) < 0

and

- 1 \leq sign (v^{*} (x)) \leq 1,

v^{*} (x) = 0 .

We have computed the solutions

v^{*}

and corresponding solutions

u_{0} \in V

for the cases in which

f (x) = 0

and

f (x) = sin (π x) / 2 .

For the solution

u_{0} (x)

for the case in which

f (x) = 0

, please see Figure 3.

For the solution

u_{0} (x)

for the case in which

f (x) = sin (π x) / 2

, please see Figure 4.

Remark 4.1.

Observe that such solutions

u_{0}

obtained are not the global solutions for the related primal optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

4.1. The algorithm through which we have obtained the numerical results

In this subsection we present the software in MATLAB through which we have obtained the last numerical results.

This algorithm is for solving the concerning Euler-Lagrange equations for the dual problem, that is, for solving the equation

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) = 0, {(v^{*})}^{'} (1) = 1 / 2 . \end{matrix}

(16)

Here the concerning software in MATLAB. We emphasize to have used the smooth approximation

| v^{*} | \approx \sqrt{{(v^{*})}^{2} + e_{1}},

where a small value for

e_{1}

is specified in the next lines.

*************************************

1.: clear all
2.: $m_{8} = 800;$ (number of nodes)
3.: $d = 1 / m_{8};$
4.: $e_{1} = 0.00001;$
5.: $f o r i = 1 : m_{8}$

$y o (i, 1) = 0.01;$

$y_{1} (i, 1) = sin (π * i / m_{8}) / 2;$

$e n d;$
6.: $f o r i = 1 : m_{8} - 1$

$d y_{1} (i, 1) = (y_{1} (i + 1, 1) - y_{1} (i, 1)) / d;$

$e n d;$
7.: $f o r k = 1 : 3000$ (we have fixed the number of iterations)

$i = 1;$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 1 + d^{2} * h_{3} + d^{2};$

$m_{50} (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (d y_{1} (i, 1) * d^{2});$
8.: $f o r i = 2 : m_{8} - 1$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 2 + h_{3} * d^{2} + d^{2} - m 50 (i - 1);$

$m 50 (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (z (i - 1) + d y_{1} (i, 1) * d^{2});$

$e n d;$
9.: $v (m_{8}, 1) = (d / 2 + z (m_{8} - 1)) / (1 - m_{50} (m_{8} - 1));$
10.: $f o r i = 1 : m_{8} - 1$

$v (m_{8} - i, 1) = m_{50} (m_{8} - i) * v (m_{8} - i + 1) + z (m_{8} - i);$

$e n d;$
11.: $v (m_{8} / 2, 1)$
12.: $v o = v;$

$e n d;$
13.: $f o r i = 1 : m_{8} - 1$

$u (i, 1) = (v (i + 1, 1) - v (i, 1)) / d + y_{1} (i, 1);$

$e n d;$
14.: $f o r i = 1 : m_{8} - 1$

$x (i) = i * d;$

$e n d;$

$p l o t (x, u (:, 1))$

********************************

5. An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation

In this section we develop an approximate primal dual formulation suitable for a large class of variational models.

Here, the applications are for the Kirchhoff-Love plate model, which may be found in Ciarlet, [17].

At this point we start to describe the primal variational formulation.

Let

Ω \subset R^{2}

be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of

Ω

, which is assumed to be regular (Lipschitzian), is denoted by

\partial Ω

. The vectorial basis related to the cartesian system

{x_{1}, x_{2}, x_{3}}

is denoted by

(a_{α}, a_{3})

, where

α = 1, 2

(in general Greek indices stand for 1 or 2), and where

a_{3}

is the vector normal to

Ω

, whereas

a_{1}

and

a_{2}

are orthogonal vectors parallel to

Ω .

Also,

n

is the outward normal to the plate surface.

The displacements will be denoted by

\hat{u} = {{\hat{u}}_{α}, {\hat{u}}_{3}} = {\hat{u}}_{α} a_{α} + {\hat{u}}_{3} a_{3} .

The Kirchhoff-Love relations are

\begin{matrix} {\hat{u}}_{α} (x_{1}, x_{2}, x_{3}) = u_{α} (x_{1}, x_{2}) - x_{3} w {(x_{1}, x_{2})}_{, α} \\ and {\hat{u}}_{3} (x_{1}, x_{2}, x_{3}) = w (x_{1}, x_{2}) . \end{matrix}

(17)

Here

- h / 2 \leq x_{3} \leq h / 2

so that we have

u = (u_{α}, w) \in U

where

\begin{matrix} U & = & \{u = (u_{α}, w) \in W^{1, 2} (Ω; R^{2}) \times W^{2, 2} (Ω), \\ u_{α} = w = \frac{\partial w}{\partial n} = 0 on \partial Ω\} \\ = & W_{0}^{1, 2} (Ω; R^{2}) \times W_{0}^{2, 2} (Ω) . \end{matrix}

It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

We also define the operator

Λ : U \to Y \times Y

, where

Y = Y^{*} = L^{2} (Ω; R^{2 \times 2})

, by

Λ (u) = {γ (u), κ (u)},

γ_{α β} (u) = \frac{u_{α, β} + u_{β, α}}{2} + \frac{w_{, α} w_{, β}}{2},

κ_{α β} (u) = - w_{, α β} .

The constitutive relations are given by

N_{α β} (u) = H_{α β λ μ} γ_{λ μ} (u),

(18)

M_{α β} (u) = h_{α β λ μ} κ_{λ μ} (u),

(19)

where:

{H_{α β λ μ}}

and

\{h_{α β λ μ} = \frac{h^{2}}{12} H_{α β λ μ}\}

, are symmetric positive definite fourth order tensors. From now on, we denote

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}^{- 1}

and

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

Furthermore

{N_{α β}}

denote the membrane force tensor and

{M_{α β}}

the moment one. The plate stored energy, represented by

(G \circ Λ) : U \to R

is expressed by

(G \circ Λ) (u) = \frac{1}{2} \int_{Ω} N_{α β} (u) γ_{α β} (u) d x + \frac{1}{2} \int_{Ω} M_{α β} (u) κ_{α β} (u) d x

(20)

and the external work, represented by

F : U \to R

, is given by

F (u) = {〈 w, P 〉}_{L^{2}} + {〈 u_{α}, P_{α} 〉}_{L^{2}},

(21)

where

P, P_{1}, P_{2} \in L^{2} (Ω)

are external loads in the directions

a_{3}

a_{1}

and

a_{2}

respectively. The potential energy, denoted by

J : U \to R

is expressed by:

J (u) = (G \circ Λ) (u) - F (u)

Define now

J_{3} : \tilde{U} \to R

J_{3} (u) = J (u) + J_{5} (w) .

where

J_{5} (w) = 10 \int_{Ω} \frac{a^{K b w}}{ln (a) K^{3 / 2}} d x + 10 \int_{Ω} \frac{a^{- K (b w - 1 / 100)}}{ln (a) K^{3 / 2}} d x .

In such a case for

a = 2.71

K = 185

b = P / | P |

Ω

and

\tilde{U} = {{u \in U : ∥ w ∥}_{\infty} \leq 0.01 and P w \geq 0 a . e . in Ω},

we get

\begin{matrix} \frac{\partial J_{3} (u)}{\partial w} & = & \frac{\partial J (u)}{\partial w} + \frac{\partial J_{5} (u)}{\partial w} \\ \approx & \frac{\partial J (u)}{\partial w} + O (\pm 3.0), \end{matrix}

(22)

and

\begin{matrix} \frac{\partial^{2} J_{3} (u)}{\partial w^{2}} & = & \frac{\partial^{2} J (u)}{\partial w^{2}} + \frac{\partial^{2} J_{5} (u)}{\partial w^{2}} \\ \approx & \frac{\partial^{2} J (u)}{\partial w^{2}} + O (850) . \end{matrix}

(23)

This new functional

J_{3}

has a relevant improvement in the convexity conditions concerning the previous functional J.

Indeed, we have obtained a gain in positiveness for the second variation

\frac{\partial^{2} J (u)}{\partial w^{2}},

which has increased of order

O (700 - 1000) .

Moreover the difference between the approximate and exact equation

\frac{\partial J (u)}{\partial w} = 0

is of order

O (\pm 3.0)

which corresponds to a small perturbation in the original equation for a load of

P = 1500 N / m^{2},

for example. Summarizing, the exact equation may be approximately solved in an appropriate sense.

5.1. A duality principle for the concerning quasi-convex envelope

In this section, denoting

V_{1} = {ϕ = ϕ (x, y) \in W^{1, 2} (Ω \times Ω; R^{2}) : ϕ = 0 on Ω \times \partial Ω},

we define the functional

J_{1} : U \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} . \end{matrix}

(24)

where

G_{1} ({w_{, α β}}) = \frac{1}{2} \int_{Ω} h_{α β λ μ} w_{, α β} w_{, λ μ} d x

and,

\begin{matrix} G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{α β λ μ} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} (x, y) + \frac{1}{2} w_{, α} w_{, β}) \\ \times (\frac{1}{2} (u_{λ, μ} + u_{λ, μ}) + ϕ_{λ, y_{μ}} (x, y) + \frac{1}{2} w_{, λ} w_{, μ}) d x d y \end{matrix}

We define also

J_{2} ({u_{α}}, ϕ) = inf_{w \in W_{0}^{2, 2} (Ω)} J_{1} (u, ϕ),

and

J_{3} ({u_{α}}) = inf_{ϕ \in V_{1}} J_{2} ({u_{α}}, ϕ) .

It is a well known result from the modern Calculus of Variations theory (please, see [18] for details) that

inf_{u \in U} J (u) = inf_{{u_{α}} \in W_{0}^{1, 2} (Ω; R^{2})} J_{3} ({u_{α}}) .

At this point we denote

Y_{1} = Y_{1}^{*} = Y_{3} = Y_{3}^{*} \equiv L^{2} (Ω \times Ω; R^{4})

and

Y_{2} = Y_{2}^{*} \equiv L^{2} (Ω \times Ω; R^{2}) .

Observe that

\begin{matrix} J (u) \\ = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ = & G_{1} ({w_{, α β}}) - {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}} \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ \geq & inf_{v_{3} \in Y_{3}} {- {〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} + G_{1} ({(v_{3})}_{α β})} \\ + inf_{w \in W_{0}^{2, 2} (Ω)} \{{〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x) Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}}\} \\ + inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{α β} v_{α β}^{*} d x d y + G_{2} ({v_{α β}})\} \\ + inf_{(v_{2}, {u_{α}}) \in Y_{2} \times W_{0}^{1, 2} (Ω; R^{2})} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} {(v_{2})}_{α} (x, y) {(v_{2})}_{β} (x, y)) \\ \times v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{α} (x, y) Q_{α} (x, y) d x d y\} \\ \geq & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

(25)

\forall u \in U, (M, Q) \in C^{*}

v = {v_{α β}} \in A^{*}

where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap B^{*},

A_{1}^{*} = {{v_{α β}^{*}} \in Y_{1}^{*} : {(v_{α β}^{*})}_{, y_{β}} = 0, in Ω},

A_{2}^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} v_{α β}^{*} d y)}_{, x_{β}} + P_{α} = 0, in Ω\},

B^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \{v_{α β}^{*} (x, y)\} is positive definite in Ω \times Ω\} .

and

C^{*} = \{(M, Q) \in Y_{3}^{*} \times Y_{2}^{*} : M_{α β, α β} - {(\int_{Ω} Q_{α} d y)}_{, x_{α}} - P = 0, in Ω\} .

Also

\{\bar{v_{α β}^{*}}\} = {\{v_{α β}^{*}\}}^{- 1},

and

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}

in an appropriate tensor sense.

Here it is worth highlighting we have denoted,

\begin{matrix} G_{1}^{*} (M) & = & sup_{v_{3} \in Y_{3}} {{〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} - G_{1} (v_{3})} \\ = & \frac{1}{2} \int_{Ω} {\bar{h}}_{α β λ μ} M_{α β} M_{λ μ} d x, \end{matrix}

(26)

where we recall that

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

in an appropriate tensorial sense.

Summarizing, defining

J^{*} : C^{*} \times A^{*} \to R

\begin{matrix} J^{*} ((M, Q), v^{*}) & = & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

(27)

we have got

inf_{u \in U} J (u) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) .

Remark 5.1.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize such results are extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if

((M_{0}, Q_{0}), v_{0}^{*}) \in C^{*} \times A^{*}

is such that

δ J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(w_{0})}_{, α β} = {\bar{h}}_{α β λ μ} {(M_{0})}_{λ μ},

and

{(v_{0}^{*})}_{α β} = H_{α β λ} (\frac{{(u_{0})}_{λ, μ} + {(u_{0})}_{μ, λ}}{2} + \frac{{(ϕ_{0})}_{λ, y_{μ}} + {(ϕ_{0})}_{μ, y_{λ}}}{2} + \frac{1}{2} {(v_{2_{0}})}_{λ} {(v_{2_{0}})}_{μ}),

{(v_{0}^{*})}_{α β, y_{β}} = 0,

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) \\ = & J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) \\ = & inf_{u \in V} J (u) . \end{matrix}

(28)

Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 0 .

Here we highlight such last results are valid just for this finite-dimensional model version.

6. A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations

In this section we develop a duality principle for a related vectorial model in the calculus of variations.

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ .

For

1 < p < + \infty

, consider a functional

J : V \to R

where

J (u) = G (\nabla u) + F (u) - {〈 u, f 〉}_{L^{2}},

where

V = \{u \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (\bar{Ω}; R^{N})

and

f \in L^{2} (Ω; R^{N}) .

We assume

G : Y \to R

and

F : V \to R

are Fréchet differentiable and F is also convex.

Also

G (\nabla u) = \int_{Ω} g (\nabla u) d x,

where

g : R^{N \times n} \to R

it is supposed to be Fréchet differentiable. Here we have denoted

Y = L^{p} (Ω; R^{N \times n})

We define also

J_{1} : V \times Y_{1} \to R

J_{1} (u, ϕ) = G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}},

where

Y_{1} = W^{1, p} (Ω \times Ω; R^{N})

and

G_{1} (\nabla u + \nabla_{y} ϕ) = \frac{1}{| Ω |} \int_{Ω} \int_{Ω} g (\nabla u (x) + \nabla_{y} ϕ (x, y)) d x d y .

Moreover, we define the relaxed functional

J_{2} : V \to R

J_{2} (u) = inf_{ϕ \in V_{0}} J_{1} (u, ϕ),

where

V_{0} = {ϕ \in Y_{1} : ϕ (x, y) = 0, on Ω \times \partial Ω} .

Now observe that

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + G_{1} (\nabla u + \nabla_{y} ϕ) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{2}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x + G_{1} (v)\} \\ + inf_{(v, ϕ) \in V \times V_{0}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}}\} \\ = & - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) \\ + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ, \end{matrix}

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\forall (u, ϕ) \in V \times V_{0}, v^{*} \in A^{*}

, where

A^{*} = {v^{*} \in Y_{2}^{*} : {div}_{y} v^{*} (x, y) = 0, in Ω} .

Here we have denoted

G_{1}^{*} (v^{*}) = sup_{v \in Y_{2}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x - G_{1} (v)\},

where

Y_{2} = L^{p} (Ω \times Ω; R^{N \times n}),

Y_{2}^{*} = L^{q} (Ω \times Ω; R^{N \times n}),

and where

\frac{1}{p} + \frac{1}{q} = 1 .

Furthermore, for

v^{*} \in A^{*}

, we have

\begin{matrix} F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) - \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ \\ = & sup_{(v, ϕ) \in V \times V_{0}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x - F (u) + {〈 u, f 〉}_{L^{2}}\}, \end{matrix}

(30)

Therefore, denoting

J_{3}^{*} : Y_{2}^{*} \to R

J_{3}^{*} (v^{*}) = - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\int_{Ω} v^{*} (x, y) d y) + f) + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ,

we have got

inf_{u \in V} J_{2} (u) \geq sup_{v^{*} \in A^{*}} J_{3}^{*} (v^{*}) .

Finally, we highlight such a dual functional

J_{3}^{*}

is convex (in fact concave).

6.1. An example in finite elasticity

In this section we develop an application of results obtained in the last section to a model in non-linear elasticity.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Concerning a standard model in non-linear elasticity, consider a functional

J : V \to R

where

\begin{matrix} J (u) \\ = & \frac{1}{2} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{1}{2} u_{m, i} u_{m, j}) (\frac{u_{k, l} + u_{j, i}}{2} + \frac{1}{2} u_{m, k} u_{m, l}) d x \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}} \end{matrix}

(31)

where

f \in L^{2} (Ω; R^{3})

and

V = W_{0}^{1, 2} (Ω; R^{3}) .

Here

{H_{i j k l}}

is a fourth-order and positive definite symmetric tensor (in an appropriate standard sense). Moreover,

u = (u_{1}, u_{2}, u_{3}) \in V

is a field of displacements resulting from the f load field action on the volume comprised by

Ω

At this point, we define the functional

J_{1} : V \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

(32)

where

V_{1} = {ϕ \in W^{1, 2} (Ω \times Ω; R^{3}) : ϕ = 0 on Ω \times \partial Ω} .

We define also the quasi-convex envelop of J, denoted by

Q_{J} : V \to R

, as

Q_{J} (u) = inf_{ϕ \in V_{1}} J_{1} (u, ϕ) .

It is a well known result from the modern calculus of variations theory (please see [18] for details), that

inf_{u \in V} J (u) = inf_{u \in V} Q_{J} (u) .

Observe now that, denoting

Y_{1} = Y_{1}^{*} = L^{2} (Ω \times Ω; R^{9})

Y_{2} = Y_{2}^{*} = L^{2} (Ω \times Ω; R^{3}),

and

\begin{matrix} G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \end{matrix}

(33)

we have that

\begin{matrix} J_{1} (u, ϕ) \\ = & G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y \\ + G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{i j} σ_{i j} d x d y - G_{1} ({v_{i j}})\} \\ + inf_{v_{2} \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{i j} Q_{i j} d x d y + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} \frac{1}{2} ({(v_{2})}_{m i} {(v_{2})}_{m j})) d x d y\} \\ + inf_{(u, ϕ) \in V \times V_{1}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} + Q_{i j}) (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2}) d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}}\} \\ \geq & - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y, \end{matrix}

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\forall (u, ϕ) \in V \times V_{1}, (σ, Q) \in A^{*}

, where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap A_{3}^{*}

A_{1}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : σ_{i j, y_{j}} + Q_{i j, y_{j}} = 0, in Ω \times Ω} .

A_{2}^{*} = \{(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} (σ_{i j}) d y)}_{x_{j}} + \frac{1}{| Ω |} {(\int_{Ω} (Q_{i j}) d y)}_{x_{j}} + f_{i} = 0, in Ω\},

A_{3}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : {σ_{i j}} is positive definite in Ω \times Ω} .

Hence, denoting

J^{*} (σ, Q) = - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y,

we have obtained

inf_{u \in V} J (u) \geq sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) .

Remark 6.1.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize again such results are also extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if

(σ_{0}, Q_{0}) \in A^{*}

is such that

δ J^{*} (σ_{0}, Q_{0}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(σ_{0})}_{i j} = H_{i j k l} (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}}))

and

{(Q_{0})}_{i j} = {(σ_{0})}_{i m} {(v_{2_{0}})}_{m j},

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} (σ_{0}, Q_{0}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{(σ, Q) A^{*}} J^{*} (σ, Q),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) \\ = & J^{*} (σ_{0}, Q_{0}) \\ = & inf_{u \in V} J (u) . \end{matrix}

(35)

Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 0 .

Here we highlight such last results are valid just for this finite-dimensional model version.

7. An exact convex dual variational formulation for a non-convex primal one

In this section we develop a convex dual variational formulation suitable to compute a critical point for the corresponding primal one.

Let

Ω \subset R^{2}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (u_{x}, u_{y}) - {〈 u, f 〉}_{L^{2}},

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Here we denote

Y = Y^{*} = L^{2} (Ω)

and

Y_{1} = Y_{1}^{*} = L^{2} (Ω) \times L^{2} (Ω) .

Defining

V_{1} = {{u \in V : ∥ u ∥}_{1, \infty} \leq K_{1}}

for some appropriate

K_{1} > 0

, suppose also F is twice Fréchet differentiable and

det \{\frac{\partial^{2} F (u_{x}, u_{y})}{\partial v_{1} \partial v_{2}}\} \neq 0,

\forall u \in V_{1} .

Define now

F_{1} : V \to R

and

F_{2} : V \to R

F_{1} (u_{x}, u_{y}) = F (u_{x}, u_{y}) + \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

and

F_{2} (u_{x}, u_{y}) = \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

where here we denote

d x = d x_{1} d x_{2} .

Moreover, we define the respective Legendre transform functionals

F_{1}^{*}

and

F_{2}^{*}

F_{1}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{1} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{2}},

and

F_{2}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} + f_{1} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{2} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} + f_{1} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{2}} .

Here

f_{1}

is any function such that

{(f_{1})}_{x} = f, in Ω .

Furthermore, we define

\begin{matrix} J^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + F_{2}^{*} (v^{*}) \\ = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x . \end{matrix}

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Observe that through the target conditions

v_{1}^{*} + f_{1} = ε u_{x},

v_{2}^{*} = ε u_{y},

we may obtain the compatibility condition

{(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0 .

Define now

A^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in B_{r} (0, 0) \subset Y_{1}^{*} : {(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0, in Ω},

for some appropriate

r > 0

such that

J^{*}

is convex in

B_{r} (0, 0) .

Consider the problem of minimizing

J^{*}

subject to

v^{*} \in A^{*} .

Assuming

r > 0

is large enough so that the restriction in r is not active, at this point we define the associated Lagrangian

J_{1}^{*} (v^{*}, φ) = J^{*} (v^{*}) + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}},

where

φ

is an appropriate Lagrange multiplier.

Therefore

\begin{matrix} J_{1}^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x \\ + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}} . \end{matrix}

(37)

The optimal point in question will be a solution of the corresponding Euler-Lagrange equations for

J_{1}^{*} .

From the variation of

J_{1}^{*}

v_{1}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + \frac{v_{1}^{*} + f}{ε} - \frac{\partial φ}{\partial y} = 0 .

(38)

From the variation of

J_{1}^{*}

v_{2}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + \frac{v_{2}^{*}}{ε} + \frac{\partial φ}{\partial x} = 0 .

(39)

From the variation of

J_{1}^{*}

φ

we have

{(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} = 0 .

From this last equation, we may obtain

u \in V

such that

v_{1}^{*} + f = ε u_{x},

and

v_{2}^{*} = ε u_{y} .

From this and the previous extremal equations indicated we have

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + u_{x} - \frac{\partial φ}{\partial y} = 0,

and

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + u_{y} + \frac{\partial φ}{\partial x} = 0 .

so that

v_{1}^{*} + f = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}},

and

v_{2}^{*} = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}} .

From this and equation (38) and (39) we have

\begin{matrix} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}})}_{x} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}})}_{y} \\ + {(v_{1}^{*} + f_{1})}_{x} + {(v_{2}^{*})}_{y} \\ = & - ε u_{x x} - ε u_{y y} + {(v_{1}^{*})}_{x} + {(v_{2}^{*})}_{y} + f = 0 . \end{matrix}

(40)

Replacing the expressions of

v_{1}^{*}

and

v_{2}^{*}

into this last equation, we have

- ε u_{x x} - ε u_{y y} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0,

so that

{(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0, in Ω .

(41)

Observe that if

\nabla^{2} φ = 0

then there exists

\hat{u}

such that u and

φ

are also such that

u_{x} - φ_{y} = {\hat{u}}_{x}

and

u_{y} + φ_{x} = {\hat{u}}_{y} .

The boundary conditions for

φ

must be such that

\hat{u} \in W_{0}^{1, 2} .

From this and equation (41) we obtain

δ J (\hat{u}) = 0 .

Summarizing, we may obtain a solution

\hat{u} \in W_{0}^{1, 2}

of equation

δ J (\hat{u}) = 0

by minimizing

J^{*}

A^{*}

Finally, observe that clearly

J^{*}

is convex in an appropriate large ball

B_{r} (0, 0)

for some appropriate

r > 0

8. Another primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

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α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(43)

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K}

for some appropriate

K > 0

to be specified.

Observe that, denoting

φ = - γ \nabla^{2} u + 2 v_{0}^{*} u - f

we have

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} = \frac{1}{α} + 4 K_{1} u^{2}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} = γ \nabla^{2} - 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}} = K_{1} (2 φ + 2 (- γ \nabla^{2} u + 2 v_{0}^{*} u)) - 2 u

so that

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}})}^{2} \\ = & \frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} - \frac{γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}}{α} \\ - 4 K_{1}^{2} φ^{2} - 8 K_{1} φ (- γ \nabla^{2} + 2 v_{0}^{*}) u + 8 K_{1} φ u \\ + 4 K_{1} (- γ \nabla^{2} u + 2 v_{0}^{*} u) u . \end{matrix}

(44)

Observe now that a critical point

φ = 0

and

(- γ \nabla^{2} u + 2 v_{0}^{*} u) u = f u \geq 0

Ω

Therefore, for an appropriate large

K_{1} > 0

, also at a critical point, we have

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & 4 K_{1} f u - \frac{δ^{2} J (u)}{α} + K_{1} \frac{{(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} > 0 . \end{matrix}

(45)

Remark 8.1.

From this last equation we may observe that

J_{1}^{*}

has a large region of convexity about any critical point

(u_{0}, {\hat{v}}_{0}^{*})

, that is, there exists a large

r > 0

such that

J_{1}^{*}

is convex on

B_{r} (u_{0}, {\hat{v}}_{0}^{*}) .

With such results in mind, we may easily prove the following theorem.

Theorem 8.2.

Assume

K_{1} ≫ max {1, K, K_{3}}

and suppose

(u_{0}, {\hat{v}}_{0}^{*}) \in V_{1} \times B^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, there exists

r > 0

such that

J_{1}^{*}

is convex in

E^{*} = B_{r} (u_{0}, {\hat{v}}_{0}^{*}) \cap (V_{1} \times B^{*})

δ J (u_{0}) = 0,

and

- J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{0}^{*}) \in E^{*}} J_{1}^{*} (u, v_{0}^{*}) .

9. A third primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(46)

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{1}{2} \int_{Ω} K u^{2} d x \\ - {〈 u, v_{1}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{(- 2 v_{0}^{*} + K)} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(47)

where

ε > 0

is a small real constant.

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K_{4}}

and

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥ \leq K_{5}},

for some appropriate real constants

K_{4}, K_{5} > 0

to be specified.

Remark 9.1.

Define now

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}

For an appropriate function (or, in a more general fashion, an appropriate bounded operator)

M_{1}

define

B_{1}^{*} = {v_{0}^{*} \in B^{*} : 2 v_{0}^{*} + M_{1} \geq ε_{1}},

for some small parameter

ε_{1} > 0 .

Moreover, define

E^{*} = {u \in V_{1} : \sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |} .

Since for

(u, v_{0}^{*}) \in V_{1} \times B_{1}^{*}

we have

u f \geq 0, in Ω

, so that for

u_{1}, u_{2} \in V_{1}

we have

sign (u_{1}) = sign (u_{2}) in Ω,

we may infer that

E^{*}

is a convex set.

Moreover if

(u, v_{0}^{*}) \in E^{*} \times B_{1}^{*}

, then

\sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |}

so that

4 α u^{2} \geq M_{1} + γ \nabla^{2}

and

2 v_{0}^{*} + M_{1} \geq ε_{1}

so that

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2} \geq ε_{1} .

Such a result we will be used many times in the next sections.

Observe that, defining

φ = v_{0}^{*} - α (u^{2} - β)

we may obtain

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u^{2}} = - γ \nabla^{2} + K + \frac{α}{α + ε} 4 u^{2} - 2 φ \frac{α}{α + ε}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial {(v_{1}^{*})}^{2}} = \frac{1}{- 2 v_{0}^{*} + K}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}} = - 1

so that

\begin{matrix} det \{\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}}\} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}})}^{2} \\ = & \frac{- γ \nabla^{2} + 2 v_{0}^{*} + 4 \frac{α^{2}}{α + ε} u^{2} - 2 \frac{α}{α + ε} φ}{- 2 v_{0}^{*} + K} \\ \equiv & H (u, v_{0}^{*}) . \end{matrix}

(48)

However, at a critical point, we have

φ = 0

so that, for a fixed

v_{0}^{*} \in B^{*}

we define the non-active but convex restriction

{(C_{1})}_{v_{0}^{*}}^{*} = {u \in V_{1} : {(φ)}^{2} \leq ε},

for a small parameter

ε > 0 .

From such results, assuming

K ≫ max {K_{3}, K_{4}, K_{5}},

and

0 < ε ≪ ε_{1} ≪ 1

, we have that

H (u, v_{0}^{*}) > 0,

for

v_{0}^{*} \in B_{1}^{*}

and

u \in E^{*} \cap {(C_{1})}_{v_{0}^{*}}^{*} .

With such results in mind, we may easily prove the following theorem.

Theorem 9.2.

Suppose

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in (E^{*} \cap {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}) \times D^{*} \times B_{1}^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, we have that

δ J (u_{0}) = 0

and

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(49)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})

may be easily made similarly as in the previous sections.

Moreover, observe that for

K > 0

sufficiently large, we have

\frac{\partial^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0, \forall v_{0}^{*} \in B^{*}

so that this and the other hypotheses, we have also

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*})

and

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*}) .

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that

\begin{matrix} J (u_{0}) & = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(50)

Moreover, observe that

\begin{matrix} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}) \\ \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x \\ + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - \frac{K}{2} \int_{Ω} u^{2} d x \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {({\hat{v}}_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \leq & sup_{v_{0}^{*} \in Y^{*}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + 〈 u^{2}, v_{0}^{*} 〉 \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \forall u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} . \end{matrix}

(51)

Summarizing, we have got

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \leq inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) .

From such results, we may infer that

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(52)

The proof is complete. □

10. An algorithm for a related model in shape optimization

The next two subsections have been previously published by Fabio Silva Botelho and Alexandre Molter in [8], Chapter 21.

10.1. Introduction

Consider an elastic solid which the volume corresponds to an open, bounded, connected set, denoted by

Ω \subset R^{3}

with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ_{0} \cup Γ_{t}

where

Γ_{0} \cap Γ_{t} = \emptyset .

Consider also the problem of minimizing the functional

\hat{J} : U \times B \to R

where

\hat{J} (u, t) = \frac{1}{2} {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

subject to

\{\begin{matrix} {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} = 0 in Ω, \\ H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3} . \end{matrix}

(53)

Here

n = (n_{1}, n_{2}, n_{3})

denotes the outward normal to

\partial Ω

and

U = {u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3}) : u = (0, 0, 0) = 0 on Γ_{0}},

B = \{t : Ω \to [0, 1] measurable : \int_{Ω} t (x) d x = t_{1} | Ω |\},

where

0 < t_{1} < 1

and

| Ω |

denotes the Lebesgue measure of

Ω .

Moreover

u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3})

is the field of displacements relating the cartesian system

(0, x_{1}, x_{2}, x_{3})

, resulting from the action of the external loads

f \in L^{2} (Ω; R^{3})

and

\hat{f} \in L^{2} (Γ_{t}; R^{3}) .

We also define the stress tensor

{σ_{i j}} \in Y^{*} = Y = L^{2} (Ω; R^{3 \times 3}),

σ_{i j} (u) = H_{i j k l} (t) e_{k l} (u),

and the strain tensor

e : U \to L^{2} (Ω; R^{3 \times 3})

e_{i j} (u) = \frac{1}{2} (u_{i, j} + u_{j, i}), \forall i, j \in {1, 2, 3} .

Finally,

{H_{i j k l} (t)} = {t H_{i j k l}^{0} + (1 - t) H_{i j k l}^{1}},

where

H^{0}

corresponds to a strong material and

H^{1}

to a very soft material, intending to simulate voids along the solid structure.

The variable t is the design one, which the optimal distribution values along the structure are intended to minimize its inner work with a volume restriction indicated through the set B.

The duality principle obtained is developed inspired by the works in [1,2]. Similar theoretical results have been developed in [7], however we believe the proof here presented, which is based on the min-max theorem is easier to follow (indeed we thank an anonymous referee for his suggestion about applying the min-max theorem to complete the proof). We highlight throughout this text we have used the standard Einstein sum convention of repeated indices.

Moreover, details on the Sobolev spaces addressed may be found in [6]. In addition, the primal variational development of the topology optimization problem has been described in [7].

The main contributions of this work are to present the detailed development, through duality theory, for such a kind of optimization problems. We emphasize that to avoid the check-board standard and obtain appropriate robust optimized structures without the use of filters, it is necessary to discretize more in the load direction, in which the displacements are much larger.

10.2. Mathematical formulation of the topology optimization problem

Our mathematical topology optimization problem is summarized by the following theorem.

Theorem 10.1.

Consider the statements and assumptions indicated in the last section, in particular those refereing to Ω and the functional

\hat{J} : U \times B \to R .

Define

J_{1} : U \times B \to R

J_{1} (u, t) = - G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

where

G (e (u), t) = \frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x,

and where

d x = d x_{1} d x_{2} d x_{3} .

Define also

J^{*} : U \to R

\begin{matrix} J^{*} (u) & = & inf_{t \in B} {J_{1} (u, t)} \\ = & inf_{t \in B} {- G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}} . \end{matrix}

(54)

Assume there exists

c_{0}, c_{1} > 0

such that

H_{i j k l}^{0} z_{i j} z_{k l} > c_{0} z_{i j} z_{i j}

and

H_{i j k l}^{1} z_{i j} z_{k l} > c_{1} z_{i j} z_{i j}, \forall z = {z_{i j}} \in R^{3 \times 3}, such that z \neq 0 .

Finally, define

J : U \times B \to R \cup {+ \infty}

J (u, t) = \hat{J} (u, t) + I n d (u, t),

where

I n d (u, t) = \{\begin{matrix} 0, & if (u, t) \in A^{*}, \\ + \infty, & otherwise, \end{matrix}

(55)

where

A^{*} = A_{1} \cap A_{2},

A_{1} = {(u, t) \in U \times B : {(σ_{i j} (u))}_{, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

A_{2} = {(u, t) \in U \times B : σ_{i j} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Under such hypotheses, there exists

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} J (u_{0}, t_{0}) & = & inf_{(u, t) \in U \times B} J (u, t) \\ = & sup_{\hat{u} \in U} J^{*} (\hat{u}) \\ = & J^{*} (u_{0}) \\ = & \hat{J} (u_{0}, t_{0}) \\ = & inf_{(t, σ) \in B \times C^{*}} G^{*} (σ, t) \\ = & G^{*} (σ (u_{0}), t_{0}), \end{matrix}

(56)

where

\begin{matrix} G^{*} (σ, t) & = & sup_{v \in Y} {{〈 v_{i j}, σ_{i j} 〉}_{L^{2} (Ω)} - G (v, t)} \\ = & \frac{1}{2} \int_{Ω} {\bar{H}}_{i j k l} (t) σ_{i j} σ_{k l} d x, \end{matrix}

(57)

{{\bar{H}}_{i j k l} (t)} = {H_{i j k l} (t)}^{- 1}

and

C^{*} = C_{1} \cap C_{2},

where

C_{1} = {σ \in Y^{*} : σ_{i j, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

C_{2} = {σ \in Y^{*} : σ_{i j} n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Proof.

Observe that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{inf_{u \in U} J (u, t)\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} 〉}_{L^{2} (Ω)} \\ - {〈 {\hat{u}}_{i}, H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ - \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{- \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (\hat{u}) d x \\ {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\} \\ = & inf_{t \in B} \{inf_{σ \in C^{*}} G^{*} (σ, t)\} . \end{matrix}

(58)

Also, from this and the min-max theorem, there exist

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{sup_{\hat{u} \in U} J_{1} (u, t)\} \\ = & sup_{u \in U} \{inf_{t \in B} J_{1} (u, t)\} \\ = & J_{1} (u_{0}, t_{0}) \\ = & inf_{t \in B} J_{1} (u_{0}, t) \\ = & J^{*} (u_{0}) . \end{matrix}

(59)

Finally, from the extremal necessary condition

\frac{\partial J_{1} (u_{0}, t_{0})}{\partial u} = 0

we obtain

{(H_{i j k l} (t_{0}) e_{k l} (u_{0}))}_{, j} + f_{i} = 0 in Ω,

and

H_{i j k l} (t_{0}) e_{k l} (u_{0}) n_{j} - {\hat{f}}_{i} = 0 on Γ_{t}, \forall i \in {1, 2, 3},

so that

G (e (u_{0})) = \frac{1}{2} {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} .

Hence

(u_{0}, t_{0}) \in A^{*}

so that

I n d (u_{0}, t_{0}) = 0

and

σ (u_{0}) \in C^{*} .

Moreover

\begin{matrix} J^{*} (u_{0}) & = & - G (e (u_{0})) + {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} \\ = & G (e (u_{0})) \\ = & G (e (u_{0})) + I n d (u_{0}, t_{0}) \\ = & J (u_{0}, t_{0}) \\ = & G^{*} (σ (u_{0}), t_{0}) . \end{matrix}

(60)

This completes the proof. □

10.3. About a concerning algorithm and related numerical method

For numerically solve this optimization problem in question, we present the following algorithm

1.: Set $t_{1} = 0.5 in Ω$ and $n = 1$ .
2.: Calculate $u_{n} \in U$ such that

$J_{1} (u_{n}, t_{n}) = sup_{u \in U} J_{1} (u, t_{n}) .$
3.: Calculate $t_{n + 1} \in B$ such that

$J_{1} (u_{n}, t_{n + 1}) = inf_{t \in B} J_{1} (u_{n}, t) .$
4.: If $∥ t_{n + 1} - t_{n} ∥_{\infty} < 10^{- 4}$ or $n > 100$ then stop, else set $n : = n + 1$ and go to item 2.

We have developed a software in finite differences for solving such a problem.

Here the software.

**************************************

1.: clear all

global P m8 d w u v Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3

m8=27;

m9=24;

c3=0.95;

d=1.0/m8;

d1=0.5/m9;

Ea= $210 * 10^{5}$ ; (stronger material)

Eb=1000; (softer material simulating voids)

w=0.30;

P=-42000000;

z1=(m8-1)*(m9-1);

A3=zeros(z1,z1);

for i=1:z1

A3(1,i)=1.0;

end;

b=zeros(z1,1);

uo=0.000001*ones(z1,1);

u1=ones(z1,1);

b(1,1)=c3*z1;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=c3;

end; end;

for i=1:z1

x1(i)=c3*z1;

end;

for i=1:2*m8*m9

xo(i)=0.000;

end;

xw=xo;

xv=Lo;

for k2=1:24

c3=0.98*c3;

b(1,1)=c3*z1;

k2

b14=1.0;

k3=0;

while $(b 14 > 10^{- 3.5})$ and $(k 3 < 5)$

k3=k3+1;

b12=1.0;

k=0;

while $(b 12 > 10^{- 4.0})$ and $(k < 120)$

k=k+1;

k2

k3

k

X=fminunc(’funbeam’,xo);

xo=X;

b12=max(abs(xw-xo));

xw=X;

end;

for i=1:m9-1

for j=1:m8-1

$E 1 = L o {(i, j)}^{2} * (E a - E b);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

$S x = E 1 * (e x + w * e y) / (1 - w^{2});$

$S y = E 1 * (w * e x + e y) / (1 - w^{2});$

Sxy=E1/(2*(1+w))*exy;

dc3(i,j)=-(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

for i=1:m9-1

for j=1:m8-1

f(j+(i-1)*(m8-1))=dc3(i,j);

end;

end;

for k1=1:1

k1

X1=linprog(f, $[]$ , $[]$ ,A3,b,uo,u1,x1);

x1=X1;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=X1(j+(m8-1)*(i-1));

end;

end;

b14=max(max(abs(Lo-xv)))

xv=Lo;

colormap(gray); imagesc(-Lo); axis equal; axis tight; axis off;pause(1e-6)

end;

end;

****************************************************

Here the auxiliary Function ’funbeam’

function S=funbeam(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2

for i=1:m9

for j=1:m8

u(i,j)=x(j+(m8)*(i-1));

v(i,j)=x(m8*m9+(i-1)*m8+j);

end;

for i=1:m9

end;

u(m9-1,1)=0;

v(m9-1,1)=0;

u(m9-1,m8-1)=0;

v(m9-1,m8-1)=0;

for i=1:m9-1

for j=1:m8-1

du1(i,j)=(u(i,j+1)-u(i,j))/d;

du2(i,j)=(u(i+1,j)-u(i,j))/d1;

dv1(i,j)=(v(i,j+1)-v(i,j))/d;

dv2(i,j)=(v(i+1,j)-v(i,j))/d1;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E 1 = L o {(i, j)}^{3} * E a + (1 - L o {(i, j)}^{3}) * E b;

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

S x = E 1 * (e x + w * e y) / (1 - w^{2});

S y = E 1 * (w * e x + e y) / (1 - w^{2});

Sxy=E1/(2*(1+w))*exy;

S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);

end;

S=S*d*d1-P*v(2,(m8)/3)*d*d1;

***********************************************

For a two dimensional beam of dimensions

1 m \times 0.5 m

and

t_{1} = 0.63

we have obtained the following results:

1.: Case A: For the optimal shape for a clamped beam at left (cantilever) and load $P = - 4 10^{6} N j$ at $(x, y) = (1, 0.25)$ , please Figure 5.
2.: Case B :For the optimal shape for a simply supported beam at $(0, 0)$ and $(1, 0)$ and load $P = - 4 10^{6} N j$ at $(x, y) = (1 / 3, 0.5)$ , please Figure 6.

In the first case the mesh was $28 \times 24$ . In the second one the mesh was $27 \times 24$

11. A duality principle for a general vectorial case in the calculus of variations

In this section we develop a duality principle for a general vectorial case in variational optimization.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

. Let

J : V \to R

be a functional where

J (u) = G (\nabla u_{1}, \dots, \nabla u_{N}) - {〈 u, f 〉}_{L^{2}},

where

V = W_{0}^{1, 2} (Ω; R^{N})

and

f = (f_{1}, \dots, f_{N}) \in L^{2} (Ω; R^{N}) .

Here we have denoted

u = (u_{1}, \dots, u_{N}) \in V

and

{〈 u, f 〉}_{L^{2}} = {〈 u_{i}, f_{i} 〉}_{L^{2}},

so that we may also denote

J (u) = G (\nabla u) - {〈 u, f 〉}_{L^{2}} .

Assume

G (\nabla u) = \int_{Ω} g (\nabla u) d x

where

g : R^{3 N} \to R

is a differentiable function such that

g (y) \to + \infty

| y | \to \infty

. Moreover, suppose there exists

α \in R

such that

α = inf_{u \in V} J (u) .

It is well known that

\begin{matrix} α & = & inf_{u \in V} J (u) \\ = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

(61)

Under some mild hypotheses, from convexity, we have that

\begin{matrix} inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} \\ = & sup_{v^{*} \in A^{*}} {- {(G \circ \nabla)}^{*} (- d i v v^{*})} = - {(G \circ \nabla)}^{*} (f), \end{matrix}

(62)

where

A^{*} = {v^{*} \in Y = Y^{*} = L^{2} (Ω; R^{3 N}) : d i v v^{*} + f = 0} .

Now observe that the restriction

v = \nabla u

for some

u \in V

is equivalent to the restriction

curl v_{i} = 0, in Ω

where

v = {v_{i}} = {v_{i j}}_{j = 1}^{3}

\forall i \in {1, \dots, N},

with appropriate boundary conditions, so that with an appropriate Lagrange multiplier

ϕ = {ϕ_{i}}

, we obtain

\begin{matrix} {(G \circ \nabla)}^{*} (- d i v v^{*}) & = & sup_{u \in V} {{〈 u, - d i v v^{*} 〉}_{L^{2}} - G (\nabla u)} \\ = & sup_{u \in V} {{〈 \nabla u, v^{*} 〉}_{L^{2}} - G (\nabla u)} \\ \leq & inf_{ϕ \in Y^{*}} \{sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G (v) + {〈 ϕ, curl v 〉}_{L^{2}}\} \\ = & inf_{ϕ \in Y^{*}} G^{*} (v^{*} + curl ϕ) . \end{matrix}

(63)

where we have denoted

curl v = {curl v_{i}}

and

curl ϕ = {curl ϕ_{i}} .

Joining the pieces, we have got

\begin{matrix} inf_{u \in V} J (u) & = & inf_{u \in V} {G (\nabla u) - {〈 u, f 〉}_{L^{2}}} \\ \geq & sup_{(v^{*}, ϕ) \in A^{*} \times Y^{*}} {- G^{*} (v^{*} + curl ϕ)}, \end{matrix}

(64)

where we recall that

Y = Y^{*} = L^{2} (Ω; R^{3 N}) .

We emphasize such a dual formulation in

(v^{*}, ϕ)

is convex (in fact concave).

12. A note on the Galerkin Functional

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \end{matrix}

(65)

Here

V = W_{0}^{1, 2} (Ω)

γ > 0, α > 0, β > 0 .

We denote also

Y = Y^{*} = L^{2} (Ω) .

At this point we define

A^{+} = {u \in V : u f \geq 0, in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

for some appropriate real constant

K_{3} > 0

and

V_{1} = A^{+} \cap V_{2} .

Observe that

J^{'} (u) = - γ \nabla^{2} u + α u^{3} - β - f,

so that we define the Galerkin functional

J_{1} : V \to R

J_{1} (u) = \frac{1}{2} {∥ J^{'} (u) ∥}_{2}^{2} = \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x .

From this, we get

\begin{matrix} \frac{\partial^{2} J_{1} (u)}{\partial u^{2}} & = & (- γ \nabla u + α u^{3} - β u - f) 6 α u \\ + {(- γ \nabla^{2} + 3 α u^{2} - β)}^{2} . \end{matrix}

(66)

Define now

φ_{2} = {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} .

At this point, for an appropriate small real constant

ε_{1} > 0

and bounded constant operator

M_{1} > ε_{1}

, we set the intended non-active restriction

\sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |},

and define

B_{1} = {u \in V_{1} : \sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |}} .

Observe that since for

u \in V_{1}

we have

u f \geq 0

Ω

so that if

u_{1}, u_{2} \in V_{1}

then

sign (u_{1}) = sign (u_{2}), in Ω,

we may infer that

B_{1}

is a convex set.

Furthermore, if

u \in B_{1}

, then

\sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |},

so that

3 α u^{2} \geq M_{1} + γ \nabla^{2} + β,

and hence

δ^{2} J (u) = - γ \nabla^{2} + 3 α u^{2} - β \geq M_{1} > ε_{1} > 0 .

For a small parameter

ε > 0

we define the intended non-active restriction

φ_{2} \leq ε, in Ω,

and define

B_{2} = {u \in V_{1} : φ_{2} \leq ε, in Ω} .

Observe that for

α > 0

and

β > 0

sufficiently large

φ_{2}

is convex in

V_{1}

(positive definite Hessian) so that

B_{2}

is a convex set. Assuming

0 < ε ≪ ε_{1} ≪ 1

, define

B_{3} = B_{1} \cap B_{2}

, which is a convex set.

Summarizing, if

u \in B_{3}

, then

δ^{2} J_{1} (u) \geq 0 .

With such results in mind, we define the following convex optimization problem for finding a critical point of J.

Minimize

J_{1} (u) = \frac{1}{2} {∥ J^{'} (u) ∥}_{2}^{2} = \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x,

subject to

u \in B_{3} .

Observe that a critical point

u_{0} \in B_{3}

J_{1}

, from such a concerning convexity of

J_{1}

on the convex set

B_{1}

, is also such that

J (u_{0}) = min_{u \in B_{3}} J_{1} (u) .

Finally, we may also define the convex optimization problem of minimizing

\begin{matrix} J_{3} (u) & = & K_{1} J_{1} (u) + J (u) \\ = & \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x \\ + \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}}, \end{matrix}

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subject to

u \in B_{3} .

Here

K_{1} > 0

is a large real constant.

Such a functional

J_{3}

is also convex on

B_{3}

so that a critical point

u_{0} \in B_{3}

of J is also a critical point of

J_{3}

, and thus

J_{3} (u_{0}) = min_{u \in B_{3}} J_{3} (u) .

13. A note on the Legendre-Galerkin functional

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \end{matrix}

(68)

Here

V = W_{0}^{1, 2} (Ω)

γ > 0, α > 0, β > 0 .

We denote also

Y = Y^{*} = L^{2} (Ω)

and

F_{1} : V \to R

F_{2} : V \to R

and

F_{3} : V \to R

F_{1} (u) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x,

F_{2} (u) = \frac{α}{4} \int_{Ω} u^{4} d x,

F_{3} (u) = \frac{β}{2} \int_{Ω} u^{2} d x .

Moreover, we define

F_{1}^{*}, F_{2}^{*}, F_{3}^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v_{1}^{*}) & = & sup_{u \in V} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{1} (u)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{- γ \nabla^{2}} d x, \end{matrix}

(69)

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{3}{4} \int_{Ω} \frac{{(v_{2}^{*})}^{4 / 3}}{α^{1 / 3}} d x, \end{matrix}

(70)

\begin{matrix} F_{3}^{*} (v_{3}^{*}) & = & sup_{u \in V} {{〈 u, v_{3}^{*} 〉}_{L^{2}} - F_{3} (u)} \\ = & \frac{1}{2 β} \int_{Ω} {(v_{3}^{*})}^{2} d x . \end{matrix}

(71)

Observe now that these three last suprema are attained through the equations,

v_{1}^{*} = \frac{\partial F_{1} (u)}{\partial u} = - γ \nabla^{2} u,

v_{2}^{*} = \frac{\partial F_{2} (u)}{\partial u} = α u^{3}

v_{3}^{*} = \frac{\partial F_{3} (u)}{\partial u} = β u .

From such results, at a critical point, we obtain the following compatibility conditions

u = \frac{v_{1}^{*}}{- γ \nabla^{2}} = {(\frac{v_{2}^{*}}{β})}^{1 / 3} = \frac{v_{3}^{*}}{β} .

From such relations we have

\frac{v_{1}^{*}}{- γ \nabla^{2}} = \frac{v_{3}^{*}}{β},

and

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3},

so that

v_{1}^{*} = - γ \nabla^{2} (\frac{v_{3}^{*}}{β}),

and

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3} .

Moreover, we define the functional

F_{4}^{*} : Y^{*} \to R,

F_{4}^{*} (v^{*}) = sup_{u \in V} {{〈 u, v_{1}^{*} + v_{2}^{*} - v_{3}^{*} 〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}}} .

Therefore

F_{4}^{*} (v^{*}) = \{\begin{matrix} 0, & if v_{1}^{*} + v_{2}^{*} - v_{3}^{*} - f = 0, in Ω, \\ + \infty, & otherwise . \end{matrix}

(72)

Hence, a critical point of J corresponds to the solution of the following system of equations

v_{1}^{*} = - γ \nabla^{2} (\frac{v_{3}^{*}}{β}),

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3},

and

v_{1}^{*} + v_{2}^{*} - v_{3}^{*} - f = 0, in Ω .

From this last equation we may obtain

v_{1}^{*} = - v_{2}^{*} + v_{3}^{*} + f,

so that the final equations to be solved are

- v_{2}^{*} + v_{3}^{*} + f + γ \nabla^{2} (\frac{v_{3}^{*}}{β}) = 0

and

v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3} = 0, in Ω,

with the boundary conditions

u = \frac{v_{3}^{*}}{β} = 0, on \partial Ω .

With such results in mind, we define the Legendre-Galerkin functional

J^{*} : {[Y^{*}]}^{2} \to R

, where

\begin{matrix} J^{*} (v^{*}) & = & \frac{1}{2} \int_{Ω} {(- v_{2}^{*} + v_{3}^{*} + f + \frac{γ \nabla^{2} v_{3}^{*}}{β})}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3})}^{2} d x . \end{matrix}

(73)

At this point, defining

φ = v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3},

we obtain

\frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{2}^{*})}^{2}} = 2;

\frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{3}^{*})}^{2}} = {(- 1 - \frac{γ \nabla^{2}}{β})}^{2} + \frac{9 α^{2} {(v_{3}^{*})}^{4}}{β^{6}} + O (φ),

\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}} = \frac{- 3 α {(v_{3}^{*})}^{2}}{β^{3}} + (- 1 - \frac{γ \nabla^{2}}{β}) .

From such results we may infer that

\begin{matrix} det (\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}}) & = & \frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{2}^{*})}^{2}} \frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{3}^{*})}^{2}} - {(\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}})}^{2} \\ = & {(- 1 - \frac{γ \nabla^{2}}{β} + 3 α \frac{{(v_{3}^{*})}^{2}}{β^{3}})}^{2} + O (φ) \end{matrix}

(74)

Observe that a critical point

φ = 0

so that

δ^{2} J^{*} (v^{*}) > 0

at a neighborhood of any critical point.

At this point we define

A^{+} = \{v^{*} = (v_{2}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{2} : \frac{v_{3}^{*}}{β} f \geq 0, in Ω\},

D^{*} = {v^{*} = (v_{2}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{2} : ∥ v^{*} ∥_{\infty} \leq K},

for an appropriate real constant

K > 0

Define now

E^{*} = A^{+} \cap D^{*}

C_{1}^{*} = {v^{*} = (v_{2}^{*}, v_{3}^{*}) \in E^{*} : φ^{2} \leq ε, in Ω},

for a small real constant

ε > 0,

C_{2}^{*} = \{v^{*} = (v_{2}^{*}, v_{3}^{*}) \in E^{*} : (- 1 - \frac{γ \nabla^{2}}{β} + 3 α \frac{{(v_{3}^{*})}^{2}}{β^{3}}) \geq ε_{1}\},

and

C^{*} = C_{1}^{*} \cap C_{2}^{*} .

Similarly as done in the previous section, we may prove that

C^{*}

is a convex set.

Furthermore, for

0 < ε ≪ ε_{1} ≪ 1,

we have that

J^{*}

is convex on

C^{*}

Summarizing, we may define the following convex optimization problem to obtain a critical point of the primal functional J,

Minimize J^{*} (v_{2}^{*}, v_{3}^{*}) subject to v^{*} = (v_{2}^{*}, v_{3}^{*}) \in C^{*} .

We call

J^{*}

the Legendre-Galerkin functional associated to J.

13.1. Numerical examples

We have obtained numerical solutions for two one-dimensional examples.

1.: For $γ = 1.0,$ $α = 3.0,$ $β = 30.0,$ $f \equiv 10, in Ω = [0, 1] .$

For the respective solution please see Figure 7.
2.: For $γ = 0.01,$ $α = 3.0,$ $β = 30.0,$ $f \equiv 10, in Ω = [0, 1] .$

For the respective solution please see Figure 8.

14. A general concave dual variational formulation for global optimization

Let

Ω \subset R^{3}

be an open, bounded and connected set a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = G (u) - {〈 u, f 〉}_{L^{2}}, \forall u \in V .

Here

V = W_{0}^{1, 2} (Ω)

f \in L^{2} (Ω)

and we also denote

Y = Y^{*} = L^{2} (Ω) .

Assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Furthermore, suppose G is three times Fréchet differentiable and there exists

K > 0

such that

\frac{\partial^{2} G (u)}{\partial u^{2}} + K > 0, \forall u \in V .

Define now

J_{1} : V \times Y \to R

where,

J_{1} (u, v) = G_{1} (u, v) + F (u),

where

G_{1} (u, v) = G (v) - \frac{ε}{2} \int_{Ω} v^{2} d x + \frac{K}{2} \int_{Ω} {(v - u)}^{2} d x,

and

F (u) = \frac{ε}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Moreover, we define the polar functionals

G_{1}^{*} : Y^{*} \times V \to R

and

F^{*} : Y^{*} \to R

, where

\begin{matrix} G_{1}^{*} (v^{*}, u) & = & sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G_{1} (u, v)} \\ = & - G_{K_{ε}}^{*} (v^{*} + K u) + \frac{K}{2} \int_{Ω} u^{2} d x, \end{matrix}

(75)

G_{K_{ε}}^{*} (v^{*} + K u) = sup_{v \in Y} \{{〈 v, v^{*} 〉}_{L^{2}} - G (v) - \frac{K}{2} \int_{Ω} v^{2} d x + \frac{ε}{2} \int_{Ω} v^{2} d x\},

and

\begin{matrix} F^{*} (- v^{*}) & = & sup_{u \in V} {- {〈 u, v^{*} 〉}_{L^{2}} - F (u)} \\ = & \frac{1}{2 ε} \int_{Ω} {(v^{*} - f)}^{2} d x . \end{matrix}

(76)

At this point we define the functional

J_{2}^{*} : Y^{*} \times V \to R

J_{2}^{*} (v^{*}, u) = - G_{K_{ε}}^{*} (v^{*} + K u) + \frac{K}{2} \int_{Ω} u^{2} d x - F^{*} (- v^{*}) .

With such results in mind we define

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

D^{*} = {v^{*} \in Y^{*} : ∥ v^{*} ∥_{\infty} \leq K_{4}},

for appropriated real constants

K_{3} > 0

and

K_{4} > 0 .

Moreover, we define also the penalized functional

J_{3}^{*} : Y^{*} \times V \to R

where

J_{3}^{*} (v^{*}, u) = J_{2}^{*} (v^{*}, u) - \frac{K_{1}}{2} \int_{Ω} {(v^{*} - \frac{\partial G (u)}{\partial u} + ε u)}^{2} d x .

Finally, we remark that for

ε > 0

sufficiently small and

K_{1} > 0

sufficiently large,

J_{3}^{*}

is concave in

D^{*} \times V_{1}

around a concerning critical point. We recall that a critical point

v^{*} - \frac{\partial G (u)}{\partial u} + ε u = 0, in Ω .

15. A related restricted problem in phase transition

In this section we develop a convex (in fact concave) dual variational for a model similar to those found in phase transition problems.

Let

Ω = [0, 1] \subset R .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{1}{2} \int_{Ω} min {{(u^{'} + 1)}^{2}, {(u^{'} - 1)}^{2}} d x \\ + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \\ = & \frac{1}{2} \int_{Ω} {(u^{'})}^{2} d x - \int_{Ω} | u^{'} | d x + 1 / 2 \\ + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(77)

Here

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2} .

We also denote

V_{1} = W_{0}^{1, 2} (Ω),

and

Y = Y^{*} = L^{2} (Ω) .

Furthermore, we define the functionals G and

F : V \times V_{1} \to R

G (u^{'}, v^{'}) = \frac{1}{2} \int_{Ω} {(u^{'} + v^{'})}^{2} d x - \int_{Ω} | u^{'} + v^{'} | d x + 1 / 2,

and

F (u, v) = \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Moreover we define

J_{1} : V \times V_{1} \to R

J_{1} (u, v) = G (u^{'}, v^{'}) + F (u, v),

and consider the problem of minimizing

J_{1}

on the set

A = {(u, v) \in V \times V_{1} : {(v^{'})}^{2} \leq K_{2}, in Ω} .

Already including the Lagrange multiplier

ϕ

concerning such restrictions, we define

J_{2} (u, v, ϕ) = J_{1} (u, v) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} .

Observe now that

\begin{matrix} J_{2} (u, v, ϕ) & = & J_{1} (u, v) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ = & G (u^{'}, v^{'}) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ + F (u, v) \\ = & - {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} - {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + G (u^{'}, v^{'}) \\ + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + F (u, v) \\ \geq & inf_{(v_{1}, v_{2}) \in Y \times Y} \{- {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} + G_{1} (v_{1}, v_{2}, ϕ) \\ + \frac{1}{2} {〈 ϕ^{2}, {(v_{2})}^{2} - K_{2} 〉}_{L^{2}}\} \\ + inf_{(u, v) \in V \times V_{1}} {{〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + F (u, v)} \\ = & - G_{1}^{*} (v_{1}^{*}, v_{2}^{*}, ϕ) - {\tilde{F}}^{*} (v_{1}^{*}, v_{2}^{*}), \forall (u, v) \in V \times V_{1}, (v_{1}^{*}, v_{2}^{*}, ϕ) \in {[Y^{*}]}^{3}, \end{matrix}

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where

G_{1} (u^{'}, v^{'}, ϕ) = G (u^{'}, v^{'}) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} .

Also,

\begin{matrix} G_{1}^{*} (v_{1}^{*}, v_{2}^{*}, ϕ) & = & sup_{(v_{1}, v_{2}) \in Y \times Y} {{〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - G_{1} (v_{1}, v_{2}, ϕ)} \\ = & \frac{1}{2} \int_{Ω} {(v_{1}^{*})}^{2} d x \\ + \int_{Ω} | v_{1}^{*} | d x + \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{ϕ^{2}} \\ + \frac{K_{2}}{2} \int_{Ω} ϕ^{2} d x, \end{matrix}

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where

{\tilde{F}}^{*} (v^{*}) = \{\begin{matrix} \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - v_{1}^{*} (1) u (1), & if {(v_{2}^{*})}^{'} = 0, in Ω, \\ + \infty, & otherwise . \end{matrix}

(80)

From this we may infer that

v_{2}^{*} = c, in Ω,

for some

c \in R .

Summarizing, denoting

v^{*} = (v_{1}^{*}, v_{2}^{*}) = (v_{1}^{*}, c)

, and

J^{*} (v^{*}, ϕ) = - G_{1}^{*} (v^{*}, ϕ) - {\tilde{F}}^{*} (v^{*})

we have got

inf_{(u, v) \in A} J_{1} (u, v) \geq sup_{(v^{*}, ϕ) \in Y^{*} \times R \times Y^{*}} J^{*} (v^{*} ϕ) .

We have developed numerical results by maximizing the dual functional

J^{*}

for two examples, namely.

1.: Example A: In this case, we consider $f (x) = cos (π x) / 2$ , $K_{2} = 10^{- 4}$ .

For the optimal

$u_{0} = {(v_{1}^{*})}^{'} + f,$

please see Figure 9.
2.: Example B: In this case, we consider $f (x) = cos (π x) / 2$ , $K_{2} = 30$ .

For the optimal

$u_{0} = {(v_{1}^{*})}^{'} + f,$

please see Figure 10.

16. One more dual variational formulation

In this section we develop one more dual variational formulation for a related model.

Let

Ω = [0, 1] \subset R

and consider the functional

J : V \to R

defined by

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2} .

We define also the relaxed functional

J_{1} : V \times V_{0} \to R

, already including a concerning restriction and corresponding non-negative Lagrange multiplier

Λ^{2}

, where

J_{1} (u, v, Λ) = \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} .

where

V_{0} = {v \in W_{0}^{1, 4} (Ω) : {(v^{'})}^{2} - K \leq 0 in Ω} .

Observe that

\begin{matrix} \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} \\ = & - {〈 v_{0}^{*}, {(u^{'} + v^{'})}^{2} - 1 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x \\ + {〈 v_{0}^{*}, {(u^{'} + v^{'})}^{2} - 1 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} - {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} - {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} \\ + {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{w \in Y} \{- {〈 v_{0}^{*}, w 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {(w)}^{2} d x\} \\ inf_{(v_{1}, v_{2}) \in Y \times Y} \{{〈 v_{0}^{*}, {(v_{1} + v_{2})}^{2} - 1 〉}_{L^{2}} + {〈 Λ^{2}, {(v_{2})}^{2} - K 〉}_{L^{2}} - {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}}\} \\ + inf_{(u, v) \in V \times V_{0}} \{{〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & - \frac{1}{2} \int_{Ω} {(v_{0}^{*})}^{2} d x - \int_{Ω} v_{0}^{*} d x \\ - \frac{1}{4} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{v_{0}^{*}} d x - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{2 Λ^{2}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - \frac{1}{2} \int_{Ω} K Λ^{2} d x + v_{1}^{*} (1) u (1) . \end{matrix}

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Here, we highlight

v_{2}^{*} = c \in R in Ω

, for some real constant c.

Hence, denoting

\begin{matrix} J_{1}^{*} (v^{*}, Λ) & = & - \frac{1}{2} \int_{Ω} {(v_{0}^{*})}^{2} d x - \int_{Ω} v_{0}^{*} d x \\ - \frac{1}{4} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{v_{0}^{*}} d x - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{2 Λ^{2}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - \frac{1}{2} \int_{Ω} K Λ^{2} d x + v_{1}^{*} (1) u (1) \end{matrix}

(82)

and

J_{2} (u, v) = \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

we have obtained

inf_{(u, v) \in V \times V_{0}} J_{2} (u, v)} \geq sup_{(v^{*}, Λ) \in A^{*} \times [Y^{*}] \times R \times Y^{*}} J_{1}^{*} (v^{*}, Λ) .

Finally, for

A^{*} = {v_{0}^{*} \in Y^{*} : v_{0}^{*} \geq ε in Ω}

we emphasize

J_{1}^{*}

is concave on

A^{*} \times [Y^{*}] \times R \times Y^{*}

Here

ε > 0

is a small regularizing real constant.

Remark 16.1.

The constraint

{(v^{'})}^{2} - K \leq 0, in Ω

is included to restrict the action of v on the region where the primal functional is non-convex, through an appropriate constant

K > 0 .

17. A model in superconductivity through an eigenvalue approach

In this section we intend to model superconductivity through a two phase eigenvalue approach.

Let

Ω = [0, 5] \subset R

be a straight wire corresponding to a one-dimensional super-conducting sample.

Consider the functional

J : V \times V \times R \to R

where

\begin{matrix} J (u, v, E) & = & \frac{γ_{1}}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α_{1}}{2} \int_{Ω} {| u |}^{4} d x \\ - \frac{ω^{2}}{2} \int_{Ω} {| u |}^{2} d x \\ + \frac{γ_{2}}{2} \int_{Ω} \nabla v \cdot \nabla v d x + \frac{α_{2}}{2} \int_{Ω} {| v |}^{4} d x \\ - \frac{ω_{1}^{2}}{2 K_{3}^{2}} \int_{Ω} {| v |}^{2} d x \\ - \frac{E}{2} (\int_{Ω} {(| u |}^{2} + {| v |}^{2}) d x - m_{T}) . \end{matrix}

(83)

Here, in atomic units,

m_{T}

is the total electronic charge,

V = W_{0}^{1, 2} (Ω)

and we set

α_{1} = 10^{4}

corresponding to higher self-interacting energy which is related to a normal phase. We also set

α_{2} = 10^{- 1}

corresponding to a lower self-interacting energy which is related to a super-conducting phase and respective super-currents.

Moreover, we set

γ_{1} = γ_{2} = 1,

and initially

ω = 1.8

which is gradually decreased to

ω = 1.0

Furthermore, we define

| ϕ_{N} |^{2} = \frac{{| u |}^{2}}{{| u |}^{2} + {| v |}^{2}}

and

| ϕ_{S} |^{2} = \frac{{| v |}^{2}}{{| u |}^{2} + {| v |}^{2}}

where

ϕ_{N}

corresponds to a normal phase and

ϕ_{S}

to a super-conducting one.

At this point we observe that the temperature

T = T (x, t)

is proportional the frequency

ω / (2 π)

of vibration for the normal phase.

We start the process with

ω = 1.8

which in atomic units corresponds to a higher temperature and gradually decreases it to the value

ω = 1.0

Between

ω = 1.2

and

ω = 1.0

the system changes from an almost total normal phase to an almost total super-conducting phase, as expected.

We highlight that the temperature is proportional to the vibrational kinetics energy

E_{1} (t) = \frac{1}{2} \int_{Ω} {| u |}^{2} \frac{\partial r_{N} (x, t)}{\partial t} \cdot \frac{\partial r_{N} (x, t)}{\partial t} d x

so that for

r_{N} (x, t) = e^{i ω t} w_{5} (x)

and for a suitable vectorial function

w_{5}

, we have

T \propto E_{1} \propto ω^{2}

so that we may model the decreasing of temperature T through the decreasing of

ω^{2}

For

ω = 1.8

, for the corresponding normal phase

ϕ_{N}

and super-conducting phase

ϕ_{S}

, please se Figure 11 and Figure 12, respectively.

For

ω = 1.0

, for the corresponding normal phase

ϕ_{N}

and super-conducting phase

ϕ_{S}

, please se Figure 13 and Figure 14, respectively.

Finally, we have set

ω_{1} / K_{3} \approx 1

which for large

ω_{1}

corresponds to the super-currents.

18. A simplified qualitative many body model for the hydrogen nuclear fusion

In this section we develop a qualitative simple model for the hydrogen nuclear fusion.

Let

Ω = {[0, L]}^{3} \subset R^{3}

be a box in which is confined a gas comprised by an amount of ionized deuterium and tritium isotopes of hydrogen.

Though a suitable increasing in temperature, we intend to develop the following nuclear reaction

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom comprises a proton and a neutron and the ionized Tritium atom comprises a proton and two neutrons.

Under certain conditions and at a suitable high temperature the ionized Deuterium and Tritium atoms react chemically resulting in an ionized Helium atom, comprised by two protons and two neutrons and resulting also in one more single energetic neutron. We emphasize the higher kinetics neutron energy level has many potential practical applications, including its conversion in electric energy.

At this point we denote by

m_{D}, m_{T}, m_{H_{e}} and m_{N}

the masses of the ionized Deuterium, Tritium and Helium atoms, and the single neutron, respectively.

Therefore, we have the following mass relation

m_{D} + m_{T} = m_{H_{e}} + m_{N} .

To simplify our analysis, in such a chemical reaction, denoting the total masses of ionized Deuterium, Tritium, Helium and single Neutrons by

{(m_{D})}_{T}, {(m_{T})}_{T}, {(m_{H_{e}})}_{T}

and

{(m_{N})}_{T}

we assume there is a real constant

c > 0

such that

{(m_{D})}_{T} = c m_{D}, {(m_{T})}_{T} = c m_{T}, {(m_{H_{e}})}_{T} = c m_{H_{e}}, {(m_{N})}_{T} = c m_{N} .

With such statements and definitions in mind, we define the following functional J, where

J (ϕ, r) = J (ϕ_{D}, ϕ_{T}, ϕ_{H_{e}}, ϕ_{N}, r) = G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r),

where, in a simplified many body context,

| ϕ_{D} {(x, y) |}^{2} = | ϕ_{p}^{D} {(y) |}^{2} + | ϕ_{N}^{D} {(x, y) |}^{2} {| ϕ_{p}^{D} (y) |}^{2} \frac{1}{m_{p}},

| ϕ_{T} {(x, y) |}^{2} = | ϕ_{p}^{T} {(y) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y) |}^{2}) | ϕ_{p}^{T} {(y) |}^{2} \frac{1}{m_{p}},

| ϕ_{H_{e}} {(x, y) |}^{2} = | ϕ_{2 P}^{H_{e}} {(y) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y) |}^{2}) | ϕ_{2 P}^{H_{e}} {(y) |}^{2} \frac{1}{2 m_{p}},

ϕ_{N} = ϕ_{N} (x) .

Here

x, y \in Ω \subset R^{3}

refers to the particle densities.

Furthermore, we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0,

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0,

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y \\ + \frac{γ_{N}^{D}}{2} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y \\ \frac{γ_{p}^{T}}{2} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y \\ + \frac{γ_{N}}{2} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x, \end{matrix}

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and,

\begin{matrix} F (ϕ) & = & \frac{α_{D}}{2} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{T}}{2} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{D T}}{2} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{H_{e}}}{2} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{N} {(x - ξ) |}^{2} {| ϕ_{N} (ξ) |}^{2}}{| x - ξ |} d x d ξ \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{N} (ξ_{j}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \end{matrix}

(85)

and the kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y, \end{matrix}

(86)

where we also assume

r_{D} \approx e^{i ω t} w_{5} (x, y),

r_{T} \approx e^{i ω t} w_{6} (x, y),

so that considering such a vibrational motion, the temperature T is proportional to

ω^{2}

, that is

T \propto ω^{2} .

Therefore, an increasing in T corresponds to a proportional increasing in

ω^{2} .

Summarizing, we have supposed

E_{c} (ϕ, r) \approx \frac{1}{2} ω^{2} \int_{Ω} | ϕ_{D} |^{2} + | ϕ_{T} |^{2} d x C_{1} + \frac{1}{2} ω_{1}^{2} \int_{Ω} {| ϕ_{N} |}^{2} d x C_{2},

so that we represent the increasing in T through an increasing in

ω^{2} .

Moreover, we denote by

m_{N}

the mass of a single neutron and by

m_{p}

the mass of a single proton.

Thus, denoting also by

λ_{1}, λ_{2}

the proportion of non-reacted and reacted masses respectively, we have the following constraints.

1.: $\int_{Ω} {| ϕ_{N}^{D} (x, y) |}^{2} d x = m_{N},$
2.: $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y) |}^{2} d x = m_{N},$
3.: $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y) |}^{2} d x = m_{N},$
4.: $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
5.: $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
6.: $\int_{Ω} {| ϕ_{p}^{D} (y) |}^{2} d y = λ_{1} c m_{p},$
7.: $\int_{Ω} {| ϕ_{p}^{T} (y) |}^{2} d y = λ_{1} c m_{p},$
8.: $\int_{Ω} {| ϕ_{2 P}^{H_{e}} (y) |}^{2} d y = λ_{2} (2 c m_{p}),$

Similar constraints are valid corresponding to the charge of a single proton.

We have also the following complementing constraints,

1.: $\int_{Ω} {| ϕ_{D} |}^{2} d x d y = λ_{1} {(m_{D})}_{T},$
2.: $\int_{Ω} {| ϕ_{T} |}^{2} d x d y = λ_{1} {(m_{T})}_{T},$
3.: $\int_{Ω} {| ϕ_{H_{e}} |}^{2} d x d y = λ_{2} {(m_{H_{e}})}_{T},$
4.: $\int_{Ω} {| ϕ_{N} |}^{2} d x d y = λ_{2} {(m_{N})}_{T},$
5.: $λ_{1} + λ_{2} = 1 .$

With such results and statements in mind and simplifying the interacting terms, we re-define the functional J now denoting it by

J_{1}

, here already including the Lagrange multipliers concerning the constraints, where

\begin{matrix} J_{1} (ϕ, ω, E, λ) & = & \frac{γ_{p}^{D}}{2} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y \\ + \frac{γ_{N}^{D}}{2} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y \\ \frac{γ_{p}^{T}}{2} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y \\ + \frac{γ_{N}}{2} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x \\ + \frac{α_{D}}{2} \int_{Ω} | ϕ_{D} |^{4} d x + \frac{α_{T}}{2} \int_{Ω} {| ϕ_{T} |}^{4} d x \\ + \frac{α_{H_{e}}}{2} \int_{Ω} | ϕ_{H_{e}} |^{4} d x + \frac{α_{N}}{2} \int_{Ω} {| ϕ_{N} |}^{4} d x \\ - ω^{2} \int_{Ω} (| ϕ_{D} |^{2} + | ϕ_{T} |^{2}) d x \\ - ω_{1}^{2} \int_{Ω} {| ϕ_{N} |}^{2} d x + J_{A u x}, \end{matrix}

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where the functional

J_{A u x}

stands for

\begin{matrix} J_{A u x} & = & - \int_{Ω} {(E_{N}^{D})}_{5} (y) (\int_{Ω} {| ϕ_{N}^{D} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y) |}^{2} d x - m_{N}) d y \\ - {(E_{D})}_{2} (\int_{Ω} {| ϕ_{p}^{D} (y) |}^{2} d y - λ_{1} c m_{p}) \\ - {(E_{T})}_{3} (\int_{Ω} {| ϕ_{p}^{T} (y) |}^{2} d y - λ_{1} c m_{p}) \\ - {(E_{H_{e}})}_{3} (\int_{Ω} {| ϕ_{2 P}^{H_{e}} (x, y) |}^{2} d y - λ_{2} 2 c m_{p}) \\ - E_{5} (\int_{Ω} {| ϕ_{D} |}^{2} d x d y - λ_{1} {(m_{D})}_{T}) \\ - E_{6} (\int_{Ω} {| ϕ_{T} |}^{2} d x d y - λ_{1} {(m_{T})}_{T}) \\ - E_{7} (\int_{Ω} {| ϕ_{H_{e}} |}^{2} d x d y - λ_{2} {(m_{H_{e}})}_{T}) \\ - E_{8} (\int_{Ω} {| ϕ_{N} |}^{2} d x d y - λ_{2} {(m_{N})}_{T}) \\ - E_{9} (λ_{1} + λ_{2} - 1) . \end{matrix}

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Remark 18.1.

In order to obtain consistent results it is necessary to set

(α_{N}, α_{H_{e}}) ≫ (α_{D}, α_{T}) .

In such a case, a higher temperature corresponding to a large

ω^{2}

, though such a nuclear reaction, will result in a small

λ_{1}

and a higher kinetics energy for the neutron field, corresponding to a large

ω_{1}^{2}

and

λ_{2}

closer to 1.

19. A more detailed mathematical description of the hydrogen nuclear fusion

In this section we develop in more details another model for the hydrogen nuclear fusion.

Remark 19.1.

Denoting by

i \in C

the imaginary unit, in this and in the subsequent sections, for the time-dependent case we generically define the gradient of a scalar function

u (x, t)

with domain in

R^{4}

, denoted by

\nabla u (x, t)

, as

\nabla u (x, t) = (i u_{t} (x, t), u_{x_{1}} (x, t), u_{x_{2}} (x, t), u_{x_{3}} (x, t)),

so that

\nabla u \cdot \nabla u = - u_{t}^{2} + \sum_{j = 1}^{3} u_{x_{j}}^{2} .

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Here such a set

Ω

stands for a control volume in which an ionized gas (plasma) flows. Such a gas comprises ionized Deuterium and Tritium atoms intended, through a suitable higher temperature, to chemically react resulting in atoms of Hellion and a field of single energetic Neutrons.

Symbolically such a reaction stands for

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

As previously mentioned, resulting from such a chemical reaction up surges also an energetic neutron which the higher kinetics energy has a great variety of applications, including its conversion in electric energy.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval

[0, t_{f}],

at this point we define the following density functions:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + | ϕ_{N}^{D} {(x, y, t) |}^{2} {| ϕ_{p}^{D} (y, t) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{T} {(x, y, t) |}^{2} = | ϕ_{p}^{T} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t) |}^{2}) | ϕ_{p}^{T} {(y, t) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t) .$

Furthermore, we define also the related densities

1.: $ρ_{D} (y, t) = \int_{Ω} {| ϕ_{D} (x, y, t) |}^{2} d x,$
2.: $ρ_{T} (y, t) = \int_{Ω} {| ϕ_{T} (x, y, t) |}^{2} d x,$

$ρ_{H_{e}} (y, t) = \int_{Ω} {| ϕ_{H_{e}} (x, y, t) |}^{2} d x,$

$ρ_{N} (x, t) = {| ϕ_{N} (x, t) |}^{2},$

$ρ_{e} (y, t) = \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x .$

For the chemical reaction in question we consider that one unit of mass of fractional proportion

α_{D}

of ionized Deuterium and

α_{T}

of ionized Tritium results in one unit of mass of fractional proportion

α_{H_{e}}

of ionized Helium and

α_{N}

of neutrons.

Symbolic, this stands for

1 = α_{D} + α_{T} = α_{H_{e}} + α_{N} .

Concerning the control volume

Ω

in question and related surface control

\partial Ω,

we assume such a volume has an initial (fot

t = 0

) amount of ionized Deuterium of

{(m_{D})}_{0}

and an initial amount of ionized Tritium of

{(m_{T})}_{0} .

The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control

\partial Ω

, we assume there is a part

Ω_{1} \subset \partial Ω

for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part

\partial Ω_{2} \subset \partial Ω

such that

\partial Ω_{1} \cap \partial Ω_{2} = \emptyset

for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

\partial Ω_{2}

is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
6.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
7.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
8.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
9.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

Here

n

denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, we define the functional

J (ϕ, ρ, r, u, E, A, B)

where

J = G (\nabla u) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3},

and where we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0, γ_{e} > 0

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0, α_{e, e} > 0, α_{H_{e}, e} < 0

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y d t \\ + \frac{γ_{N}^{D}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y d t \\ \frac{γ_{p}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y d t \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y d t \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y d t \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y d t \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y d t \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y d t \\ + \frac{γ_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{e}) \cdot (\nabla ϕ_{e}) d x d y d t, \end{matrix}

(89)

and

\begin{matrix} F (ϕ) & = & \frac{α_{D}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{T}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{D T}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{N} {(x - ξ, t) |}^{2} {| ϕ_{N} (ξ) |}^{2}}{| x - ξ, t |} d x d ξ d t \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{N} (ξ_{j}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}, e}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{e, e}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{e} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \end{matrix}

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and the internal kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{e} |}^{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} d x d y d t, \end{matrix}

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Here it is worth highlighting we have approximated the initially discrete set of indices s of particles as a continuous positive real variable s.

Moreover,

F_{1} = \frac{1}{4 π} \int_{0}^{t_{f}} {∥ curl A - B_{0} ∥}_{2} d t,

\begin{matrix} F_{2} & = & \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d x d y d t, \end{matrix}

(92)

where

K_{p}

and

K_{e}

are appropriate real constants related to the respective charges.

Here

u = (u_{1}, u_{2}, u_{3})

is the fluid velocity field and

r_{D}, r_{T}, r_{H_{e}}, r_{N}, r_{e}

are fields of displacements for the corresponding atom fields.

Also

A

denotes the magnetic potential,

B_{0}

an external magnetic field and

B

is the total magnetic field.

Moreover,

E_{i n d}

is an induced electric field.

Finally,

\begin{matrix} F_{3} & = & \frac{C_{D}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{D} \cdot \nabla_{(x, y)} r_{D} d x d y d t + \frac{C_{T}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{T} \cdot \nabla_{(x, y)} r_{T} d x d y d t \\ \frac{C_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{H_{e}} \cdot \nabla_{(x, y)} r_{H_{e}} d x d y d t + \frac{C_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{N} \cdot \nabla_{(x, y)} r_{N} d x d y d t \\ \frac{C_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{e} \cdot \nabla_{(x, y)} r_{e} d x d y d t, \end{matrix}

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for appropriate real positive constants

C_{D} C_{T}, C_{H_{e}}, C_{N}, C_{e} .

Such a functional J is subject to the following constraints:

1.

The momentum conservation equation for the fluid motion

ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) = ρ f_{k} - \frac{\partial P}{\partial x_{k}} + τ_{k j, j} + {(F_{E})}_{k} + {(F_{M})}_{k},

\forall k \in {1, 2, 3} .

Here

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e}

is the total density and P is the fluid pressure field.

Furthermore,

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} δ_{i j} \sum_{k = 1}^{3} \frac{\partial u_{k}}{\partial x_{k}}),

\forall i, j \in {1, 2, 3},

F_{E} = {{(F_{E})}_{k}} = (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} |}^{2} d x) E,

and

\begin{matrix} F_{M} & = & {{(F_{M})}_{k}} \\ = & (K_{p} (| ϕ_{p}^{D} |^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ | ϕ_{p}^{T} |^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + | ϕ_{2 p}^{H_{e}} |^{2} (u + \frac{\partial r_{H_{e}}}{\partial t})) \\ + K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t})) \times B . \end{matrix}

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2.

Mass conservation equation:

\frac{\partial ρ}{\partial t} + div (ρ u) = 0 .

3.

Energy equation

ρ \frac{D e}{D t} + P (div u) = \frac{\partial Q}{\partial t} - div q,

where we assume the Fourier law

q = - K \nabla T,

where

T = T (x, t)

is the scalar field of temperature.

Also,

\begin{matrix} e & = & \frac{ρ}{2} u \cdot u + \frac{ρ_{D}}{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} \\ + \frac{ρ_{T}}{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} \\ + \frac{ρ_{H_{e}}}{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} \\ + \frac{ρ_{N}}{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} \\ + \frac{ρ_{e}}{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} \end{matrix}

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and

\frac{D e}{D t} = \frac{\partial e}{\partial t} + u_{j} \frac{\partial e}{\partial x_{j}} .

4.

P = F_{7} (ρ, T),

for an appropriate scalar function

F_{7}

5.

Mass relations

(a): $m_{D} (t) = \int_{Ω} ρ_{D} (x, t) d x,$
(b): $m_{T} (t) = \int_{Ω} ρ_{T} (x, t) d x,$
(c): $m_{H e} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
(d): $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
(e): $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x,$
(f): ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
(g): ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
(h): $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$

where,

(a): ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$
(b): $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
(c): $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
(d): $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
(e): ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$
(f): $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

6.

Other mass constraints

(a): $\int_{Ω} {| ϕ_{N}^{D} (x, y, t) |}^{2} d x = m_{N},$
(b): $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t) |}^{2} d x = m_{N},$
(c): $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t) |}^{2} d x = m_{N},$
(d): $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t) |}^{2} d x = m_{N},$
(e): $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t) |}^{2} d x = m_{N} .$

7.

For the induced electric field, we must have

\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x)) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0}) = 0, \end{matrix}

(96)

where

{\hat{K}}_{p}

and

{\hat{K}}_{e}

are appropriate real constants related to the respective charges.

8.

A Maxwell equation:

div B = 0,

where

B = B_{0} - curl A .

9.

Another Maxwell equation:

div E = 4 π (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x),

where the total electric field

E

stands for

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, ξ) d x d ξ,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, ξ)}{\partial x_{k}} d ξ\} .

At this point we generically denote

{〈 h_{1}, h_{2} 〉}_{L^{2}} = \int_{0}^{t_{f}} \int_{Ω} h_{1} h_{2} d x d y d t .

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional

J_{3}

stands for

\begin{matrix} J_{3} = J_{3} (ϕ, u, r, P, A, B, E, Λ, E) \\ = & G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} \\ + {〈Λ_{k}, ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) - ρ f_{k} + \frac{\partial P}{\partial x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{\partial ρ}{\partial t} + div (ρ u)〉}_{L^{2}} + J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}}, \end{matrix}

(97)

where,

\begin{matrix} J_{A u x_{1}} & = & {〈Λ_{5}, ρ \frac{D e}{D t} + P (div u) - \frac{\partial Q}{\partial t} + div q〉}_{L^{2}} \\ + {〈Λ_{6}, P - F_{7} (ρ, T)〉}_{L^{2}}, \end{matrix}

(98)

\begin{matrix} J_{A u x_{2}} & = & {〈Λ_{7}, m_{D} (t) - \int_{Ω} ρ_{D} (x, t) d x〉}_{L^{2}} \\ + {〈Λ_{8}, m_{T} (t) - \int_{Ω} ρ_{T} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{9}, m_{H_{e}} (t) - \int_{Ω} ρ_{H_{e}} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{10}, m_{N} (t) - \int_{Ω} ρ_{N} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{11}, m_{e} (t) - \int_{Ω} ρ_{e} (x, t) d x〉}_{L^{2}} \\ \int_{0}^{t_{f}} E_{1} 2 (t) {(α_{N} m_{H_{e}})}_{T} (t) - α_{H_{e}} {(m_{N})}_{T} (t)) d t, \end{matrix}

(99)

\begin{matrix} J_{A u x_{3}} & = & - \int_{0}^{t_{f}} \int_{Ω} {(E_{N}^{D})}_{5} (y, t) (\int_{Ω} {| ϕ_{N}^{D} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y, t) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y, t) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y, t) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y, t) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t) |}^{2} d x - m_{N}) d y d t, \end{matrix}

(100)

\begin{matrix} J_{A u x_{4}} & = & 〈Λ_{12}, curl E_{i n d} \\ + \frac{1}{c} curl ({\hat{K}}_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x)) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0})〉}_{L^{2}} \\ + {〈Λ_{13}, div B〉}_{L^{2}} \\ + {〈Λ_{14}, div E - 4 π (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x)〉}_{L^{2}} . \end{matrix}

(101)

Here we recall the following definitions and relations:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + | ϕ_{N}^{D} {(x, y, t) |}^{2} {| ϕ_{p}^{D} (y, t) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{D} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{D} {(x, y, t) |}^{2}) | ϕ_{p}^{D} {(y, t) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t) .$

1.: $ρ_{D} (y, t) = \int_{Ω} {| ϕ_{D} (x, y, t) |}^{2} d x,$
2.: $ρ_{T} (y, t) = \int_{Ω} {| ϕ_{T} (x, y, t) |}^{2} d x,$

$ρ_{H_{e}} (y, t) = \int_{Ω} {| ϕ_{H_{e}} (x, y, t) |}^{2} d x,$

$ρ_{N} (x, t) = {| ϕ_{N} (x, t) |}^{2},$

$ρ_{e} (y, t) = \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x .$

Also,

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e},

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

Finally,

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, ξ) d x d ξ,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, ξ)}{\partial x_{k}} d ξ\} .

and,

B = B_{0} - curl A .

20. A final mathematical description of the hydrogen nuclear fusion

In this section we develop in even more details another model for the hydrogen nuclear fusion.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Here such a set

Ω

Symbolically such a reaction stands for

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval

[0, t_{f}],

at this point we define the following density functions:

1.: For a single Deuterium atom indexed by s:

$| ϕ_{D} {(x, y, t, s) |}^{2} = | ϕ_{p}^{D} {(y, t, s) |}^{2} + | ϕ_{N}^{D} {(x, y, t, s) |}^{2} {| ϕ_{p}^{D} (y, t, s) |}^{2} \frac{1}{m_{p}},$
2.: For a single Tritium atom indexed by s:

$| ϕ_{T} {(x, y, t, s) |}^{2} = | ϕ_{p}^{T} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t, s) |}^{2}) | ϕ_{p}^{T} {(y, t, s) |}^{2} \frac{1}{m_{p}},$
3.: For a single Helium atom indexed by s:

$| ϕ_{H_{e}} {(x, y, t, s) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t, s) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field:

$ϕ_{N} = ϕ_{N} (x, t, s),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

Furthermore, we define also the related densities

1.: $ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
2.: $ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

For the chemical reaction in question we consider that one unit of mass of fractional proportion

α_{D}

of ionized Deuterium and

α_{T}

of ionized Tritium results in one unit of mass of fractional proportion

α_{H_{e}}

of ionized Helium and

α_{N}

of neutrons.

Symbolically, this stands for

1 = α_{D} + α_{T} = α_{H_{e}} + α_{N} .

Concerning the control volume

Ω

in question and related surface control

\partial Ω,

we assume such a volume has an initial (fot

t = 0

) amount of ionized Deuterium of

{(m_{D})}_{0}

and an initial amount of ionized Tritium of

{(m_{T})}_{0} .

The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control

\partial Ω

, we assume there is a part

Ω_{1} \subset \partial Ω

for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part

\partial Ω_{2} \subset \partial Ω

such that

\partial Ω_{1} \cap \partial Ω_{2} = \emptyset

for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

\partial Ω_{2}

is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

(102)

Here

n

denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, denoting by

N_{D} (t) N_{T} (t), N_{H_{e}} (t), N_{N} (t), N_{e} (t)

the respective indexed number of particles at time t, we define the functional

J (ϕ, ρ, r, u, E, A, B, {N_{D}, N_{T}, N_{H_{e}}, N_{N}, N_{e}})

where

J = G (\nabla u) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} + F_{4},

and where we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0, γ_{e} > 0

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0, α_{e, e} > 0, α_{H_{e}, e} < 0

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y d s d t \\ + \frac{γ_{N}^{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y d s d t \\ \frac{γ_{p}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y d s d t \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y d s d t \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y d s d t \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y d s d t \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y d s d t \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y d s d t \\ + \frac{γ_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x d s d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} (\nabla ϕ_{e}) \cdot (\nabla ϕ_{e}) d x d y d s d t, \end{matrix}

(103)

and

\begin{matrix} F (ϕ) = \\ \frac{α_{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{0}^{N_{D} (t)} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{0}^{N_{T} (t)} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{D T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{0}^{N_{T} (t)} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}, s_{1}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{0}^{N_{N} (t)} \int_{Ω} \frac{| ϕ_{N} (x - ξ, t, s - s_{1}) |^{2} {| ϕ_{N} (ξ, t, s_{1}) |}^{2}}{| x - ξ |} d x d ξ d s d s_{1} d t \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{D} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{N} (ξ_{j}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{H_{e}, e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{e} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{e, e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{0}^{N_{e} (t)} \int_{Ω} \frac{| ϕ_{e} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \end{matrix}

(104)

and the internal kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} |}^{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} d x d y d s d t, \end{matrix}

(105)

Moreover,

F_{1} = \frac{1}{4 π} \int_{0}^{t_{f}} {∥ curl A - B_{0} ∥}_{2} d t,

\begin{matrix} F_{2} & = & \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} E_{i n d} \cdot K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d x d y d s d t, \end{matrix}

(106)

where

K_{p}

and

K_{e}

are appropriate real constants related to the respective charges.

Here

u = (u_{1}, u_{2}, u_{3})

is the fluid velocity field and

r_{D}, r_{T}, r_{H_{e}}, r_{N}, r_{e}

are fields of displacements for the corresponding particle fields.

Also

A

denotes the magnetic potential,

B_{0}

an external magnetic field and

B

is the total magnetic field.

Moreover,

E_{i n d}

is an induced electric field.

Also,

\begin{matrix} F_{3} & = & \frac{C_{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} \nabla_{(x, y)} r_{D} \cdot \nabla_{(x, y)} r_{D} d x d y d s d t \\ + \frac{C_{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} \nabla_{(x, y)} r_{T} \cdot \nabla_{(x, y)} r_{T} d x d y d s d t \\ + \frac{C_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} \nabla_{(x, y)} r_{H_{e}} \cdot \nabla_{(x, y)} r_{H_{e}} d x d y d s d t \\ + \frac{C_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} \nabla_{(x, y)} r_{N} \cdot \nabla_{(x, y)} r_{N} d x d y d s d t \\ \frac{C_{e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} \nabla_{(x, y)} r_{e} \cdot \nabla_{(x, y)} r_{e} d x d y d s d t, \end{matrix}

(107)

for appropriate real positive constants

C_{D} C_{T}, C_{H_{e}}, C_{N}, C_{e} .

Finally,

\begin{matrix} F_{4} & = & \frac{ε_{D}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{D} (t)}{\partial t})}^{2} d t + \frac{ε_{T}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{D} (t)}{\partial t})}^{2} d t \\ + \frac{ε_{N}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{N} (t)}{\partial t})}^{2} d t + \frac{ε_{H_{e}}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{H_{e}} (t)}{\partial t})}^{2} d t \\ + \frac{ε_{e}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{e} (t)}{\partial t})}^{2} d t, \end{matrix}

(108)

where

ε_{D}, ε_{T}, ε_{N}, ε_{H_{e}}, ε_{e}

are small real positive constants.

Such a functional J is subject to the following constraints:

1.

The momentum conservation equation for the fluid motion

ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) = ρ f_{k} - \frac{\partial P}{\partial x_{k}} + τ_{k j, j} + {(F_{E})}_{k} + {(F_{M})}_{k},

\forall k \in {1, 2, 3} .

Here

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e}

is the total density and P is the fluid pressure field.

Furthermore,

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} δ_{i j} \sum_{k = 1}^{3} \frac{\partial u_{k}}{\partial x_{k}}),

\forall i, j \in {1, 2, 3},

\begin{matrix} F_{E} = {{(F_{E})}_{k}} = \\ (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} d s) E, \end{matrix}

and

\begin{matrix} F_{M} & = & {{(F_{M})}_{k}} \\ = & (K_{p} (\int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s) \\ + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d s) \times B . \end{matrix}

(109)

2.

Mass conservation equation:

\frac{\partial ρ}{\partial t} + div (ρ u) = 0 .

3.

Energy equation

ρ \frac{D e}{D t} + P (div u) = \frac{\partial Q}{\partial t} - div q,

where we assume the Fourier law

q = - K \nabla T,

where

T = T (x, t)

is the scalar field of temperature.

Also,

\begin{matrix} e & = & \frac{ρ}{2} u \cdot u + \frac{ρ_{D}}{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} \\ + \frac{ρ_{T}}{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} \\ + \frac{ρ_{H_{e}}}{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} \\ + \frac{ρ_{N}}{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} \\ + \frac{ρ_{e}}{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} \end{matrix}

(110)

and

\frac{D e}{D t} = \frac{\partial e}{\partial t} + u_{j} \frac{\partial e}{\partial x_{j}} .

4.

P = F_{7} (ρ, T),

for an appropriate scalar function

F_{7}

5.

Mass relations

(a): $m_{D} (t) = \int_{Ω} ρ_{D} (x, t) d x,$
(b): $m_{T} (t) = \int_{Ω} ρ_{T} (x, t) d x,$
(c): $m_{H e} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
(d): $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
(e): $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x,$

where,

(a): ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$
(b): $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
(c): $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
(d): $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
(e): ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
(f): ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
(g): $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
(h): ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$
(i): $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

(111)

6.

Other mass constraints

(a): $\int_{Ω} {| ϕ_{N}^{D} (x, y, t, s) |}^{2} d x = m_{N},$
(b): $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t, s) |}^{2} d x = m_{N},$
(c): $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t, s) |}^{2} d x = m_{N},$
(d): $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t, s) |}^{2} d x = m_{N},$
(e): $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t, s) |}^{2} d x = m_{N},$
(f): $\int_{Ω} {| ϕ_{p}^{D} (x, t, s) |}^{2} d x = m_{p},$
(g): $\int_{Ω} {| ϕ_{p}^{T} (x, t, s) |}^{2} d x = m_{p},$
(h): $\int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t, s) |}^{2} d x = 2 m_{p},$

7.

m_{D} (t) = m_{p} N_{D} (t) + m_{N} N_{D} (t)

m_{T} (t) = m_{p} N_{T} (t) + m_{N} N_{T} (t),

m_{H_{e}} (t) = 2 m_{p} N_{H_{e}} (t) + 2 m_{N} N_{H_{e}} (t),

m_{e} (t) = m_{e} N_{D} (t) + m_{e} N_{T} (t) + 2 m_{e} N_{H_{e}} (t) .

8.

For the induced electric field, we must have

\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} \int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x) d s) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0}) = 0, \end{matrix}

(112)

where

{\hat{K}}_{p}

and

{\hat{K}}_{e}

are appropriate real constants related to the respective charges.

9.

A Maxwell equation:

div B = 0,

where

B = B_{0} - curl A .

10.

Another Maxwell equation:

\begin{matrix} div E & = & 4 π (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) \\ + K_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s), \end{matrix}

(113)

where the total electric field

E

stands for

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, t ξ, s) d x d ξ d s,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, t, ξ, s)}{\partial x_{k}} d ξ d s\} .

At this point we generically denote

{〈 h_{1}, h_{2} 〉}_{L^{2}} = \int_{0}^{t_{f}} \int_{Ω} h_{1} h_{2} d x d y d t .

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional

J_{3}

stands for

\begin{matrix} J_{3} = J_{3} (ϕ, u, r, P, A, B, E, Λ, E, {N_{D}, N_{T}, N_{H_{e}}, N_{N}, N_{e}}) \\ = & G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} + F_{4} \\ + {〈Λ_{k}, ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) - ρ f_{k} + \frac{\partial P}{\partial x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{\partial ρ}{\partial t} + div (ρ u)〉}_{L^{2}} + J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}} + J_{A u x_{5}}, \end{matrix}

(114)

where,

\begin{matrix} J_{A u x_{1}} & = & {〈Λ_{5}, ρ \frac{D e}{D t} + P (div u) - \frac{\partial Q}{\partial t} + div q〉}_{L^{2}} \\ + {〈Λ_{6}, P - F_{7} (ρ, T)〉}_{L^{2}}, \end{matrix}

(115)

\begin{matrix} J_{A u x_{2}} & = & {〈Λ_{7}, m_{D} (t) - \int_{Ω} ρ_{D} (x, t) d x〉}_{L^{2}} \\ + {〈Λ_{8}, m_{T} (t) - \int_{Ω} ρ_{T} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{9}, m_{H_{e}} (t) - \int_{Ω} ρ_{H_{e}} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{10}, m_{N} (t) - \int_{Ω} ρ_{N} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{11}, m_{e} (t) - \int_{Ω} ρ_{e} (x, t) d x〉}_{L^{2}} \\ \int_{0}^{t_{f}} E_{12} (t) {(α_{N} m_{H_{e}})}_{T} (t) - α_{H_{e}} {(m_{N})}_{T} (t)) d t, \end{matrix}

(116)

\begin{matrix} J_{A u x_{3}} & = & - \int_{0}^{t_{f}} \int_{Ω} {(E_{N}^{D})}_{5} (y, t, s) (\int_{Ω} {| ϕ_{N}^{D} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y, t, s) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y, t, s) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y, t, s) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y, t, s) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t, s) |}^{2} d x - m_{N}) d y d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{p}^{D}) (t, s) (\int_{Ω} {| ϕ_{p}^{D} (y, t, s) |}^{2} d y - m_{p}) d s d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{p}^{T}) (t, s) (\int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y - m_{p}) d s d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{2 p}^{H_{e}}) (t, s) (\int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y - 2 m_{p}) d s d t, \end{matrix}

(117)

\begin{matrix} J_{A u x_{4}} & = & 〈Λ_{12}, curl E_{i n d} \\ + \frac{1}{c} curl ({\hat{K}}_{p} \int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t, s)}{\partial t} d x) d s) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0})〉}_{L^{2}} \\ + {〈Λ_{13}, div B〉}_{L^{2}} \\ + 〈Λ_{14}, div E - 4 π (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) \\ {+ K_{e} \int_{Ω} {| ϕ_{e} |}^{2} d x d s)〉}_{L^{2}} . \end{matrix}

(118)

\begin{matrix} J_{A u x_{5}} & = & {〈 Λ_{15}, m_{D} (t) - (m_{p} N_{D} (t) + m_{N} N_{D} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{16}, m_{T} (t) - (m_{p} N_{T} (t) + m_{N} N_{T} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{17}, m_{H_{e}} (t) - (2 m_{p} N_{H_{e}} (t) + 2 m_{N} N_{H_{e}} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{18}, m_{e} (t) - (m_{e} N_{D} (t) + m_{e} N_{T} (t) + 2 m_{e} N_{H_{e}} (t)) 〉}_{L^{2}} . \end{matrix}

(119)

Here we recall the following definitions and relations:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t, s) |}^{2} = | ϕ_{p}^{D} {(y, t, s) |}^{2} + | ϕ_{N}^{D} {(x, y, t, s) |}^{2} {| ϕ_{p}^{D} (y, t, s) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{T} {(x, y, t, s) |}^{2} = | ϕ_{p}^{T} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t, s) |}^{2}) | ϕ_{p}^{D} {(y, t, s) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t, s) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t, s) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t, s),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

1.: $ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
2.: $ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

Also,

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e},

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} {(m_{H_{e}})}_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d Γ d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

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Finally,

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, t, ξ, s) d x d ξ d s,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, t, ξ, s)}{\partial x_{k}} d ξ d s\} .

and,

B = B_{0} - curl A .

21. A qualitative modeling for a general phase transition process

In this section we develop a general qualitative modeling for a phase transition process.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Such a set

Ω

is supposed to a be a fixed volume in which an amount of mass of a substance A with a density function u will develop phase a transition for another phase with corresponding density function

v .

The total mass

m_{T}

is suppose to be kept constant throughout such a process.

We model such transition in phase through a functional

J : V \times V \to R

where

\begin{matrix} J (u, v) & = & \frac{γ_{1}}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α_{1}}{2} \int_{Ω} u^{4} d x \\ \frac{γ_{2}}{2} \int_{Ω} \nabla v \cdot \nabla v d x + \frac{α_{2}}{2} \int_{Ω} v^{4} d x \\ - \frac{1}{2} \int_{Ω} ω^{2} (u^{2} + v^{2}) d x - \frac{E}{2} (\int_{Ω} (u^{2} + v^{2}) d x - m_{T}) . \end{matrix}

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Here

γ_{1} > 0, γ_{2} > 0, α_{1} > 0, α_{2} > 0

and

V = W^{1, 2} (Ω) .

The phases corresponding to u and v are connected through a Lagrange multiplier E, which represents the chemical potential of the chemical process in question.

We assume the temperature is directly proportional to the internal kinetics

E_{C}

energy where

E_{C} = \frac{1}{2} \int_{Ω} u^{2} \frac{\partial r_{u}}{\partial t} \cdot \frac{\partial r_{u}}{\partial t} d x .

For a internal vibrational motion, we assume approximately

r_{u} \approx e^{i ω t} w_{5} (x),

for an appropriate frequency

ω

and vectorial function

w_{5} .

Thus, the temperature

T = T (x, t)

is indeed proportional to

ω^{2}

, that is, symbolically, we may write

T \propto E_{1} \propto ω^{2} .

Therefore, we start with the system with a phase corresponding to

u \approx 1

and

v \approx 0

ω = 1

. Gradually increasing the temperature to a corresponding

ω = 15

, we obtain a transition to a phase corresponding to

u \approx 0

and

v \approx 1

At this point, we also define the index normalized corresponding densities

ϕ_{u} = \frac{u^{2}}{u^{2} + v^{2}}

and

ϕ_{v} = \frac{v^{2}}{u^{2} + v^{2}} .

Finally, we have obtained some numerical results for the following parameters:

Ω = [0, 1] \subset R,

γ_{1} = γ_{2} = 1

α = 0.1

α_{2} = 10^{3} .

1.: We start with $ω = 1$ corresponding to $ϕ_{u} \approx 1$ and $ϕ_{v} \approx 0$ in $Ω$ .

For the corresponding solutions $ϕ_{u}$ and $ϕ_{v}$ , please see Figure 15 and Figure 16, respectively.
2.: We end the process with $ω = 15$ corresponding to $ϕ_{u} \approx 0$ and $ϕ_{v} \approx 1$ in $Ω$ .

For the corresponding solutions $ϕ_{u}$ and $ϕ_{v}$ , please see Figure 17 and Figure 18, respectively.

22. A mathematical description of a hydrogen molecule in a quantum mechanics context

In this section we develop a mathematical description for a hydrogen molecule.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Observe that a single hydrogen molecule comprises two hydrogen atoms physically linked through their electrons.

We recall that each hydrogen atom comprises one proton, one neutron and one electron.

Since the electric charge interaction effects are much higher than those related to the respective masses, in a first analysis we neglect the single neutron densities.

Denoting

(x, y, z) \in Ω \times Ω \times Ω

and time

t \in [0, t_{f}]

, generically, for a particle

p_{j k l}

at the atom

A_{k l}

in the molecule

M_{l}

, we define the following general density:

| ϕ_{{(p_{j k l})}_{T}} {(x, y, z, t) |}^{2} = \frac{| ϕ_{p_{j k l}} {(x, y, z, t) |}^{2} | ϕ_{A_{k l}} {(y, z, t) |}^{2} {| ϕ_{M_{l}} (z, t) |}^{2}}{m_{A_{j k}} m_{M_{l}}} .

Here we have the particle density

| ϕ_{p_{j k l}} {(x, y, z, t) |}^{2}

in the atom

A_{k l}

with density

| ϕ_{A_{k l}} {(y, z, t) |}^{2}

, at the molecule

M_{l}

with a global density

| ϕ_{M_{l}} {(z, t) |}^{2}

Here we have also denoted,

m_{p_{j k l}}

the particle mass,

m_{A_{k l}}

the mass of atom

A_{k l}

and

m_{M_{l}}

the mass of molecule

M_{l}

, so that we set the following constraints:

1.: $\int_{Ω} {| ϕ_{p_{j k l}} (x, y, z, t) |}^{2} d x = m_{p_{j k l}},$
2.: $\int_{Ω} {| ϕ_{A_{k l}} (y, z, t) |}^{2} d y = m_{A_{k l}},$
3.: $\int_{Ω} {| ϕ_{M_{l}} (z, t) |}^{2} d z = m_{M_{l}} .$

At this point we denote for the atoms

A_{1}

A_{2}

of a hydrogen molecule:

1.: $m_{e_{j}} = m_{e}$ : mass of electron $e_{j}$ in the atom $A_{j}$ , where $j \in {1, 2} .$
2.: $m_{p_{j}} = m_{p}$ : mass of proton $p_{j}$ in the atom $A_{j}$ , where $j \in {1, 2} .$

Therefore, considering the respective indexed densities for the particles in question, we define the total hydrogen molecule density, denoted by

| ϕ_{H_{2}} {(x, y, z, t) |}^{2}

\begin{matrix} | ϕ_{H_{2}} {(x, y, z, t) |}^{2} & = & \frac{| ϕ_{p_{1}} {(x, y, z, t) |}^{2} | ϕ_{A_{1}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{1}} m_{M}} \\ + \frac{| ϕ_{e_{1}} {(x, y, z, t) |}^{2} | ϕ_{A_{1}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{1}} m_{M}} \\ + \frac{| ϕ_{p_{2}} {(x, y, z, t) |}^{2} | ϕ_{A_{2}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{2}} m_{M}} \\ + \frac{| ϕ_{e_{2}} {(x, y, z, t) |}^{2} | ϕ_{A_{2}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{2}} m_{M}} . \end{matrix}

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Such system is subject to the following constraints:

From the proton $p_{1}$ in the atom $A_{1}$ :

$\int_{Ω} {| ϕ_{p_{1}} (x, y, z, t) |}^{2} d x = m_{p},$
For the proton $p_{2}$ in the atom $A_{2}$ :

$\int_{Ω} {| ϕ_{p_{2}} (x, y, z, t) |}^{2} d x = m_{p},$
For the atom $A_{1}$ :

$\int_{Ω} {| ϕ_{A_{1}} (y, z, t) |}^{2} d y = m_{A_{1}},$
For the atom $A_{2}$ :

$\int_{Ω} {| ϕ_{A_{2}} (y, z, t) |}^{2} d y = m_{A_{2}},$
For the electrons $e_{1}$ and $e_{2}$ , concerning the physical electronic link between the atoms:

$\int_{Ω} | ϕ_{e_{1}} {(x, y, z, t) |}^{2} d x + \int_{Ω} {| ϕ_{e_{2}} (x, y, z, t) |}^{2} d x = 2 m_{e} .$
For the total molecular density:

$\int_{Ω} {| ϕ_{M} (z, t) |}^{2} d z = m_{M} .$

Therefore, already including the Lagrange multipliers, the corresponding variational formulation for such a system stands for

J : V \to R

, where

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + J_{A u x} (ϕ, E) .

Here we denote

| {(ϕ_{p_{j}})}_{T} |^{2} = \frac{| ϕ_{p_{j}} {(x, y, z, t) |}^{2} | ϕ_{A_{j}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{j}} m_{M}},

| {(ϕ_{e_{j}})}_{T} |^{2} = \frac{| ϕ_{e_{j}} {(x, y, z, t) |}^{2} | ϕ_{A_{j}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{j}} m_{M}}, \forall j \in {1, 2}

we assume

γ_{(p_{j})} > 0, γ_{e_{j}} > 0, γ_{A_{j}} > 0, γ_{M} > 0, α_{{(p_{j})}_{T}} > 0, α_{{(e_{j})}_{T}} > 0 α_{{(p_{j} e_{k})}_{T}} < 0, \forall j, k \in {1, 2},

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p_{j}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p_{j}}) \cdot (\nabla ϕ_{p_{j}}) d x d y d z d t \\ + \frac{γ_{e_{j}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{e_{j}}) \cdot (\nabla ϕ_{e_{j}}) d x d y d z d t \\ \frac{γ_{A_{j}}}{2} \int_{Ω} (\nabla ϕ_{A_{j}}) \cdot (\nabla ϕ_{A_{j}}) d y d z d t \\ + \frac{γ_{M}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{M}) \cdot (\nabla ϕ_{M}) d z d t \end{matrix}

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and

\begin{matrix} F (ϕ) = \\ \frac{α_{{(p_{j})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(p_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(p_{j})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z; d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{α_{{(e_{j})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(e_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(e_{j})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{α_{{(p_{j} e_{k})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(p_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(e_{k})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \end{matrix}

Finally,

\begin{matrix} J_{A u x} (ϕ, E) & = & \int_{0}^{t_{f}} \int_{Ω} {(E_{p})}_{j} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} (E_{e}) (y, z, t) (\int_{Ω} (| ϕ_{e_{1}} {(x, y, z, t) |}^{2} + | ϕ_{e_{2}} (x, y, z, t) |^{2}) d x - 2 m_{e}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} {(E_{A})}_{j} (z, t) (\int_{Ω} {| ϕ_{A_{j}} (y, z, t) |}^{2} d y - m_{A_{j}}) d z d t \\ \int_{0}^{t_{f}} (E_{M}) (t) (\int_{Ω} {| ϕ_{M} (z, t) |}^{2} d z - m_{M}) d t . \end{matrix}

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Remark 22.1.

We highlight the two electrons which link the atoms are at same level of energy

E_{e}

. Morever, each atom has its energy level

E_{A_{j}}

and the molecule as a whole has also its energy level

E_{M} .

23. A mathematical model for the water hydrolysis

In this section we develop a modeling for a chemical reaction known as the water hydrolysis.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

In such a volume

Ω

containing a total mass

m_{T}

of water initially at the temperature 25 C with pressure 1 atm, we intend to model the following reaction

H_{2} O ⇌ O H^{-} + H^{+}

which as previously mentioned is the well known water hydrolysis.

We highlight

H_{2} O

stand for a water molecule which subject to an appropriate electric potential is decomposed into a ionized

O H^{-}

molecule and ionized

H^{+}

atom.

It is also well known that the water symbol

H_{2} O

corresponds to a molecule with two hydrogen (H) atoms and one oxygen (O) atom.

Moreover, the oxygen atom O has 8 protons, 8 neutrons and 8 electrons whereas the hydrogen atom H has one proton, one neutron and one electron.

Remark 23.1.

Here we have assumed that a unit mass of

H_{2} O

reacts into a fractional mass

α_{B}

O H^{-}

and a fractional mass

α_{C}

H^{+} .

Symbolically, we have:

1 = α_{B} + α_{C} .

To clarify the notation we set the conventions:

$H_{2} O$ molecule generically corresponds to wave function $ϕ_{1}$ .
$O H^{-}$ molecule corresponds to wave function $ϕ_{2} .$
$H^{+}$ hydrogen atom corresponds to wave function $ϕ_{3} .$

At this point we define the following densities:

For the $H_{2} O$ water density (for charges), denoted by $| ϕ_{1} |^{2}$ , we have

$\begin{matrix} | ϕ_{1} {(x, y, z, t) |}^{2} & = & K_{p} \sum_{j = 1}^{2} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A_{j}}^{H} {(m_{1})}_{M}} \\ + K_{e} \sum_{j = 1}^{2} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m_{1})}_{A_{j}}^{H} {(m_{1})}_{M}} \\ + K_{p} \sum_{j = 1}^{8} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{1})}_{M}} \\ + K_{e} \sum_{j = 1}^{8} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{1})}_{M}} \end{matrix}$

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where ${(m_{1})}_{M}$ is the mass of a single water molecule and generically $| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |^{2}$ refers to the hydrogen proton $p_{j}$ at the hydrogen atom $A_{j}$ concerning the $H_{2} O$ molecular density and so on.
For the $O H^{-}$ density, denoted by $| ϕ_{2} |^{2}$ , we have

$\begin{matrix} | ϕ_{2} {(x, y, z, t) |}^{2} & = & K_{p} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{H})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{H} {(m_{2})}_{M}} \\ + K_{e} {| {(ϕ_{2}^{H})}_{e_{1}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{H})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{H} {(m_{2})}_{M}} \\ + K_{e} {| {(ϕ_{2}^{O H^{-}})}_{e_{2}} (x, z, t) |}^{2} \frac{| {(ϕ_{2})}_{M} (z, t) |^{2}}{{(m_{2})}_{M}} \\ + K_{p} \sum_{j = 1}^{8} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{2})}_{M}} \\ + K_{e} \sum_{j = 1}^{8} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{2})}_{M}}, \end{matrix}$

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where ${(m_{2})}_{M}$ is the mass of a single molecule of $O H^{-}$ .
For the ionized hydrogen atom have

$| ϕ_{3} {(x, y, t) |}^{2} = K_{p} {| {(ϕ_{3}^{H})}_{p} (x, y, t) |}^{2} \frac{| {(ϕ_{3}^{H})}_{A} (y, t) |^{2}}{{(m_{3})}_{A}} .$

where we have denoted

{(m_{3})}_{A}

is the mass of a single atom of

H^{+}

Here

K_{p} > 0

and

K_{e} < 0

are appropriate real constants concerning a proton and an electron charge, respectively.

The system is subject to the following constraints:

$\int_{Ω} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} d x = m_{p},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{e_{1}} (x, y, z, t) |}^{2} d x = m_{e},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{e_{2}} (x, y, z, t) |}^{2} d x = m_{e},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{3}^{H})}_{p} (x, z, t) |}^{2} d x = m_{p},$
$\int_{Ω} {| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |}^{2} d y = m_{A}^{H}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{A} (y, z, t) |}^{2} d y = m_{A}^{O},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{A} (y, z, t) |}^{2} d y = m_{A}^{H},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{A} (y, z, t) |}^{2} d y = m_{A}^{O},$
$\int_{Ω} {| {(ϕ_{3}^{H})}_{A} (y, z, t) |}^{2} d y = m_{A}^{H},$
$\int_{Ω} (| {(ϕ_{1})}_{M} (z, t) |^{2} + | {(ϕ_{2})}_{M} (z, t) |^{2} + | {(ϕ_{3})}_{M} (z, t) |^{2}) d z = m_{T},$
$\int_{Ω} (α_{C} | {(ϕ_{2})}_{M} (z, t) |^{2} - α_{B} | {(ϕ_{3})}_{M} (z, t) |^{2}) d z = 0 .$

Already including the Lagrange multipliers for the constraints, the variational formulation for such system. denoted by the functional

J (ϕ, E)

stands for

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + F_{1} (ϕ) - J_{A u x} (ϕ, E),

where

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{O})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{O})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{p} \cdot \nabla {(ϕ_{2}^{H})}_{p} d x d y d z d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{e_{1}} \cdot \nabla {(ϕ_{2}^{H})}_{e_{1}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O H^{-}})}_{e_{2}} \cdot \nabla {(ϕ_{1}^{O H^{-}})}_{e_{2}} d x d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{p_{j}} \cdot \nabla {(ϕ_{2}^{O})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{e_{j}} \cdot \nabla {(ϕ_{2}^{O})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{p} \cdot \nabla {(ϕ_{2}^{O})}_{p} d x d y d t \\ + \frac{γ_{A^{H}}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{A_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{A_{j}} d y d z d t \\ + \frac{γ_{A^{O}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{A} \cdot \nabla {(ϕ_{1}^{O})}_{A} d y d z d t \\ + \frac{γ_{A^{H}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{A} \cdot \nabla {(ϕ_{2}^{H})}_{A} d y d z d t \\ + \frac{γ_{A^{O}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{A} \cdot \nabla {(ϕ_{2}^{O})}_{A} d y d z d t \\ + \frac{γ_{M_{1}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1})}_{M} \cdot \nabla {(ϕ_{1})}_{M} d z d t \\ + \frac{γ_{M_{2}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2})}_{M} \cdot \nabla {(ϕ_{2})}_{M} d z d t \\ \frac{γ_{A_{3}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{3})}_{A} \cdot \nabla {(ϕ_{3})}_{A} d y d t . \end{matrix}

Here

γ_{p} > 0

γ_{e} > 0

γ_{A}^{H} > 0,

γ_{A}^{O} > 0,

γ_{M_{1}} > 0

γ_{M_{2}} > 0

γ_{A_{3}} > 0 .

Moreover,

\begin{matrix} F (ϕ) \\ = & \frac{α_{1}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{1} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{1} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \\ + \frac{α_{2}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{2} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \\ + \frac{α_{3}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{3} (x - ξ_{1}, z - ξ_{3}, t) |^{2} {| ϕ_{3} (ξ_{1}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{3} d t \\ + \frac{α_{23}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{3} (ξ_{1}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \end{matrix}

where

α_{1} > 0

α_{2} > 0

α_{3} > 0

and

α_{23} > 0 .

Furthermore,

\begin{matrix} F_{1} (ϕ) & = & \int_{0}^{t_{f}} \int_{Ω} V (x, y, z, t) (| ϕ_{1} |^{2} + | ϕ_{2} |^{2} + | ϕ_{3} |^{2}) d x d y d z d t, \end{matrix}

(127)

where

V = V (x, y, z, t)

is an electric potential originated from an external electric field

E

applied on

Ω .

Finally,

\begin{matrix} J_{A u x} (ϕ, E) \\ = & \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{p_{j}}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{e_{j}}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{p_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{e_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{p}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{p_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{e_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{3})}_{p}^{H} (y, t) (\int_{Ω} {| {(ϕ_{3}^{H})}_{p} (x, y, t) |}^{2} d x - m_{p}) d y d t \\ + \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{4})}_{A_{j}}^{H} (z, t) (\int_{Ω} (| {(ϕ_{1})}_{A_{j}}^{H} (y, z, t) |^{2} d y - m_{A_{j}}^{H}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} \int_{Ω} {(E_{4})}_{A}^{O} (z, t) (\int_{Ω} (| {(ϕ_{1})}_{A}^{O} (y, z, t) |^{2} d y - m_{A}^{O}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{5})}_{A}^{H} (z, t) (\int_{Ω} (| {(ϕ_{2})}_{A}^{H} (y, z, t) |^{2} d y - m_{A}^{H}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{5})}_{A}^{O} (z, t) (\int_{Ω} (| {(ϕ_{2})}_{A}^{O} (y, z, t) |^{2} d y - m_{A}^{O}) d z d t \\ \int_{0}^{t_{f}} {(E_{6})}_{A}^{H} (t) (\int_{Ω} (| {(ϕ_{3})}_{A}^{H} (y, t) |^{2} d y - m_{A}^{H}) d t \\ + \int_{0}^{t_{f}} (E_{7}) (t) (\int_{Ω} (| {(ϕ_{1})}_{M} (z, t) |^{2} + | {(ϕ_{2})}_{M} (z, t) |^{2} + | {(ϕ_{3})}_{M} (z, t) |^{2}) d z - m_{T}) d t \\ + \int_{0}^{t_{f}} (E_{8}) (t) (\int_{Ω} (α_{C} | {(ϕ_{2})}_{M} (z, t) |^{2} - α_{B} | {(ϕ_{3})}_{M} (z, t) |^{2}) d z) d t . \end{matrix}

(128)

24. A mathematical model for the Austenite and Martensite phase transition

In this section we consider a phase transition of a solid solution of

γ - F e

(

γ - i r o n

) and carbon with a

0.75 / 100

proportion of carbon, known as austenite, initially at a temperature above and close to

723 C

and rapidly cooled to a temperature of about

25 C

, developing a phase transition which generates a solid solution of

α - F e

(

α - i r o n

) and carbon known as martensite.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω

which contains an amount of austenite at

723 C

and which, as previously mentioned, is rapidly cooled to a temperature

25 C

on a time interval

[0, t_{f}],

resulting a phase known as martensite.

We recall the

γ - F e

of austenite phase presents a multi-faced cubic crystalline structure in a micro-structure with carbon atoms.

On the other hand,

α - F_{e}

structure of the martensite phase has a

C C C

cubic centralized crystalline structure in a micro-structure with carbon atoms.

At this point, we also recall that the

F_{e}

(iron) atom has 26 protons, 26 electrons and 30 neutrons.

On the other hand a

C a r b o n_{12}

atom has 6 protons and this same number of electrons and neutrons.

Here we define the density function

ϕ_{1}

, representing the Austenite phase, where:

\begin{matrix} | ϕ_{1} {(x, y, z, t) |}^{2} & = & \sum_{j = 1}^{26} | ϕ_{p_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{26} | ϕ_{e_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{30} | ϕ_{N_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} . \end{matrix}

(129)

Similarly, we define the density function for the Martensite phase, which is denoted by

ϕ_{2}

, where:

\begin{matrix} | ϕ_{2} {(x, y, z, t) |}^{2} & = & \sum_{j = 1}^{26} | ϕ_{p_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{α} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{26} | ϕ_{e_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{α - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{30} | ϕ_{N_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{α - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{α} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |^{2} | {(ϕ_{2}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{2}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |^{2} | {(ϕ_{2}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{2}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} . \end{matrix}

(130)

For the

C F C

γ - F_{e}

(γ - i r o n)

corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

C_{γ} = {(ε_{1}, 0, 0), (0, ε_{2}, 0), (0, 0, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

{(C_{γ})}_{1} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

and

{(C_{γ})}_{2} = {(ε_{1}, ε_{2}, 0), (ε_{1}, 0, ε_{3}), (0, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

we must have

ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in C_{γ},

where

δ_{z} \in R^{+}

is a small real parameter related to

γ - F_{e}

crystalline structure dimensions.

We must have also,

ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in {(C_{γ})}_{1}

and,

{(ϕ_{1}^{C})}_{A} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = {(ϕ_{1}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in {(C_{γ})}_{2} .

For the

C C C

α - F_{e}

(α - i r o n)

corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

C_{α} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

{(C_{α})}_{1} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{1}, ε_{2} \in {+ 1, - 1} and ε_{3} = 0},

{(C_{α})}_{2} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{1} = ε_{2} = 0 and ε_{3} \in {+ 1, - 1}},

we must have

ϕ_{A}^{α - F_{e}} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) = ϕ_{A}^{α - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t),

\forall ε, \tilde{ε} \in C_{α},

where

{\hat{δ}}_{z} \in R^{+}

is a small real parameter related to

α - F_{e}

crystalline structure dimensions.

We must have also,

{(ϕ_{2}^{C})}_{A} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) = {(ϕ_{2}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t),

\forall ε, \tilde{ε} \in {(C_{α})}_{1} \cup {(C_{α})}_{2} .

The other constraints for the densities are given by:

1.

For the Austenite phase:

(a): $\int_{Ω} {| ϕ_{p_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 26},$
(b): $\int_{Ω} {| ϕ_{e_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 26},$
(c): $\int_{Ω} {| ϕ_{N_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 30},$
(d): $\int_{Ω} {| ϕ_{A}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{A}^{γ},$
(e): $\int_{Ω} {| {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 6},$
(f): $\int_{Ω} {| {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 6},$
(g): $\int_{Ω} {| {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 6},$
(h): $\int_{Ω} {| {(ϕ_{1}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C},$

2.

For the Martensite phase:

(a): $\int_{Ω} {| ϕ_{p_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 26},$
(b): $\int_{Ω} {| ϕ_{e_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 26},$
(c): $\int_{Ω} {| ϕ_{N_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 30},$
(d): $\int_{Ω} {| ϕ_{A}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{A}^{α},$
(e): $\int_{Ω} {| {(ϕ_{2}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 6},$
(f): $\int_{Ω} {| {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 6},$
(g): $\int_{Ω} {| {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 6},$
(h): $\int_{Ω} {| {(ϕ_{2}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C} .$

3.

For the total

F_{e}

(iron) mass,

\int_{Ω} | ϕ_{1}^{γ} {(z, t) |}^{2} d z + \int_{Ω} {| ϕ_{2}^{γ} (z, t) |}^{2} d z = {(m_{F_{e}})}_{T},

4.

For the total Carbon mass

\int_{Ω} | ϕ_{1}^{C} {(z, t) |}^{2} d z + \int_{Ω} {| ϕ_{2}^{C} (z, t) |}^{2} d z = {(m_{C})}_{T} .

At this point we define the functional J which models such a pahse transition in question, where

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + F_{1} (ϕ) + J_{A u x} (ϕ, E)

where

\begin{matrix} G (\nabla ϕ) & = & \sum_{j = 1}^{26} \frac{{\hat{γ}}_{p}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{p_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{p_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{e}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{e_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{e_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{30} \frac{{\hat{γ}}_{N}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{N_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{N_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{p}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{p_{j}}^{α - F_{e}} \cdot \nabla ϕ_{p_{j}}^{α - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{e}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{e_{j}}^{α - F_{e}} \cdot \nabla ϕ_{e_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{30} \frac{{\hat{γ}}_{N}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{N_{j}}^{α - F_{e}} \cdot \nabla ϕ_{N_{j}}^{α - F_{e}} d x d y d z d t \\ + \frac{{\hat{γ}}_{A}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{A}^{γ - F_{e}} (y, z, t) \cdot \nabla ϕ_{A}^{γ - F_{e}} (y, z, t)) d y d z d t \\ + \frac{{\hat{γ}}_{A}^{α}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{A}^{α - F_{e}} (y, z, t) \cdot \nabla ϕ_{A}^{α - F_{e}} (y, z, t)) d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{p}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{p_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{e}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{e_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{N}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{N_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{N_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{p}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{p_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{p_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{e}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{e_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{e_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{N}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{N_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{N_{j}} d x d y d z d t \\ + \frac{{\hat{γ}}_{A}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla {(ϕ_{1}^{C})}_{A} \cdot \nabla {(ϕ_{1}^{C})}_{A}) d y d z d t + \frac{{\hat{γ}}_{A}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla {(ϕ_{2}^{C})}_{A} \cdot \nabla {(ϕ_{2}^{C})}_{A}) d y d z d t \\ + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{γ}) \cdot \nabla (ϕ_{1}^{γ})) d z d t + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{α}) \cdot \nabla (ϕ_{1}^{α})) d z d t \\ + \frac{{\hat{γ}}_{T}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{C}) \cdot \nabla (ϕ_{1}^{C})) d z d t + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{2}^{C}) \cdot \nabla (ϕ_{2}^{C})) d z d t \end{matrix}

(131)

Also,

\begin{matrix} F (ϕ) \\ = & \frac{{\hat{α}}_{1}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{1} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} | | ϕ_{1} (ξ_{1}, ξ_{2}, ξ_{3}, t) |^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{{\hat{α}}_{2}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} | | ϕ_{2} (ξ_{1}, ξ_{2}, ξ_{3}, t) |^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t, \end{matrix}

F_{1} (ϕ) = - \int_{0}^{t_{f}} \int_{Ω} w^{2} (z, t) (| ϕ_{1} {(z, t) |}^{2} + ϕ_{2} (z, t) |^{2}) d z d t,

Finally,

J_{A u x} = J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}} + J_{A u x_{5}}

, where

\begin{matrix} J_{A u x_{1}} & = & \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{p_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{e_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{e_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 1}^{30} \int_{0}^{t_{f}} \int_{Ω} E_{N_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{N_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{p_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{e_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{e_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 1}^{30} \int_{0}^{t_{f}} \int_{Ω} E_{N_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{N_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{A}^{γ - F_{e}} (y, t) (\int_{Ω} {| ϕ_{A}^{γ - F_{e}} (y, z, t) |}^{2} d y - m_{A}^{γ}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{A}^{α - F_{e}} (y, t) (\int_{Ω} {| ϕ_{A}^{α - F_{e}} (y, z, t) |}^{2} d y - m_{A}^{α}) d z d t \end{matrix}

(132)

\begin{matrix} J_{A u x_{2}} & = & \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{p_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{e_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{N_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{p_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{e_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{N_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{A} (y, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{A} (y, z, t) |}^{2} d y - m_{A}^{C}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{A} (y, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{A} (y, z, t) |}^{2} d y - m_{A}^{C}) d z d t \end{matrix}

(133)

and,

\begin{matrix} J_{A u x_{3}} & = & \int_{0}^{t_{f}} E_{3}^{γ, α} (t) (\int_{Ω} (| ϕ_{1}^{γ} {(z, t) |}^{2} + | ϕ_{2}^{α} (z, t) |^{2}) d z - {(m_{F_{e}})}_{T}) d t \\ + \int_{0}^{t_{f}} E_{3}^{C} (t) (\int_{Ω} (| ϕ_{1}^{C} {(z, t) |}^{2} + | ϕ_{2}^{C} (z, t) |^{2}) d z - {(m_{C})}_{T}) d t . \end{matrix}

(134)

\begin{matrix} J_{A u x_{4}} \\ = & + \sum_{ε, \tilde{ε} \in C_{γ}} \int_{0}^{t_{f}} \int_{Ω} E_{4}^{ε, \tilde{ε}} (y, z, t) (ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ \sum_{ε, \tilde{ε} \in {(C_{γ})}_{1}} \int_{0}^{t_{f}} \int_{Ω} E_{5}^{ε, \tilde{ε}} (y, z, t) ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in {(C_{γ})}_{2}} \int_{0}^{t_{f}} \int_{Ω} E_{6}^{ε, \tilde{ε}} (y, z, t) {(ϕ_{1}^{C})}_{A} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - {(ϕ_{1}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in (C_{α})} \int_{0}^{t_{f}} \int_{Ω} E_{7}^{ε, \tilde{ε}} (y, z, t) (ϕ_{A}^{α - F_{e}} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) \\ - ϕ_{A}^{α - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in {(C_{α})}_{1} \cup {(C_{α})}_{2}} \int_{0}^{t_{f}} \int_{Ω} E_{8}^{ε, \tilde{ε}} (y, z, t) ({(ϕ_{2}^{C})}_{A} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) \\ - {(ϕ_{2}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t)) d y d z d t . \end{matrix}

(135)

Finally, for a field of displacements

u = (u_{1}, u_{2}, u_{3})

resulting from the action of a external load field

f = (f_{1}, f_{2}, f_{3})

and temperature variations, we define

\begin{matrix} J_{A u x_{5}} \\ = & \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} (Λ_{1} (x, t) H_{i j k l}^{1} ((e_{i j} (u) - e_{i j}^{1} (w)) (e_{k l} (u) - e_{k l}^{1} (w))) \\ + Λ_{2} (z, t) H_{i j k l}^{2} ((e_{i j} (u) - e_{i j}^{2} (w)) (e_{k l} (u) - e_{k l}^{2} (w)))) d x d t \\ - \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} ρ (x, t) u_{t} (x, t) \cdot u_{t} (x, t) d x d t \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

(136)

where

e_{i j} (u) = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}),

ρ_{1} (z, t) = \int_{Ω} {| ϕ_{1} (x, y, z, t) |}^{2} d x d y,

ρ_{2} (z, t) = \int_{Ω} {| ϕ_{2} (x, y, z, t) |}^{2} d x d y,

ρ (z, t) = ρ_{1} (z, t) + ρ_{2} (z, t),

and

Λ_{1} (z, t) = \frac{ρ_{1} (z, t)}{ρ_{1} (z, t) + ρ_{2} (z, t)},

Λ_{2} (z, t) = \frac{ρ_{2} (z, t)}{ρ_{1} (z, t) + ρ_{2} (z, t)} .

Remark 24.1.

The system temperature is supposed to be directly proportional to

w {(z, t)}^{2}

, which in this model is a known function obtained experimentally. Finally, the strain tensors

{e_{i j}^{1} (w)}

and

{e_{i j}^{2} (w)}

refer to austenite and martensite phases, respectively. Such tensors also depend on the temperature and must be also obtained experimentally.

25. A note on classical free fields through a variational perspective

This section is strongly based on the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov.

Therefore, the credit for this section is of these mentioned authors. This section is a kind of review of such a book chapter indicated. In fact, what we have done is simply to open more and clarify some calculations, specially about the first variation of the functional L, in order to improve their understanding.

Let

Ω = \hat{Ω} \times [0, T] \subset R^{4}

where

\hat{Ω} \subset R^{3}

is a bounded, open and connected set with a regular boundary denoted by

\partial \hat{Ω} .

Consider the Lagrangian density

L : R^{N} \times R^{N \times n} \to R

and an action

A : V \to R

where

A (u) = \int_{Ω} L (u, \nabla u) d x,

V = W_{0}^{1, 2} (Ω; R^{N})

We denote

\nabla u = \{\frac{\partial u_{i}}{\partial x_{j}}\}

and

\frac{\partial u_{i}}{\partial x_{j}} = {(u_{i})}_{x_{j}} .

Assume

u \in V

is such that

δ L (u, \nabla u) = 0,

so that

\frac{\partial L (u, \nabla u)}{\partial u_{i}} - \sum_{k = 1}^{n} \frac{d}{d x_{k}} (\frac{\partial L (u, \nabla u)}{\partial {(u_{i})}_{x_{k}}}) = 0, in Ω, \forall i \in {1, \dots, N} .

We define a change of variables

{(x^{'})}_{k} = x_{k} + δ x_{k},

where

x_{k} = (x_{0}, x_{1}, x_{2}, x_{3})

and

x_{0} = t

(here t denotes time).

Also

g_{j k} = 0, if j \neq k, g_{00} = - 1 and g_{11} = g_{22} = g_{33} = 1, {g^{j k}} = {g_{j k}}^{- 1},

δ x_{k} = \sum_{j = 1}^{N} X_{j}^{k} ε w^{j},

where

| ε | ≪ 1

denotes a small real parameter.

We define also

u_{i}^{'} (x^{'}) = u_{i} (x) + δ u_{i} (x),

where

δ u_{i} (x) = \sum_{j = 1}^{N} ψ_{i j} ε w^{j},

and

\bar{δ u_{i}} = u_{i}^{'} (x) - u_{i} (x) .

Observe that

\begin{matrix} δ u_{i} (x) & = & u_{i}^{'} (x) - u_{i} (x) \\ = & u_{i}^{'} (x^{'}) - u_{i}^{'} (x) + u_{i}^{'} (x) - u_{i} (x), \end{matrix}

(137)

so that

\begin{matrix} \bar{δ u_{i} (x)} & = & u_{i}^{'} (x) - u_{i} (x) \\ = & δ u_{i} (x) - (u_{i}^{'} (x^{'}) - u_{i}^{'} (x)) \\ \sum_{j = 1}^{N} ψ_{i j} ε w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i}^{'} ({\tilde{x}}^{i})}{d x_{k}} δ x_{k} \\ = & \sum_{j = 1}^{N} ψ_{i j} ε w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} δ x_{k} + O (ε^{2}) . \end{matrix}

(138)

Summarizing, we have got

\bar{δ u_{i} (x)} = ε (\sum_{j = 1}^{N} (ψ_{i j} w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} X_{j}^{K} w^{j})) + O (ε^{2}) .

Define now

\tilde{A} (u, φ_{1}, φ_{2}, ε) = \int_{Ω} L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] det J (x) d x .

where we have generically denoted

L [u] \equiv L (u, \nabla u),

L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] \equiv L (u (x + ε φ_{2} (x)) + ε φ_{1} (x), \nabla u (x + ε φ_{2} (x)) + ε \nabla φ_{1} (x)),

and

\begin{matrix} J (x) & = & \{\frac{\partial x_{j}^{'}}{\partial x_{k}}\} \\ = & \{\frac{\partial (x_{j} + ε {(φ_{2})}_{j} (x))}{\partial x_{k}}\} \\ = & \{δ_{j k} + ε \frac{\partial {(φ_{2})}_{j} (x)}{\partial x_{k}}\} . \end{matrix}

(139)

From such a last definition we have

det J (x) = 1 + ε \sum_{k = 1}^{n} \frac{\partial {(φ_{2})}_{k} (x)}{\partial x_{k}} + O (ε^{2}) .

so that

\frac{\partial det J (x)}{\partial ε} |_{ε = 0} = \sum_{k = 1}^{n} \frac{\partial {(φ_{2})}_{k} (x)}{\partial x_{k}},

At this point we define

δ A (u, φ_{1}, φ_{2}) = \frac{d}{d ε} (\tilde{A} (u, φ_{1}, φ_{2}, ε)) |_{ε = 0},

so that

\begin{matrix} δ A (u, φ_{1}, φ_{2}) & = & \int_{Ω} (\sum_{i = 1}^{N} (\frac{\partial L (u, \nabla u)}{\partial u_{i}} {(φ_{1})}_{i} \\ + \sum_{k = 1}^{n} (\frac{\partial L (u, \nabla u)}{\partial {(u_{i})}_{x_{k}}} {({(φ_{1})}_{i})}_{x_{k}}) \\ + \sum_{k = 1}^{n} \frac{δ L [u]}{δ u_{i}} \frac{\partial u_{i}}{\partial x_{k}} {(φ_{2})}_{k}) + \sum_{k = 1}^{n} L [u] \frac{\partial {(φ_{2})}_{k}}{\partial x_{k}}) d x . \end{matrix}

(140)

From this and

\frac{\partial L (u, \nabla u)}{\partial u_{i}} - \frac{d}{d x_{k}} (\frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}}) = 0, in Ω, \forall i \in {1, \dots, N},

we obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) & = & \sum_{i = 1}^{N} \sum_{k = 1}^{n} (\int_{Ω} \frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} {(φ_{1})}_{k})) d x \\ + \sum_{k = 1}^{n} \int_{Ω} \frac{d (L [u] {(φ_{2})}_{k})}{d x_{k}} d x . \end{matrix}

(141)

In particular, for

{(φ_{2})}_{k} = \sum_{j = 1}^{N} X_{j}^{k} w^{j}

and

{(φ_{1})}_{i} = \sum_{j = 1}^{N} (ψ_{i j} w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i}}{\partial x_{k}} X_{j}^{k} w^{j}),

we obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) \\ = & \sum_{i = 1}^{N} \sum_{k = 1}^{n} \int_{Ω} (\frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{l}} X_{j}^{l} w^{j})))) d x \\ + \sum_{k = 1}^{n} \int_{Ω} \frac{\partial L [u] X_{j}^{k} w^{j}}{d x_{k}} d x \\ = & \sum_{j = 1}^{N} (\sum_{k = 1}^{n} (\int_{Ω} \frac{d}{d x_{k}} (\sum_{i = 1}^{N} \frac{\partial L [u]}{\partial {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{l}} X_{j}^{l} w^{j})) \\ + L [u] X_{j}^{k} w^{j}) d x)) . \end{matrix}

(142)

Moreover, we define

θ_{k}^{j} = \sum_{i = 1}^{N} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (- ψ_{i j} + \sum_{l = 1}^{n} \frac{\partial u_{i}}{\partial x_{l}} X_{j}^{l})) - L (u) X_{j}^{k}

so that

δ A (u, φ_{1}, φ_{2}) = - \int_{Ω} \sum_{j = 1}^{N} \sum_{k = 1}^{n} \frac{d (θ_{j}^{k} w^{j})}{d x_{k}} d x,

\forall {w^{j}} \in C_{c}^{\infty} (Ω; R^{N}) .

In particular, for

ψ_{i j} = 0

and

X_{j}^{k} = δ_{j}^{k}

we obtain the Energy-Momentum tensor

T_{k}^{j}

, where

T_{k}^{j} \equiv θ_{k}^{j} = \sum_{i = 1}^{N} \sum_{l = 1}^{n} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} \frac{\partial u_{i}}{\partial x_{l}} δ_{j}^{l}) - L [u] δ_{j}^{k} .

25.1. The Angular-Momentum tensor

In this subsection we define the following change of variables

x_{k}^{'} = x_{k} + \sum_{m \neq k} g^{m m} x_{m} ε w^{k m},

where

w^{k m} = - w^{m k} .

With such relations in mind, we set

\begin{matrix} δ x_{k} & = & x_{k}^{'} - x_{k} \\ = & ε \sum_{l = 1}^{n} \sum_{m < l} w^{m l} (g^{l l} x_{l} g_{m}^{k} - g^{m m} x_{m} g_{l}^{k}) . \end{matrix}

(143)

We define also,

u_{i}^{'} (x^{'}) = u_{i} (x) + δ u_{i} (x)

where

δ u_{i} (x) = \sum_{l = 1}^{n} \sum_{j, p < l} A_{i (p l)}^{j} u_{j} (x) ε w^{p l} .

Moreover, we define

ψ_{i (m n)} = \sum_{j = 1}^{n} A_{i (m n)}^{j},

where

A_{i (p l)}^{j} = g_{i p} δ_{l}^{j} - g_{i}^{l} δ_{p}^{j} .

Hence,

ψ_{i (m n)} = \sum_{j = 1}^{n} A_{i (m n)}^{j} u_{j} (x) = g_{i n} u_{m} (x) - g_{j m} u_{n} (x) .

For the general variation, we define again

\tilde{A} (u, φ_{1}, φ_{2}, ε) = \int_{Ω} L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] det J (x) d x .

where we have generically denoted

L [u] \equiv L (u, \nabla u),

L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] \equiv L (u (x + ε φ_{2} (x)) + ε φ_{1} (x), \nabla u (x + ε φ_{2} (x)) + ε \nabla φ_{1} (x)),

\begin{matrix} J (x) & = & \{\frac{\partial x_{j}^{'}}{\partial x_{k}}\} \\ = & \{\frac{\partial (x_{j} + ε {(φ_{2})}_{j} (x))}{\partial x_{k}}\} \\ = & \{δ_{j k} + ε \frac{\partial {(φ_{2})}_{j} (x)}{\partial x_{k}}\} . \end{matrix}

(144)

and

δ A (u, φ_{1}, φ_{2}) = \frac{d}{d ε} (\tilde{A} (u, φ_{1}, φ_{2}, ε)) |_{ε = 0},

Moreover, we set

{(φ_{2})}_{k}^{m l} = w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}),

and

\bar{δ u_{i}} = u_{i}^{'} (x) - u_{i} (x) .

Thus,

\begin{matrix} δ u_{i} (x) & = & u_{i}^{'} (x) - u_{i} (x) \\ = & u_{i}^{'} (x^{'}) - u_{i}^{'} (x) + u_{i}^{'} (x) - u_{i} (x), \end{matrix}

(145)

so that

\begin{matrix} \bar{δ u_{i} (x)} = u_{i}^{'} (x) - u_{i} (x) \\ = & δ u_{i} (x) - (u_{i}^{'} (x^{'}) - u_{i}^{'} (x)) \\ δ u_{i} (x) - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} δ x_{k} + O (ε^{2}) \\ = & δ u_{i} (x) - \sum_{l = 1}^{n} \sum_{m < l} \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{k}} ε w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}) + O (ε^{2}) \\ = & ε (\sum_{l = 1}^{n} \sum_{j, k < l} A_{i (k l)}^{j} u_{j} (x) w^{k l} - \sum_{l = 1}^{n} \sum_{m < l} \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{k}} w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k})) + O (ε^{2}), \end{matrix}

With such results in mind, we define

\begin{matrix} {(φ_{1})}_{i}^{m l} & = & \sum_{j, k < l} A_{i (k l)}^{j} u_{j} (x) w^{m l} \\ - \sum_{k = 1}^{n} (\frac{\partial u_{i} (x)}{\partial x_{k}} w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k})) . \end{matrix}

(146)

Similarly as in the previous section, we may obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) \\ = & \frac{d \tilde{A} (u, φ_{1}, φ_{2}, ε)}{d ε} |_{ε = 0} \\ = & \sum_{l = 1}^{n} \sum_{j, m < l} \sum_{k = 1}^{n} \sum_{i = 1}^{N} \int_{Ω} \frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (A_{i {(l, m)}^{j}} u_{j} (x) + \frac{\partial u_{i}}{\partial x_{p}} g^{m m} x_{m} δ_{l}^{p} - \frac{\partial u_{i}}{\partial x_{p}} g^{l l} x_{l} δ_{m}^{p}) w^{m l}) d x \\ + \sum_{k = 1}^{n} \sum_{l = 1}^{n} \sum_{j, m < l} \sum_{i = 1}^{N} \int_{Ω} \frac{d}{d x_{k}} (L [u] (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}) w^{m l}) d x \end{matrix}

(147)

Thus,

δ A (u, φ_{1}, φ_{2}) = - \sum_{k = 1}^{n} \sum_{m < l} \int_{Ω} \frac{d}{d x_{k}} (M_{m l}^{k} w^{m l}) d x,

where

\begin{matrix} M_{m l}^{k} & = & \sum_{i = 1}^{N} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (A_{i l m}^{j} u_{j} - \frac{\partial u_{i}}{\partial x_{l}} g^{m m} x_{m} + \frac{\partial u_{i}}{\partial x_{m}} g^{l l} x_{l}) \\ + L [u] (g^{l l} x_{l} δ_{m}^{k} + g^{m m} x_{m} δ_{l}^{k}), \end{matrix}

(148)

so that

\begin{matrix} M_{l m}^{k} & = & (g^{m m} x_{m} T_{l}^{k} - g^{l l} x_{l} T_{m}^{k}) \\ - \sum_{i = 1}^{n} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} A_{i (l m)}^{j} u_{j} (x) \\ = & L_{m l}^{k} + S_{m l}^{k}, \end{matrix}

(149)

where

L_{m l}^{k} = (g^{m m} x_{m} T_{l}^{k} - g^{l l} x_{l} T_{m}^{k})

and

S_{m l}^{k} = - \sum_{i = 1}^{N} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} A_{i (l m)}^{j} u_{j} (x) .

The tensor

{L_{m l}^{k}}

is said to be the Orbital angular momentum tensor and

{S_{m l}^{k}}

is said to be Spin one.

25.2. A note on the solution of the Klein-Gordon equation

For

Ω = R^{4}

Ω_{1} = R^{3}

and denoting as usual by

i \in C

the imaginary unit, consider the Klein-Gordon equation in distributional sense

- \frac{\partial^{2} u}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} u}{\partial x_{j}^{2}} - m^{2} u = 0, in Ω,

where

u \in V = W^{1, 2} (Ω) .

Defining the Fourier transform of u, by

ϕ (p) = \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} e^{- i p \cdot x} u (x) d x,

in the momenta space, the last equation is equivalent to

(p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) = 0, in Ω,

where we have denoted

p = (p_{0}, p_{1}, p_{2}, p_{3}) \in R^{4},

and

x = (x_{0}, x_{1}, x_{2}, x_{3}) \in R 4

Observe that a general solution for this last equation is given by the wave function

\hat{ϕ} (p) = δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p),

where

ϕ \in W^{1, 2} (Ω) .

Indeed,

\begin{matrix} (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) \hat{ϕ} (p) & = & (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) \\ = & 0, in Ω . \end{matrix}

(150)

Here, we recall that generically for the Dirac delta function

δ (t),

we have

δ (t) = \{\begin{matrix} 0, & if t \neq 0, \\ + \infty, & if t = 0 . \end{matrix}

(151)

Observe that, for the scalar case in the previous section, we have

2 T^{00} = \sum_{j = 0}^{3} {(\frac{\partial u}{\partial x_{j}})}^{2} + m^{2} u .

Also, from

- \frac{\partial^{2} u}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} u}{\partial x_{j}^{2}} - m^{2} u = 0, in Ω,

we get

\int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x - \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x - m^{2} \int_{Ω} u^{2} d x = 0,

so that

\int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x = \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x + m^{2} \int_{Ω} u^{2} d x .

From such results, we may infer that

\begin{matrix} \int_{Ω} T^{00} d x & = & \int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x \\ = & {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2} \\ = & \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x + m^{2} \int_{Ω} u^{2} d x . \end{matrix}

(152)

On the other hand,

\begin{matrix} \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x \\ = & \frac{1}{{(2 π)}^{3}} \sum_{j = 1}^{3} \int_{Ω} (\int_{Ω} i p_{j} \hat{ϕ} (p) e^{i p \cdot x} d p) (\int_{Ω} i p_{j}^{'} \hat{ϕ} (p^{'}) e^{i p^{'} \cdot x} d p^{'}) d x \\ = & \frac{1}{{(2 π)}^{3}} \sum_{j = 1}^{3} \int_{Ω} \int_{Ω} (- p_{j} p_{j}^{'} \hat{ϕ} (p) \hat{ϕ} (p^{'}) \int_{Ω} e^{i (p + p^{'}) \cdot x} d x) d p d p^{'} \\ = & \frac{1}{{(2 π)}^{3 / 2}} \sum_{j = 1}^{3} \int_{Ω} \int_{Ω} (- p_{j} p_{j}^{'} \hat{ϕ} (p) \hat{ϕ} (p^{'}) δ (p + p^{'})) d p d p^{'} \\ = & \frac{1}{{(2 π)}^{3 / 2}} \sum_{j = 1}^{3} \int_{Ω} (p_{j}^{2} \hat{ϕ} (p) \hat{ϕ} (- p)) d p . \end{matrix}

(153)

Thus, denoting

\hat{p} = (p_{1}, p_{2}, p_{3}),

d \hat{p} = d p_{1} d p_{2} d p_{3},

and

p_{0} (\hat{p}) = \sqrt{\sum_{j = 1}^{3} p_{j}^{2} + m^{2}},

we may infer that

\begin{matrix} \int_{Ω} T^{00} d x & = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\sum_{j = 1}^{3} p_{j}^{2} + m^{2}) \hat{ϕ} (p) \hat{ϕ} (- p)) d p \\ = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\sum_{j = 1}^{3} p_{j}^{2} + m^{2}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) ϕ (- p)) d p \\ = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (p_{0} {(\hat{p})}^{2} ϕ (p_{0} (\hat{p}), \hat{p}) ϕ (- p_{0} (\hat{p}), - \hat{p})) d \hat{p} . \end{matrix}

(154)

Summarizing we have got

\begin{matrix} \int_{Ω} T^{00} d x & = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (p_{0} {(\hat{p})}^{2} ϕ (p_{0} (\hat{p}), \hat{p}) ϕ (- p_{0} (\hat{p}), - \hat{p})) d \hat{p} \\ = & {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2}, \end{matrix}

(155)

so that

\int_{Ω} T^{00} d x = {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2}

may be expressed as a kind of average expectance of

p_{0}^{2}

related to the function

ϕ (p) .

25.3. A note on the Dirac equation

In this subsection we denote

Δ^{2} = \sum_{j = 0}^{3} g^{j j} L_{j} L_{j},

where

L_{j} = i g^{j j} \frac{\partial}{\partial x_{j}}, \forall j \in {0, 1, 2, 3} .

We recall that the relativistic Klein-Gordon equation may be written as

(Δ^{2} - m^{2}) u = 0, in Ω = R^{4} .

Moreover, for

4 \times 4

matrices

γ^{k}

indicated in the subsequent lines, we may obtain

{D_{i j}} u = [- i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) - m] [- i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) + m] u,

where

D_{i i} = Δ^{2} - m^{2}

and

D_{i j} = 0, if i \neq j, \forall i, j \in {0, 1, 2, 3} .

Here

u = {(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in V = W^{1, 2} (Ω; C^{4}) .

In such a case the fundamental Dirac equation stands for

[i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) - m] u = 0 \in R^{4}, in Ω .

Summarizing, if

{(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in V

is a solution of this last Dirac equation, then

u_{0}, u_{1}, u_{2}, u_{3}

are four solutions of the Klein-Gordon equation.

In the momentum configuration space, through the Fourier transform proprieties, the Dirac equation stands for

(\hat{p} + m) \hat{u} (p) = 0, in R^{4},

where

\hat{p} = \sum_{j = 0}^{3} g^{j j} p_{j} γ^{j} .

Observe that

\tilde{u} (p) = δ (\hat{p} + m) u (p)

corresponds to a general solution of the Dirac equation.

Indeed,

(\hat{p} + m) \tilde{u} (p) = (\hat{p} + m) δ (\hat{p} + m) u (p) = 0 \in R^{4}, in Ω .

On the other hand

\hat{u} (p) = δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) u (p)

correspond to four solutions of the Klein-Gordon equation.

At this point, we assume such a

\hat{u} (p)

corresponds to a solution of the Dirac equation as well.

Furthermore, here we recall that (please see the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov for details):

γ^{0} = \{\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}\},

(156)

γ^{1} = \{\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{matrix}\},

(157)

γ^{2} = \{\begin{matrix} 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ - i & 0 & 0 & 0 \end{matrix}\},

(158)

γ^{3} = \{\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \\ - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}\}

(159)

and

γ^{5} = \{\begin{matrix} 0 & 0 & - i & 0 \\ 0 & 0 & 0 & - i \\ - i & 0 & 0 & 0 \\ 0 & - i & 0 & 0 \end{matrix}\}

(160)

where we also denote

α_{j} = γ^{0} γ^{j}, \forall j \in {1, 2, 3},

σ_{j} = i γ^{5} γ^{0} γ^{j}, \forall j \in {1, 2, 3},

and

β = γ^{0} .

On the other hand, a variational formulation for the Dirac equation corresponds to the functional

A : V \to R

where

A (u) = \frac{1}{2} \int_{Ω} L (u, \nabla u) d x,

where

L (u, \nabla u) = i \sum_{j = 0}^{3} (u^{*} γ^{j} \frac{\partial u}{\partial x_{j}} - \frac{\partial u^{*}}{\partial x_{j}} γ^{j} u) - m^{2} u^{*} u,

where here

u = {(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in W^{1, 2} (Ω; C^{4}) .

From such statements and definitions, similarly as in the previous sections (please see [20] for details), we may obtain

T^{k l} = \frac{i}{2} g^{l l} (u^{*} γ^{k} \frac{\partial u}{\partial x_{l}} - \frac{\partial u^{*}}{\partial x_{l}} γ^{k} u),

and

S^{k, l m} = - (\frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}} A^{u, l m} u - u^{*} A^{u^{*}, l m} \frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}}),

where

A^{u, l m} = \frac{i}{2} σ^{m l},

A^{u^{*}, l m} = \frac{i}{2} σ^{l m},

and where

σ^{l m} = \frac{γ^{l} γ^{k} - γ^{k} γ^{l}}{2},

so that

S^{k, l m} = \frac{1}{4} u^{*} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u .

Thus,

\begin{matrix} \int_{Ω} S^{k, l m} d x \\ = & \frac{1}{4} \int_{Ω} (u^{*} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u) d x \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3}} \int_{Ω} (\int_{Ω} \int_{Ω} (\hat{u} (p) e^{i p \cdot x} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) \hat{u} (p^{'}) e^{i p^{'} \cdot x}) d p d p^{'}) d x \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} \int_{Ω} (\hat{u} (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) δ (p + p^{'}) \hat{u} (p^{'})) d p d p^{'} \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\hat{u} (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) \hat{u} (- p)) d p \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (u (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) u (- p)) d p \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (u (p_{0} (\hat{p}), \hat{p}) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u (- p_{0} (\hat{p}), - \hat{p})) d \hat{p}, \end{matrix}

(161)

where

p_{0} (\hat{p}) = \sqrt{\sum_{j = 1}^{3} p_{j}^{2} + m^{2}} .

Summarizing, we have got

\int_{Ω} S^{k, l m} d x = \frac{1}{4 {(2 π)}^{3 / 2}} \int_{Ω_{1}} (u (p_{0} (\hat{p}), \hat{p}) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u (- p_{0} (\hat{p}), - \hat{p})) d \hat{p},

where

Ω_{1} = R^{3}

\hat{p} = (p_{1}, p_{2}, p_{3})

and

d \hat{p} = d p_{1} d p_{2} d p_{3} .

26. A note on quantum field operators

This section is strongly based on the chapter 3, page 53 of the book [21], by G.B. Folland.

Therefore, here we have done a kind of review of these pages of such a book chapter indicated. In fact, we have simply opened more and clarified some calculations, in order to improve their understanding.

Let

Ω = \hat{Ω} \times [0, T] \subset R^{4}

where

\hat{Ω} \subset R^{3}

is a open, bounded and connected set with a regular boundary denoted

\partial \hat{Ω} .

Define

V = W^{1, 2} (Ω)

and

V_{0} = W_{0}^{1, 2} (Ω) .

Consider an operator

H : V_{1} = V_{0} \cap W^{2, 2} (Ω) \to Y

where in a distributional sense,

H (u) = - \frac{\partial^{2} u}{\partial t^{2}} + \nabla^{2} u - m^{2} u,

and where

Y = Y^{*} = L^{2} (Ω) .

Suppose there exists operators

B_{1} : Y \to Y

and

B_{2} : Y \to Y

such that

B_{1} B_{2} (u) = H (u) + \frac{1}{2} u

and

B_{2} B_{1} (u) = H (u) - \frac{1}{2} u, \forall u \in V_{1} .

Assume also

ϕ_{0} \in V_{1}

is such that

∥ ϕ_{0} ∥_{L^{2}} = 1,

and

B_{1} ϕ_{0} = 0 .

Now define

ϕ_{k} = \frac{B_{2}^{k} (ϕ_{0})}{\sqrt{k!}}, \forall k \in N .

Observe that

[B_{1} B_{2}] = B_{1} B_{2} - B_{2} B_{1} = I_{d} .

We shall prove by induction that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

(162)

Indeed, for

k = 1

[B_{1}, B_{2}] = I_{d} = 1 B_{2}^{0},

so that (162) holds for

k = 1

Suppose now (162) holds for

k \in N,

so that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1} .

In order to complete the induction, it suffices to prove that (162) holds for

k + 1 .

Observe that

\begin{matrix} [B_{1}, B_{2}^{k + 1}] & = & (B_{1} B_{2}^{k + 1} - B_{2}^{k + 1} B_{1}) \\ = & (B_{1} B_{2}^{k}) B_{2} - B_{2}^{k + 1} B_{1} \\ = & (B_{2}^{k} B_{1} + k B_{2}^{k - 1}) B_{2} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k} (B_{1} B_{2}) + k B_{2}^{k} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k} (B_{2} B_{1} + I_{d}) + k B_{2} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k + 1} B_{1} + B_{2}^{k} + k B_{2}^{k} - B_{2}^{k + 1} B_{1} \\ = & (k + 1) B_{2}^{k} . \end{matrix}

(163)

Thus, the induction is complete, so that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

Moreover, we recall that

B_{1} ϕ_{0} = 0,

so that

\begin{matrix} B_{1} ϕ_{k} & = & B_{1} (\frac{B_{2}^{k} ϕ_{0}}{\sqrt{k!}}) \\ = & \frac{(B_{2}^{k} B_{1} + k B_{2}^{k - 1}) ϕ_{0}}{\sqrt{k!}} \\ = & \frac{k ϕ_{k - 1} \sqrt{(k - 1)!}}{\sqrt{k!}} \\ = & \frac{k ϕ_{k - 1}}{\sqrt{k}} \\ = & \sqrt{k} ϕ_{k - 1}, \forall k \in N . \end{matrix}

(164)

Summarizing, we have got

B_{1} ϕ_{k} = \sqrt{k} ϕ_{k - 1}, \forall k \in N .

Now, we shall prove that

B_{2} ϕ_{k} = \sqrt{k + 1} ϕ_{k + 1}, \forall k \in N .

Observe that

\begin{matrix} B_{2}^{k + 1} ϕ_{0} & = & ϕ_{k + 1} (\sqrt{(k + 1)!} \\ = & B_{2} (B_{2}^{k} ϕ_{0}) \\ = & (B_{2} ϕ_{k}) \sqrt{k!} . \end{matrix}

(165)

Summarizing, we have got

(B_{2} ϕ_{k}) \sqrt{k!} = ϕ_{k + 1} (\sqrt{(k + 1)!},

so that

(B_{2} ϕ_{k}) = \sqrt{k + 1} ϕ_{k + 1}, \forall k \in N .

Finally, from such results, we may infer that

\begin{matrix} B_{1} B_{2} ϕ_{k} & = & B_{1} (\sqrt{k + 1} ϕ_{k + 1}) \\ = & \sqrt{k + 1} B_{1} ϕ_{k + 1} \\ = & \sqrt{k + 1} \sqrt{k + 1} ϕ_{k} \\ = & (k + 1) ϕ_{k}, \forall k \in N . \end{matrix}

(166)

Similarly,

\begin{matrix} B_{2} B_{1} ϕ_{k} & = & B_{2} (\sqrt{k} ϕ_{k - 1}) \\ = & \sqrt{k} B_{2} ϕ_{k - 1} \\ = & \sqrt{k} \sqrt{k} ϕ_{k} \\ = & k ϕ_{k} . \end{matrix}

(167)

Therefore we have got

H ϕ_{k} = B_{1} B_{2} ϕ_{k} - \frac{1}{2} ϕ_{k} = (k + 1) ϕ_{k} - \frac{1}{2} ϕ_{k} = (k + \frac{1}{2}) ϕ_{k},

that is

H ϕ_{k} = (k + \frac{1}{2}) ϕ_{k}, \forall k \in N .

Thus, for each

k \in N

k + \frac{1}{2}

is an eigenvalue of H with corresponding eigenvector

ϕ_{k}

26.1. An application concerning the harmonic oscillator operator in quantum mechanics

In this section we have the aim of representing the relativistic Klein-Gordon equation through the creation and annihilation operations related to the harmonic oscillator in quantum mechanics.

Consider first the one-dimensional Hamiltonian, corresponding to the harmonic oscillator, namely

H = - \frac{ℏ}{2 m} \frac{d^{2}}{d x^{2}} + K \frac{x^{2}}{2},

which through an appropriate re-scale results into the following related Hamiltonian

H_{0}

, where

H_{0} = \frac{1}{2} (- \frac{d^{2}}{d x^{2}} + x^{2}) .

Define now the operators

B_{1} = A = \frac{1}{\sqrt{2}} (x + \frac{d}{d x}),

and

B_{2} = A^{*} = \frac{1}{\sqrt{2}} (x - \frac{d}{d x}) .

Clearly,

H_{0} = B_{1} B_{2} - \frac{I_{d}}{2} = B_{2} B_{1} + \frac{I_{d}}{2},

so that

[A, A^{*}] = [B_{1}, B_{2}] = B_{1} B_{2} - B_{2} B_{1} = I_{d} .

Similarly, as in the previous sections, by induction, we may obtain

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

For

ϕ_{0} = π^{- 1 / 4} e^{- \frac{x^{2}}{2}},

we define

ϕ_{k} = \frac{1}{\sqrt{k}} B_{2}^{k} ϕ_{0}, \forall k \in N .

Also from the previous section, we may obtain

B_{2} ϕ_{k} = A^{*} ϕ_{k} = \sqrt{k + 1} ϕ_{k + 1},

B_{1} ϕ_{k} = A ϕ_{k} = \sqrt{k} ϕ_{k - 1}, \forall k \in N .

B_{2} B_{1} = A^{*} A ϕ_{k} = k ϕ_{k},

and

B_{1} B_{2} ϕ_{k} = A A^{*} ϕ_{k} = (k + 1) ϕ_{k}, \forall k \in N \cup {0} .

so that

H_{0} ϕ_{k} = (k + 1 / 2) ϕ_{k}, \forall k \in N .

Here we recall that

B_{1} ϕ_{0} = A ϕ_{0} = 0,

and

∥ ϕ_{0} ∥_{L^{2}} = 1 .

In reference [21], page 54 it is proven that such a sequence

{ϕ_{k}}

is an ortho-normal basis for

L^{2} (R) .

Finally, observe that for

R^{4}

we may define

{(B_{1})}_{j} = A_{j} = \frac{1}{\sqrt{2}} (\frac{\partial}{\partial x_{j}} + x_{j}),

and

{(B_{2})}_{j} = A_{j}^{*} = \frac{1}{\sqrt{2}} (- \frac{\partial}{\partial x_{j}} + x_{j}), \forall j \in {0, 1, 2, 3} .

Here generically,

x = (x_{0}, x_{1}, x_{2}, x_{3}) \in R^{4} .

Observe that clearly

\frac{\partial}{\partial x_{j}} = \frac{\sqrt{2}}{2} (A_{j} - A_{j}^{*}),

and

x_{j} I_{d} = \frac{\sqrt{2}}{2} (A_{j} + A_{j}^{*}), \forall j \in {0, 1, 2, 3} .

Denoting

x_{0} = t

where t stands for time, consider the relativistic Klein-Gordon equation,

- \frac{\partial^{2} ϕ}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} ϕ}{\partial x_{j}^{2}} - m^{2} ϕ = 0 .

From the previous results, we may represent such an equation by

(- \frac{1}{2} {(A_{0} - A_{0}^{*})}^{2} + \sum_{j = 1}^{3} \frac{1}{2} {(A_{j} - A_{j}^{*})}^{2} - m^{2} I_{d}) ϕ = 0 .

We highlight from the previous results we know the action of

A_{j}

and

A_{j}^{*}

on an appropriate basis of

L^{2} (R^{4})

obtained though an appropriate tensorial product of the bases

{{ϕ_{k} (x_{j})}, for j \in {0, 1, 2, 3}} .

We shall call the operators

A_{j}^{*}

and

A_{j}

as the creation and annihilation operators concerning the original harmonic operator in quantum mechanics.

To justify such a nomenclature, we recall that

A_{j}^{*} ϕ_{0} (x_{j}) = ϕ_{1} (x_{j})

and

A_{j} ϕ_{0} (x_{j}) = 0, \forall j \in {0, 1, 2, 3} .

27. A dual variational formulation for a related model

In this section we develop a concave dual variational formulation for a Ginzburg-Landau type equation.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

defined by

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(168)

where

γ > 0,

α > 0

β > 0

f \in L^{2} (Ω),

and

V = W_{0}^{1, 2} (Ω) .

We also denote

Y = Y^{*} = L^{2} (Ω) .

Define now

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

for some appropriate

K_{3} > 0

and,

J_{1} : V \times Y \to R

J_{1} (u, v_{0}^{*}) = J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x,

where

K_{1} = \frac{1}{4 α K_{3}^{2} + ε}

for some small parameter

0 < ε ≪ 1 .

Observe that

\begin{matrix} J (u, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ \geq & inf_{u \in V_{1}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ + inf_{v \in Y} \{- {〈 v, v_{0}^{*} 〉}_{L^{2}} + \frac{α}{2} \int_{Ω} {(v - β)}^{2} d x\} \\ = & - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}) \\ \equiv & J^{*} (v_{0}^{*}), \forall u \in V_{1}, v_{0}^{*} \in Y^{*}, \end{matrix}

(169)

where we have denoted

F^{*} (v_{0}^{*}) = sup_{u \in V_{1}} {- {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - F (u, v_{0}^{*})},

F (u, v_{0}^{*}) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}},

and

G (v) = \frac{α}{2} \int_{Ω} {(v - β)}^{2} d x,

\begin{matrix} G^{*} (v_{0}^{*}) & = & sup_{v \in Y} {{〈 v, v_{0}^{*} 〉}_{L^{2}} - G (v)} \\ = & \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(170)

Observe that

\frac{\partial F (u, v_{0}^{*})}{\partial u^{2}} = - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- \nabla^{2} + 2 v_{0}^{*})}^{2},

so that we define

B^{*} = {v_{0}^{*} \in Y^{*} : - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- \nabla^{2} + 2 v_{0}^{*})}^{2} > 0} .

With such assumptions and definitions in mind, we may prove the following theorem:

Theorem 27.1.

For

J^{*} (v_{0}^{*}) = - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}),

suppose

{\hat{v}}_{0}^{*} \in B^{*}

is such that

δ J^{*} ({\hat{v}}_{0}^{*}) = 0 .

Let

u_{0} \in Y

be such that

\frac{\partial H (u_{0}, {\hat{v}}_{0}^{*})}{\partial u} = 0,

where

H (u, v_{0}^{*}) = F (u, v_{0}^{*}) + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} .

Suppose

u_{0} \in V_{1} .

Under such hypotheses,

F^{*} ({\hat{v}}_{0}^{*}) = H (u_{0}, {\hat{v}}_{0}^{*}),

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) \\ = & sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) \\ = & J^{*} ({\hat{v}}_{0}^{*}) . \end{matrix}

(171)

Proof.

The proof that

F^{*} ({\hat{v}}_{0}^{*}) = H (u_{0}, {\hat{v}}_{0}^{*}),

is immediate from

{\hat{v}}_{0} \in B^{*} .

Moreover, the proof that

δ J (u_{0}) = 0,

and

J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = J^{*} ({\hat{v}}_{0}^{*})

may be done similarly as in the previous sections.

Observe that

J^{*} (v_{0}^{*}) = - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}) = inf_{u \in V_{1}} {H (u, v_{0}^{*}) - G^{*} (v_{0}^{*})},

so that

J^{*}

is concave in

v_{0}^{*}

as the infimum of a family of concave functionals in

v_{0}^{*}

From this and

δ J^{*} ({\hat{v}}_{0}^{*}) = 0

we get

J^{*} ({\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) .

Furthermore observe that

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ \leq & J_{1} (u, v_{0}^{*}) \\ = & F (u, {\hat{v}}_{0}^{*}) + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - G^{*} ({\hat{v}}_{0}^{*}) \\ \leq & F (u, {\hat{v}}_{0}^{*}) + sup_{v_{0}^{*} \in Y^{*}} \{{〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - G^{*} ({\hat{v}}_{0}^{*})\} \\ = & F (u, {\hat{v}}_{0}^{*}) + G (u^{2}) \\ = & J_{1} (u, {\hat{v}}_{0}^{*}), \forall u \in V_{1} . \end{matrix}

(172)

Hence

J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) .

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) \\ = & sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) \\ = & J^{*} ({\hat{v}}_{0}^{*}) . \end{matrix}

(173)

The proof is complete. □

28. The generalized method of lines applied to fourth order differential equations

In this sections we develop an application of the generalized method of lines to a fourth order equation.

We start by addressing the following ordinary differential equation (ode):

ε \frac{d^{4} u (x)}{d x^{4}} - f = 0, in [0, 1],

with the boundary conditions

u (0) = u^{'} (0) = 0

and

u (1) = u^{'} (1) = 0 .

In terms of linear elasticity, such a boundary conditions corresponds to a bi-clamped beam.

In a finite difference context, this last equation corresponds to

ε (\frac{u_{n + 2} - 4 u_{n + 1} + 6 u_{n} - 4 u_{n - 1} + u_{n - 2}}{d^{4}}) - f_{n} = 0, \forall n \in {1, \dots N - 2},

where N is the number of nodes and

d = 1 / N

Considering that, from the boundary conditions,

u_{- 1} = u_{0} = 0,

for

n = 1

we get

6 u_{1} - 4 u_{2} + u_{3} = \frac{f_{1} d^{4}}{ε},

so that

u_{1} = a_{1} u_{2} + b_{1} u_{3} + c_{1},

where

a_{1} = 2 / 3, b_{1} - 1 / 6 and c_{1} = \frac{f_{1} d^{4}}{6 ε} .

Similarly, for

n = 2

, we obtain

- 4 u_{1} + 6 u_{2} - 4 u_{3} + u_{4} = \frac{f_{2} d^{4}}{ε} .

Hence, replacing the value of

u_{1}

previously obtained in this last equation, we have

- 4 (a_{1} u_{2} + b_{1} u_{3} + c_{1}) + 6 u_{2} - 4 u_{3} + u_{4} = \frac{f_{2} d^{4}}{ε},

so that

u_{2} = a_{2} u_{3} + b_{2} u_{4} + c_{2},

where defining

m_{12} = (6 - 4 a_{1})

, we have also

a_{2} = \frac{4 b_{1} + 4}{m_{12}},

b_{2} = - \frac{1}{m_{12}},

c_{2} = \frac{1}{m_{12}} (\frac{f_{2} d^{4}}{ε} + 4 c_{1}) .

Now reasoning inductively, for n, having

u_{n - 1} = a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1},

and

u_{n - 2} = a_{n - 2} u_{n - 1} + b_{n - 2} u_{n} + c_{n - 2}

we obtain

u_{n - 2} = a_{n - 2} (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + b_{n - 2} u_{n} + c_{n - 2},

so that from this and

u_{n + 2} - 4 u_{n + 1} + 6 u_{n} - 4 u_{n - 1} + u_{n - 2} = \frac{f_{n} d^{4}}{ε},

we obtain

\begin{matrix} a_{n - 2} (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + b_{n - 2} u_{n} + c_{n - 2} \\ - 4 (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + 6 u_{n} - 4 u_{n + 1} + u_{n + 2} = \frac{f_{n} d^{4}}{ε}, \end{matrix}

(174)

so that

u_{n} = a_{n} u_{n + 1} + b_{n} u_{n + 1} + c_{n}

where defining

m_{12} = (a_{n - 2} (a_{n - 1}) + b_{n - 2} - 4 a_{n - 1} + 6)

we obtain

a_{n} = - \frac{1}{m_{12}} (a_{n - 2} b_{n - 1} - 4 b_{n - 1} - 4)

b_{n} = - \frac{1}{m_{12}},

and

c_{n} = \frac{1}{m_{12}} (a_{n - 2} c_{n - 1} + c_{n - 2} - 4 c_{n - 1} - \frac{f_{n} d^{4}}{ε}) .

Summarizing, we have got

u_{n} = a_{n} u_{n + 1} + b_{n} u_{n + 2} + c_{n}, \forall n \in {1, \cdot N - 2} .

Observe now that from the boundary conditions,

u_{N - 1} = u_{N} = 0 .

From these last two equations, we may obtain

u_{N - 2} = c_{N_{2}},

and

u_{N - 3} = a_{N - 3} u_{N - 2} + b_{N - 3} u_{N - 1} + c_{N - 3},

and so on up to obtaining

u_{1} = a_{1} u_{2} + b_{1} u_{3} + c_{1} .

The problem is then solved.

28.1. A numerical example

We develop a numerical example considering

ε = 1,

and

f \equiv 1, in [0, 1] .

Thus, we have solved the equation

ε \frac{d^{4} u (x)}{d x^{4}} - f = 0, in [0, 1],

with the boundary conditions

u (0) = u^{'} (0) = 0

and

u (1) = u^{'} (1) = 0 .

In a finite differences context, we have used

N = 100

nodes and

d = 1 / N .

For a solution

u (x)

, please see Figure 19.

In the next lines, we present the concerning software in MAT-LAB

**************

1.: clear all

m8=100;

d=1/m8;

e1=1.0;

for i=1:m8

f(i,1)=1.0;

end;

a(1)=2/3;

b(1)=-1/6;

c(1)=f(1,1)* $d^{4}$ /(6e1);

m12=(6-4*a(1));

a(2)=(4*b(1)+4)/m12;

b(2)=-1/m12;

c(2)=1/m12*(4*c(1)+f(2,1)* $d^{4}$ /e1);

for i=3:m8-2

m12=(a(i-2)*a(i-1)+b(i-2)-4*a(i-1)+6);

a(i)=-1/m12*(a(i-2)*b(i-1)-4*b(i-1)-4);

b(i)=-1/m12;

c(i)=1/m12*(f(i,1)* $d^{4}$ /e1-c(i-2)-a(i-2)*c(i-1)+4*c(i-1));

end;

u(m8,1)=0;

u(m8-1,1)=0;

for i=2:m8-1;

u(m8-i,1)=a(m8-i)*u(m8-i+1,1)+b(m8-i)*u(m8-i+2,1)+c(m8-i);

end;

for i=1:m8

x(i)=i*d;

end;

plot(x,u)

******************

29. A note on hyper-finite differences for the generalized method of lines

In this section we develop an application of the hyper finite differences method through an approximation of the generalized method of lines.

Consider the equation

\{\begin{matrix} - ε u^{''} (x) + α u^{3} - β u - f = 0, & in Ω = [0, 1], \\ u (0) = 0, & u (1) = 0 \end{matrix}

(175)

ε > 0

is small, in order to decrease the error concerning the approximations used we propose to divide the domain

Ω = [0, 1]

into

N_{1}

sub-intervals of same measure. Thus we define

x_{k} = \frac{k}{N_{1}}, \forall k \in {0, 1, \dots, N_{1}} .

For each sub-interval

I_{k} = [x_{k - 1}, x_{k}]

we are going to obtain an approximate solution of the equation in question with the general boundary conditions

u ((k - 1) / N_{1}) = U [k - 1],

and

u (k / N_{1}) = U [k] .

Denoting such a solution by

{u [i, k]}

where

x_{i} = \frac{k - 1}{N_{1}} + i d,

and

d = \frac{1}{m_{8} N_{1}},

where

m_{8}

is the fixed number of nodes in each interval

I_{k}

Observe that in a finite differences context, linearizing it about a initial solution

{u_{0} [i, k]},

the equation in question stands for:

\begin{matrix} - ε \frac{(u [i + 1, k] - 2 u [i, k] + u [i - 1, k])}{d^{2}} + 3 α u_{0} {[i, k]}^{2} u [i, k] - 2 α u_{0} {[i, k]}^{3} \\ - β u [i, k] - f [i, k] = 0, \forall i \in {1, \dots, m_{8} - 1} . \end{matrix}

(176)

In particular, for

i = 1

, we obtain

\begin{matrix} - ε \frac{(u [2, k] - 2 u [1, k] + u [0, k])}{d^{2}} + 3 α u_{0} {[1, k]}^{2} u [1, k] - 2 α u_{0} {[1, k]}^{3} \\ - β u [1, k] - f [1, k] = 0, \end{matrix}

(177)

so that

\begin{matrix} u [1, k] & = & a [1, k] u [2, k] + b [1, k] u [0, k] + c [1, k] T [1, k] \\ + e [1, k] + E_{r} [1, k], \end{matrix}

(178)

where

a [1, k] = 1 / 2,

b [1, k] = 1 / 2,

c [1, k] = 1 / 2,

e [1, k] = f [1, k] \frac{d^{2}}{2 ε},

T [1, k] = (- 3 α u_{0} {[1, k]}^{2} u [i, k] + 2 α u_{0} {[1, k]}^{3} - β u [1, k]) \frac{d^{2}}{ε},

and

E_{r} [1, k] = 0 .

Now reasoning inductively, having

\begin{matrix} u [i - 1, k] & = & a [i - 1, k] u [i, k] + b [i - 1, k] u [0, k] + c [i - 1, k] T [i - 1, k] \\ + e [i - 1, k] + E_{r} [i - 1, k], \end{matrix}

(179)

and

\begin{matrix} - ε \frac{(u [i + 1, k] - 2 u [i, k] + u [i - 1, k])}{d^{2}} + 3 α u_{0} {[i, k]}^{2} u [i, k] - 2 α u_{0} {[i, k]}^{3} \\ - β u [i, k] - f [i, k] = 0, \end{matrix}

(180)

so that

(u [i + 1, k] - 2 u [i, k] + u [i - 1, k]) + T [i, k] + f [i, k] \frac{d^{2}}{ε} = 0,

where,

T [i, k] = (- 3 α u_{0} {[i, k]}^{2} u [i, k] + 2 α u_{0} {[i, k]}^{3} + β u [i, k]) \frac{d^{2}}{ε},

we obtain

\begin{matrix} u [i, k] & = & a [i, k] u [i, k] + b [i, k] u [0, k] + c [i, k] T [i, k] \\ + e [i, k] + E_{r} [i, k], \end{matrix}

(181)

where,

a [i, k] = {(2 - a [i - 1, k])}^{- 1},

b [i, k] = a [i, k] b [i - 1, k],

c [i, k] = a [i, k] (c [i - 1, k] + 1),

e [i, k] = a [i, k] (e [i - 1, k] + \frac{f [i, k] d^{2}}{ε}),

and

E_{r} [i, k] = a [i, k] (E_{r} [i - 1, k]) + c [i, k] (T [i - 1, k] - T [i, k]) .

Observe that in particular for

i = m_{8} - 1

, we have

u [m 8, k] = U [k]

and

u [0, k] = U [k - 1],

so that from above, neglecting

E_{r} [1, k]

, we also obtain

\begin{matrix} u [m_{8} - 1, k] \approx a [m_{8} - 1] u [m_{8}, k] + b [m_{8} - 1, k] u [0, k] \\ + c [m_{8} - 1, k] T [m 8 - 1, k] (u [m_{8}, k], u [0, k]) + e [m_{8} - 1, k] \\ = & H_{m_{8} - 1} (U [k], U [k - 1]) . \end{matrix}

(182)

Similarly, for

i = m 8 - 2

we may obtain

\begin{matrix} u [m_{8} - 2, k] \approx a [m_{8} - 2] u [m_{8} - 1, k] + b [m_{8} - 2, k] u [0, k] \\ + c [m_{8} - 2, k] T [m 8 - 2, k] (u [m_{8} - 1, k], u [0, k]) + e [m_{8} - 2, k] \\ = & H_{m_{8} - 2} (U [k], U [k - 1]), \end{matrix}

(183)

and so on, up to finding

u [1, k] = H_{1} (U [k], U [k - 1]), \forall k \in {1, \dots, N_{1}} .

At this point we connect the sub-intervals by setting

U [0] = U [N_{1}] = 0

and obtaining

{U [1], \dots, U [N_{1} - 1]},

by solving the equations

\begin{matrix} - ε \frac{(u [m_{8} - 1, k] - 2 U [k] + u [1, k + 1])}{d^{2}} + 3 α u_{0} {[m 8, k]}^{2} U [k] - 2 α u_{0} {[m 8, k]}^{3} \\ - β U [k] - f [m_{8}, k] = 0, \forall k \in {1, \dots, N_{1} - 1} . \end{matrix}

(184)

Having obtained

{U [k], \forall k \in {1, \dots, N_{1} - 1}}

we may obtain the solution

{u [i, k]}

where

i \in {0, \dots, m_{8}}

and

k \in {1, \dots, N_{1}} .

The next step is to replace

{u_{0} [i, k]}

{u [i, k]}

and then to repeat the process until an appropriate convergence criterion is satisfied.

The problem is then approximately solved.

We have obtained numerical results for

ε = 0.001

f \equiv 1,

Ω

N_{1} = 10

m_{8} = 100

and

α = β = 1 .

For the related software in MATHEMATICA we have obtained

U [1], \dots, U [9],

Here the software and results:

**************************

1.: Clear[u, U, z, N1];

m8 = 100;

N1 = 10;

d = 1/m8/N1;

e1 = 0.001;

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = 1.01]];

A = 1.0;

B = 1.0;

a[1] = 1.0/2;

b[1] = 1.0/2;

c[1] = 1/2.0;

e[1] = $d^{2} / e 1 / 2.0$ ;

For[i = 2, i < m8, i++,

a[i] = 1/(2.0 - a[i - 1]);

b[i] = b[i - 1]*a[i];

c[i] = a[i]*(c[i - 1] + 1.0);

e[i] = $a [i] * (e [i - 1] + d^{2} / e 1)$ ;

];

For[k1 = 1, k1 < 10, k1++,

Print[k1];

Clear[U, z];

For[k = 1, k < N1 + 1, k++,

u[0, k] = U[k - 1];

u[m8, k] = U[k];

For[i = 1, i < m8, i++,

z = a[m8 - i]*u[m8 - i + 1, k] + b[m8 - i]*u[0, k] +

c[m8 - i]*(-3*A*uo[m8 - i + 1, k] $^{2}$ *u[m8 - i + 1, k] +

2*A*uo[m8 - i + 1, k] $^{3}$ + B*u[m8 - i + 1, k])* $d^{2} / e 1$ +

e[m8 - i];

u[m8 - i, k] = Expand[z]]];

U[0] = 0.0;

U[N1] = 0.0;

S = 0;

For[k = 1, k < N1, k++,

S = S + (e1*(-u[m8 - 1, k] + 2*U[k] - u[1, k + 1])/ $d^{2}$ +

3*A*U[k]*uo[m8, k] $^{2}$ - 2*A*uo[m8, k] $^{3}$ - B*U[k] - 1) $^{2}$ ];

Sol = NMinimize[S, U[1], U[2], U[3], U[4], U[5], U[6], U[7], U[8], U[9]];

For[k = 1, k < N1, k++,

w4[k] = U[k] Ṡol[[2, k]]];

For[k = 1, k < N1, k++,

U[k] = w4[k]];

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = u[i, k]]];

Print[U[5]]];

For[k = 0, k < N1 + 1, k++,

Print["U[", k, "]=", U[k]]]

U[0]=0.

U[1]=1.27567

U[2]=1.32297

U[3]=1.32466

U[4]=1.32472

U[5]=1.32472

U[6]=1.32472

U[7]=1.32472

U[8]=1.32472

U[9]=1.32471

U[10]=0.

**********************

Remark 29.1.

Observe that along the domain we have obtained approximately the constant value

u = 1.32472

. This is expected since

ε = 0.001

is small and such a value u is approximately the solution of equation

α u^{3} - β u - 1 = 0 .

30. Applications to the optimal shape design for a beam model

In this section, we present a numerical procedure for the shape optimization concerning the Bernoulli beam model.

Let

Ω = [0, 1] \subset R

corresponds to the horizontal axis of a straight beam with rectangular cross section

b \times h (x)

, that is, the beam has a variable thickness

h (x)

distributed along such a horizontal axis x, where

x \in [0, 1] .

Define now

V = {w \in W^{2, 2} (Ω) : w (0) = w (1) = 0},

which corresponds to a simply supported beam.

Consider the problem of minimizing in

V \times B

the functional

J (w, h) = \frac{1}{2} \int_{Ω} H (x) w_{x x} {(x)}^{2} d x

subject to

{(H (x) w_{x x} (x))}_{x x} - P (x) = 0, in Ω,

where

H (x) = \frac{h {(x)}^{3} b}{12} E,

h (x)

is variable beam thickness,

A (x) = b h (x)

corresponds to a rectangular cross section perpendicular to the x axis, and E is the young elasticity model.

Also, we define

B = \{h : [0, 1] \to R measurable : h_{m i n} \leq h (x) \leq h_{m a x} and \int_{0}^{1} h (x) \leq c_{0} h_{m a x}\},

where

0 < c_{0} < 1

and

C^{*} = {w \in V : {(H (x) w_{x x} (x))}_{x x} - P (x) = 0, in Ω} .

Observe that

\begin{matrix} inf_{(w, h) \in C^{*} \times B} J (w, h) \\ = & inf_{h \in B} \{inf_{w \in C^{*}} J (w, h)\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{\frac{1}{2} \int_{Ω} H (x) w_{x x} {(x)}^{2} d x - {〈 \hat{w}, {(H (x) w_{x x} (x))}_{x x} - P (x) 〉}_{L^{2}}\}\}\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{- \frac{1}{2} \int_{Ω} H (x) {\hat{w}}_{x x}^{2} d x + {〈 \hat{w}, P 〉}_{L^{2}}\}\} \\ = & inf_{h \in B} \{inf_{M \in D^{*}} \{\frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x\}\} . \end{matrix}

(185)

where

D^{*} = {M \in Y^{*} : M_{x x} - P = 0, in Ω, and M (0) = M (1) = 0} .

Summarizing, we have got

inf_{(w, h) \in C^{*} \times B} J (w, h) = inf_{(M, h) \in D^{*} \times B} \{\frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x\} .

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n = 1$ and

$h_{n} (x) = c_{0} h_{m a x} .$
Calculate $w_{n} \in V$ solution of equation

${(H_{n} (x) {(w_{n})}_{x x})}_{x x} = P (x),$

where

$H_{n} (x) = \frac{E b h_{n} {(x)}^{3}}{12} .$
Calculate $h_{n + 1} (x) \in B$ such that

$J^{*} (M_{n}, h_{n + 1}) = inf_{h \in B} J^{*} (M_{n}, h),$

where

$M_{n} = H_{n} {(w_{n})}_{x x},$

$J^{*} (M, h) = \frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x .$
Set $n : = n + 1$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for

c_{0} = 0.65

E = 210 10^{7}

b = 0.1 m

P (x) = 36 10^{2} N

h_{m i n} = 0.072 m

and

h_{m a x} = 0.18 m .

We have also defined

h (x) = t (x) h_{m a x},

where

0.4 \leq t (x) \leq 1, a . e . in Ω .

For the optimal solution

w = w (x)

, please see Figure 20.

For a corresponding optimal solution

t = t (x)

, please see Figure 21.

Remark 30.1.

For such a simply-supported beam model, for the numerical solution of equation

{(H (x) w_{x x})}_{x x} = P,

with the boundary conditions

w (0) = w (1) = w^{''} (0) = w^{''} (1) = 0

firstly we have solved the equation

v_{x x} - P = 0

with the boundary conditions

v (0) = v (1) = 0 .

Subsequently, we have solved the equation

H (x) w_{x x} = v

with the boundary conditions

w (0) = w (1) = 0 .

Here we present the software developed in MAT-LAB.

******************

1.: clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210* $10^{7}$ ;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0* $10^{2}$ ;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)* $d^{2}$ +z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ and $(k < 10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5* $L {(i, 1)}^{3} * h o^{3}$ /12*e1;

f1(i,1)=v(i,1)/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ +z1(i-1));

end;

w(m8,1)=0;

for i=1:m8-1

w(m8-i,1)=m70(m8-i)*w(m8-i+1,1)+z1(m8-i);

end;

d2wo(1,1)=(-2*w(1,1)+w(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2wo(i,1)=(w(i+1,1)-2*w(i,1)+w(i-1,1))/ $d^{2}$ ;

end;

k9=1;

b14=1.0;

while $(b 14 > 10^{- 4}) and (k 9 < 120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

***************

With the auxiliary function "beamNov2023":

********************

1.: function S=beamNov2023(x)

global m8 d d2wo H e1 ho h1 xo b5

S=0;

for i=1:m8-1

S=S+1/ $(x {(i, 1)}^{3}) / h o^{3}$ /b5/e1*(H(i,1)* ${d 2 w o (i, 1))}^{2}$ *12;

end;

*****************************

We develop numerical results also for

V = W_{0}^{2, 2} (Ω) = {w \in W^{2, 2} (Ω) such that w (0) = w (1) = w^{'} (0) = w^{'} (1) = 0} .

Such boundary conditions corresponds to bi-clamped beam. The remaining data is equal to the previous example

For the optimal solution

w = w (x)

, please see Figure 22.

For a corresponding optimal solution

t = t (x)

, please see Figure 23.

Remark 30.2.

For such a bi-clamped beam model, for the numerical solution of equation

{(H (x) w_{x x})}_{x x} = P,

with the boundary conditions

w (0) = w (1) = w^{'} (0) = w^{'} (1) = 0,

firstly we have solved the equation

v_{x x} - P = 0

with the boundary conditions

v (0) = v (1) = 0 .

Subsequently, we solved the equation

H (x) w_{x x} = v + a x + b

with the boundary conditions

w (0) = w (1) = 0,

obtaining

a, b \in R

such that the boundary conditions

w^{'} (0) = w^{'} (1) = 0

are also satisfied.

Here we present the software developed in MAT-LAB.

*************************

1.: clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210* $10^{7}$ ;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0* $10^{2}$ ;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)* $d^{2}$ +z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4}) and (k < 10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5* $L {(i, 1)}^{3} * h o^{3}$ /12*e1;

f1(i,1)=v(i,1)/H(i,1);

f2(i,1)=i*d/H(i,1);

f3(i,1)=1/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ );

z2(i)=m70(i)*(-f2(i,1)* $d^{2}$ );

z3(i)=m70(i)*(-f3(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ +z1(i-1));

z2(i)=m70(i)*(-f2(i,1)* $d^{2}$ +z2(i-1));

z3(i)=m70(i)*(-f3(i,1)* $d^{2}$ +z3(i-1));

end;

w1(m8,1)=0;

w2(m8,1)=0;

w3(m8,1)=0;

for i=1:m8-1

w1(m8-i,1)=m70(m8-i)*w1(m8-i+1,1)+z1(m8-i);

w2(m8-i,1)=m70(m8-i)*w2(m8-i+1,1)+z2(m8-i);

w3(m8-i,1)=m70(m8-i)*w3(m8-i+1,1)+z3(m8-i);

end;

m3(1,1)=w2(1,1);

m3(1,2)=w3(1,1);

m3(2,1)=w2(m8-1,1);

m3(2,2)=w3(m8-1,1);

h3(1,1)=-w1(1,1);

h3(2,1)=-w1(m8-1,1);

h5(:,1)=inv(m3)*h3;

for i=1:m8

wo(i,1)=w1(i,1)+h5(1,1)*w2(i,1)+h5(2,1)*w3(i,1);

end;

d2wo(1,1)=(-2*wo(1,1)+wo(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2wo(i,1)=(wo(i+1,1)-2*wo(i,1)+wo(i-1,1))/ $d^{2}$ ;

end;

k9=1;

b14=1.0;

while $(b 14 > 10^{- 4}) and (k 9 < 120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

*****************************

Remark 30.3.

About the numerical results obtained for these two beam models, a final word of caution is necessary.

Indeed, the full convergence in such cases is hard to obtain so that we have obtained just approximations of critical points with the functionals close to their optimal values. It is also worth emphasizing we have fixed the number of iterations so that the solutions and shapes obtained are just approximate ones.

31. Applications to the optimal shape design for a plate model

In this section, we present a numerical procedure for the shape optimization concerning a thin plate model.

Let

Ω = [0, 1] \times [0, 1] \subset R^{2}

corresponds to the middle surface of a thin plate with a variable thickness

h (x, y)

Define now

V = {w \in W^{2, 2} (Ω) : w = 0 on \partial Ω},

which corresponds to a simply supported plate.

Consider the problem of minimizing in

V \times B

the functional

J (w, h) = \frac{1}{2} \int_{Ω} H (x, y) {(\nabla^{2} w (x, y))}^{2} d x

subject to

\nabla^{2} [(H (x, y) \nabla^{2} w (x, y))] - P (x, y) = 0, in Ω,

where

H (x, y) = \frac{h {(x, y)}^{3}}{12} E / (1 - w_{5}^{2}),

h = h (x, y)

is variable plate thickness, E is the young elasticity model and

w_{5} = 0.3

Also, we define

B = \{h : Ω \to R measurable : h_{m i n} \leq h (x, y) \leq h_{m a x} and \int_{Ω} h (x, y) \leq c_{0} h_{m a x}\},

where

0 < c_{0} < 1

and

C^{*} = {w \in V : \nabla^{2} [H (x, y) \nabla^{2} w (x, y))] - P (x, y) = 0, in Ω} .

Observe that

\begin{matrix} inf_{(w, h) \in C^{*} \times B} J (w, h) \\ = & inf_{h \in B} \{inf_{w \in C^{*}} J (w, h)\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{\frac{1}{2} \int_{Ω} H (x, y) {[\nabla^{2} w (x, y)]}^{2} d x - {〈 \hat{w}, \nabla^{2} [H (x, y) \nabla^{2} w (x, y)] - P (x, y) 〉}_{L^{2}}\}\}\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{- \frac{1}{2} \int_{Ω} H (x, y) {[\nabla^{2} \hat{w} (x, y)]}^{2} d x + {〈 \hat{w}, P 〉}_{L^{2}}\}\} \\ = & inf_{h \in B} \{inf_{\tilde{M} \in D^{*}} \{\frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x\}\} . \end{matrix}

(186)

where

D^{*} = {\tilde{M} \in Y^{*} \nabla^{2} \tilde{M} - P = 0, in Ω, and \tilde{M} = 0, on Ω} .

Summarizing, we have got

inf_{(w, h) \in C^{*} \times B} J (w, h) = inf_{(\tilde{M}, h) \in D^{*} \times B} \{\frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x\} .

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n = 1$ and

$h_{n} (x) = c_{0} h_{m a x} .$
Calculate $w_{n} \in V$ solution of equation

$\nabla^{2} (H_{n} (x, y) \nabla^{2} w_{n} (x, y)) = P (x, y),$

where

$H_{n} (x, y) = \frac{E h_{n} {(x)}^{3}}{12 (1 - w_{5}^{2})} .$
Calculate $h_{n + 1} \in B$ such that

$J^{*} ({\tilde{M}}_{n}, h_{n + 1}) = inf_{h \in B} J^{*} ({\tilde{M}}_{n}, h),$

where

${\tilde{M}}_{n} = H_{n} (x, y) \nabla^{2} w_{n},$

$J^{*} (\tilde{M}, h) = \frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x .$
Set $n : = n + 1$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for

c_{0} = 0.75

E = 200 10^{5}

P (x, y) = 2 10^{2} N

h_{m i n} = 0.45 * (0.12) m

and

h_{m a x} = 0.12 m .

We have also defined

h (x, y) = t (x, y) h_{m a x},

where

0.45 \leq t (x, y) \leq 1, a . e . in Ω .

For the optimal solution

w = w (x, y)

, please see Figure 24.

For a corresponding optimal solution

t = t (x, y)

, please see Figure 22.

Remark 31.1.

For such a simply-supported plate model, for the numerical solution of equation

\nabla^{2} [H (x, y) \nabla^{2} w (x, y)] = P,

with the boundary conditions

w = 0 on \partial Ω,

firstly we have solved the equation

\nabla^{2} v - P = 0

with the boundary conditions

v = 0 on \partial Ω .

Subsequently, we have solved the equation

H (x, y) \nabla^{2} w (x, y) = v (x, y)

with the boundary conditions

w = 0 on \partial Ω .

Here we present the software developed in MAT-LAB.

*********************

1.: clear all

global m8 d d2xwo d2ywo H e1 ho xo b5

m8=40;

d=1.0/m8;

w5=0.3;

e1=200* $10^{5}$ / $(1 - w 5^{2})$ ;

ho=0.12;

A=zeros( ${(m 8 - 1)}^{2}, {(m 8 - 1)}^{2}$ );

for i=1: ${(m 8 - 1)}^{2}$

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.45*ones( ${(m 8 - 1)}^{2}$ ,1);

ub=ones( ${(m 8 - 1)}^{2}$ ,1);

b=zeros( ${(m 8 - 1)}^{2}$ ,1);

b(1,1)=0.75* ${(m 8 - 1)}^{2};$

for i=1:(m8-1)

for j=1:m8-1

f(i,j,1)=1.0;

L(i,j,1)=1/2;

P(i,j,1)=2* $10^{2}$ ; end;

end;

for i=1:m8

wo(:,i)=0.001*ones(m8-1,1);

end;

m2=zeros(m8-1,m8-1);

for i=2:m8-2

m2(i,i)=-2.0;

m2(i,i-1)=1.0;

m2(i,i+1)=1.0;

end;

m2(1,1)=-2.0;

m2(1,2)=1.0;

m2(m8-1,m8-1)=-2.0;

m2(m8-1,m8-2)=1.0;

Id=eye(m8-1);

i=1;

m12=2*Id-m2* $d^{2} / d^{2}$ ; m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)* $d^{2}$ );

for i=2:m8-1

m12=2*Id-m2* $d^{2} / d^{2}$ -m50(:,:,i-1);

m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)* $d^{2}$ +z(:,i-1));

end; v(:,m8)=zeros(m8-1,1);

for i=1:m8-1

v(:,m8-i)=m50(:,:,m8-i)*v(:,m8-i+1)+z(:,m8-i);

end;

k=1;

b12=1.0;

while ( $b 12 > 10^{- 4}$ ) and ( $k < 12$ )

k

k=k+1;

for i=1:m8-1

for j=1:m8-1

H(j,i,1)= $L {(j, i, 1)}^{3} * h o^{3}$ /12*e1;

f1(j,i,1)=v(j,i)/H(j,i,1);

end;

end;

i=1;

m12=2*Id-m2* $d^{2} / d^{2}$ ;

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)* $d^{2}$ );

for i=2:m8-1

m12=2*Id-m2* $d^{2} / d^{2}$ -m70(:,:,i-1);

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)* $d^{2}$ +z1(:,i-1));

end;

w(:,m8)=zeros(m8-1,1);

for i=1:m8-1

w(:,m8-i)=m70(:,:,m8-i)*w(:,m8-i+1)+z1(:,m8-i);

end;

d2xwo(:,1)=(-2*w(:,1)+w(:,2))/ $d^{2}$ ;

for i=2:m8-1

d2xwo(:,i)=(w(:,i+1)-2*w(:,i)+w(:,i-1))/ $d^{2}$ ;

end;

for i=1:m8-1

d2ywo(:,i)=m2*w(:,i)/ $d^{2}$ ;

end;

k9=1; b14=1.0;

while ( $b 14 > 10^{- 4}$ ) and ( $k 9 < 30$ )

k9

k9=k9+1;

X=fmincon(’beamNov2023A3’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(max(abs(w-wo)))

wo=w;

x3=xo;

for i=1:m8-1

for j=1:m8-1

L(j,i,1)=xo((i-1)*(m8-1)+j,1);

end;

end;

end;

for i=1:m8-1

x8(i,1)=i*d;

end;

mesh(x8,x8,L);

*********************

With the auxiliary function "beamNov2023A3’, where

****************************

1.: function S=beamNov2023A3(x)

global m8 d d2xwo d2ywo H e1 ho xo b5

S=0;

for i=1:m8-1

for j=1:m8-1

x1(j,i)=x((m8-1)*(i-1)+j,1);

end;

end;

for i=1:m8-1

for j=1:m8-1

S=S+ $1 / ({(x 1 (j, i))}^{3}) / h o^{3} / e 1 * {(H (j, i, 1))}^{2} * {(d 2 x w o (j, i) + d 2 y w o (j, i))}^{2} * 12$ ;

end;

end;

********************************

Remark 31.2.

About the numerical results obtained for this plate model, a final word of caution is necessary.

Indeed, the full convergence in such a case is hard to obtain so that we have obtained just approximations of critical points with the functional close to its optimal value. It is also worth emphasizing we have fixed the number of iterations so that the solution and shape obtained are just approximate ones.

32. A note on the first Maxwell equation of electromagnetism

Let

Ω_{1} \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω_{1}

Suppose

E : Ω_{1} \to R^{3}

is an electric field of

C^{1}

class in

Ω .

Let

Ω \subset Ω_{1}

be also an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

S = \partial Ω

Observe that there exists a scalar field

V : Ω \to R

such that

\nabla^{2} V = div E, in Ω,

and

\nabla V \cdot n = 0, on S = \partial Ω .

Here

n

denotes the normal outward field to S.

Observe also that

\nabla^{2} V = div \nabla V = div E,

so that defining

h = \nabla V - E,

we have that

div h = 0, in Ω .

Hence, from such results and the divergence Theorem, we get

\begin{matrix} \int_{S} E \cdot n d S & = & \int_{S} (\nabla V) \cdot n d S - \int_{S} h \cdot n d S \\ = & - \int_{Ω} div h d V = 0 . \end{matrix}

(187)

Summarizing, we have got

\int_{S} E \cdot n d S = 0 .

Consider now a charge

q_{0}

localized at the center of a sphere

Ω_{2}

of radius

R > 0

and boundary

S_{2} = \partial Ω_{2} .

The electric field on the sphere surface generated by

q_{0}

is given by

E_{2} = \frac{1}{4 π ε_{0}} \frac{q_{0}}{R^{2}} n_{2},

where

n_{2}

is the normal outward field to

S_{2}

Clearly

\int_{S_{2}} E_{2} \cdot n_{2} d S_{2} = \frac{1}{4 π ε_{0}} \frac{q_{0}}{R^{2}} (4 π R^{2}) = \frac{q_{0}}{ε_{0}} .

Consider again the set

Ω

but now with a charge

q_{0}

localized at a point

x

inside the interior of

Ω

, which is denoted by

Ω^{0}

At first the electric field

E

generated by

q_{0}

is not of

C^{1}

class on

Ω .

However, there exists

R > 0

such that

B_{R} (x) \subset Ω = Ω^{0} .

Define

Ω_{3} = Ω ∖ B_{R} (x)

Therefore,

E

is of

C^{1}

class on

Ω_{3} .

Denoting the boundary of

Ω_{3}

S_{3}

, from the previous results, we may infer that

\int_{S_{3}} E \cdot n d S_{3} = 0,

so that

\begin{matrix} \int_{S_{3}} E \cdot n d S_{3} & = & \int_{S} E \cdot n d S - \int_{\partial B_{R} (x)} E \cdot n d S_{2} \\ = & \int_{S} E \cdot n d S - \frac{q_{0}}{ε_{0}} \\ = & 0 . \end{matrix}

(188)

Therefore, we have got

\int_{S} E \cdot n d S = \frac{q_{0}}{ε_{0}} .

Assume now on

Ω

we have a density of charges

ρ (x)

For a small volume

Δ V

consider a punctual charge

q_{0}

localized in

x \in Ω

such that

q_{0} \approx ρ (x) Δ V .

Denoting by

Δ E

the electric field generated by

q_{0}

, from the previous results we may infer that

\int_{S} Δ E \cdot n d S = \frac{q_{0}}{ε_{0}} \approx \frac{ρ (x) Δ V}{ε_{0}} .

Such an equation in its differential form, stands for:

\int_{S} d E \cdot n d S = \frac{ρ (x) d V}{ε_{0}} .

Integrating in

Ω

we may obtain

\begin{matrix} \int_{S} E \cdot n d S & = & \int_{S} \int_{Ω} d E \cdot n d V d S \\ = & \int_{Ω} \frac{ρ (x)}{ε_{0}} d V, \end{matrix}

(189)

so that

\int_{S} E \cdot n d S = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

From this and the Divergence Theorem, we have

\int_{S} E \cdot n d S = \int_{Ω} div E d V = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

Summarizing, we have got

\int_{Ω} div E d V = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

This is the integral form of the first Maxwell equation of electromagnetism.

For this last equation, the set

Ω \subset Ω_{1}

is rather arbitrary so that for

Ω

as a ball of small radius

r > 0

with center at a point

x \in Ω_{1}

, from the Mean Value Theorem fot integrals and letting

r \to 0^{+}

, we obtain

div E = \frac{ρ}{ε_{0}}, in Ω_{1} .

This last equation stands for the differential form of the first Maxwell equation of electromagnetism.

Remark 32.1.

Summarizing, in this section we have formally obtained a mathematical deduction of the first Maxwell equation of electromagnetism.

33. A note on relaxation for a general model in the vectorial calculus of variations

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a function

g : R^{N \times n} \to R

twice differentiable and such that

g (y) \to + \infty, as | y | \to + \infty .

Define a functional

G : V \to R

G (\nabla u) = \frac{1}{2} \int_{Ω} g (\nabla u) d x,

where

V = {W^{1, 2} (Ω; R^{N}) : u = u_{0} on \partial Ω} .

Moreover, for

f \in L^{2} (Ω; R^{N})

, define also

J (u) = G (\nabla u) - {〈 u, f 〉}_{L^{2}} .

We assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Observe that from the convex analysis basic theory, we have that

\begin{matrix} α & = & inf_{u \in^{V}} J (u) \\ = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

(190)

On the other hand

\begin{matrix} {(G \circ \nabla)}^{* *} (u) & \leq & H (u) \\ \equiv & inf_{(λ, (v, w)) \in [0, 1] \times B (u, λ)} {λ G (\nabla w) + (1 - λ) G (\nabla v)} \\ \leq & G (\nabla u), \end{matrix}

(191)

where

B (u, λ) = {(v, w) \in V : λ w + (1 - λ) v = u} .

From such results, we may infer that

inf_{u \in V} J^{* *} (u) = inf_{u \in V} {H (u) - {〈 u, f 〉}_{L^{2}}} = inf_{u \in V} J (u) .

Furthermore, observe that

λ \nabla w + (1 - λ) \nabla v = \nabla u,

so that

\begin{matrix} \nabla v & = & \nabla u + λ (\nabla v - \nabla w) \\ = & \nabla u + λ \nabla ϕ, \end{matrix}

(192)

where

ϕ = v - w \in W_{0}^{1, 2} (Ω; R^{N})

so that

\nabla ϕ = \nabla v - \nabla w,

and

\nabla w = \nabla v - \nabla ϕ .

Therefore,

\nabla w = \nabla v - \nabla ϕ = \nabla u + λ \nabla ϕ - \nabla ϕ = \nabla u - (1 - λ) \nabla ϕ .

Replacing such results into the expression of H, we have

H (u) = inf_{(λ, ϕ) \in [0, 1] \times V_{0}} {λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ)},

where

V_{0} = W_{0}^{1, 2} (Ω; R^{N}) .

Joining the pieces, we have got

\begin{matrix} inf_{u \in V} J (u) & = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {H (u) - {〈 u, f 〉}_{L^{2}}} \\ = & inf_{(λ, ϕ, u) \in [0, 1] \times V_{0} \times V} {λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

This last functional corresponds to a relaxation for the original non-convex functional.

The note is complete.

33.1. Some related numerical results

In this subsection we present numerical results for an one-dimensional model and related relaxed formulation.

For

Ω = [0, 1] \subset R

, consider the functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x,

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2},

f \in Y = Y^{*} = L^{2} (Ω) .

Based on the results of the previous section, denoting

V_{0} = W_{0}^{1, 2} (Ω),

we define the following relaxed functional

J_{1} : [0, 1] \times V \times V_{0} \to R,

where

J_{1} (λ, u, ϕ) = \frac{λ}{2} \int_{Ω} {({(u^{'} - (1 - λ) ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1 - λ}{2} \int_{Ω} {({(u^{'} + λ ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x .

Indeed, we have developed an algorithm for minimizing the following regularized functional

J_{2} : [0, 1] \times V \times V_{0} \to R,

where

J_{2} (λ, u, ϕ) = J_{1} (λ, u, ϕ) + \frac{ε_{3}}{2} \int_{Ω} {(u^{''})}^{2} d x,

for a small parameter

ε_{3} > 0

For the case in which

f (x) = sin (π x) / 2

, for the optimal solution u, please see Figure 26.

For the case in which

f (x) = cos (π x) / 2

, for the optimal solution u, please see Figure 27.

For the case in which

f (x) = 0

, for the optimal solution u, please see Figure 28.

We highlight to obtain the solution for this last case which

f = 0

is harder. A good solution was possible only using

x_{0} = 0

as the initial solution concerning the iterative process.

Here we present the software in MAT-LAB developed.

*****************

1.: clear all

global m8 d u e3

m8=100;

d=1/m8;

e3=0.0005;

for i=1:2*m8+1

xo(i,1)=0.36;

end;

b12=1.0;

k=1;

while $(b 12 > 10^{- 7}) and (k < 60)$

k

k=k+1;

X=fminunc(’funDecember2023’,xo);

b12=max(abs(xo-X))

xo=X;

u(m8/2)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

***********************

With the main function "funDecember2023"

*********************

1.: function S=funDecember2023(x)

global m8 d u e3

for i=1:m8

u(i,1)=x(i,1);

v(i,1)=x(i+m8,1);

yo(i,1)=sin(pi*i*d)/2;

end;

L=(1+sin(x(2*m8+1,1)))/2;

u(m8,1)=1/2;

v(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

for i=2:m8

du(i,1)=(u(i,1)-u(i-1,1))/d;

dv(i,1)=(v(i,1)-v(i-1,1))/d;

end;

d2u(1,1)=(-2*u(1,1)+u(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2u(i,1)=(u(i-1,1)-2*u(i,1)+u(i+1,1))/ $d^{2}$ ;

end;

S=0;

for i=1:m8

S=S+ $1 / 2 * L * {({(d u (i, 1) - (1 - L) * d v (i, 1))}^{2} - 1)}^{2}$ ;

S=S+ $1 / 2 * (1 - L) * {({(d u (i, 1) + L * d v (i, 1))}^{2} - 1)}^{2}$ ;

S=S+ ${(u (i, 1) - y o (i, 1))}^{2}$ ;

end;

for i=1:m8-1

S=S+e3* $d 2 u {(i, 1)}^{2}$ ;

end;

*******************

33.2. A related duality principle and concerning convex dual formulation

With the notation and statements of the previous sections in mind, consider the functionals

J : V \to R

and

J_{3} : [0, 1] \times V \times V_{0} \to R

where

J (u) = G (\nabla u) + \frac{1}{2} \int_{Ω} u \cdot u d x - {〈 u, f 〉}_{L^{2}},

and

\begin{matrix} J_{3} (λ, u, ϕ) & = & λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ) \\ + \frac{λ}{2} \int_{Ω} (u - (1 - λ) ϕ) \cdot (u - (1 - λ) ϕ) d x \\ + \frac{(1 - λ)}{2} \int_{Ω} (u + λ ϕ) \cdot (u + λ ϕ) d x \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}} . \end{matrix}

(193)

Here we have denoted

V = {u \in W^{1, 2} (Ω; R^{N}) : u = u_{0} on \partial Ω = S},

V_{0} = W_{0}^{1, 2} (Ω; R^{N}),

Y = Y^{*} = L^{2} (Ω; R^{N \times n})

and

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{N}) .

Observe that

J^{* *} (u) \leq min_{(λ, ϕ) \in [0, 1] \times V_{0}} J_{3} (λ, u, ϕ) .

Moreover,

\begin{matrix} J_{3} (λ, u, ϕ) & = & - {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + λ G (\nabla u - (1 - λ) \nabla ϕ) \\ - {〈 \nabla u - (1 - λ) \nabla ϕ, v_{2}^{*} 〉}_{L^{2}} + (1 - λ) G (\nabla u + λ \nabla ϕ) \\ - {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} (u - (1 - λ) ϕ) \cdot (u - (1 - λ) ϕ) d x \\ - {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} (u + λ ϕ) \cdot (u + λ ϕ) d x \\ + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} \\ + {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}} . \end{matrix}

(194)

Therefore,

\begin{matrix} J_{3} (λ, u, ϕ) & \geq & inf_{v_{1} \in Y} {- {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + λ G (v_{1})} \\ + inf_{v_{2} \in Y} {- {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} + (1 - λ) G (v_{2})} \\ + inf_{v_{3} \in Y_{1}} \{- {〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} (v_{3}) \cdot (v_{3}) d x\} \\ + inf_{v_{4} \in Y_{1}} \{- {〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} (v_{4}) \cdot (v_{4}) d x\} \\ + inf_{(u, ϕ) \in V \times V_{0}} {{〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} \\ + {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}}} \\ = & - λ G^{*} (\frac{v_{1}^{*}}{λ}) - (1 - λ) G^{*} (\frac{v_{2}^{*}}{(1 - λ)}) \\ - F_{3}^{*} (v_{3}^{*}, λ) - F_{4}^{*} (v_{4}^{*}, λ) \\ + \int_{S} {(v_{1}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S + \int_{S} {(v_{2}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S, \\ \forall λ \in (0, 1), u \in V, ϕ \in V_{0}, v^{*} \in A^{*}, \end{matrix}

(195)

where

G^{*} (v^{*}) = sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G (v)},

\begin{matrix} F_{3}^{*} (v_{3}^{*}, λ) & = & sup_{v_{3} \in Y_{1}} \{{〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} - \frac{λ}{2} \int_{Ω} v_{3} \cdot v_{3} d x\} \\ = & \frac{1}{2 λ} \int_{Ω} v_{3}^{*} \cdot v_{3}^{*} d x, \end{matrix}

(196)

\begin{matrix} F_{4}^{*} (v_{4}^{*}, λ) & = & sup_{v_{4} \in Y_{1}} \{{〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} - \frac{(1 - λ)}{2} \int_{Ω} v_{4} \cdot v_{4} d x\} \\ = & \frac{1}{2 (1 - λ)} \int_{Ω} v_{4}^{*} \cdot v_{4}^{*} d x . \end{matrix}

(197)

Furthermore,

A^{*} = A_{1}^{*} \cap A_{2}^{*}

where

A_{1}^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y^{*}]}^{2} \times {[Y_{1}^{*}]}^{2} : - div {(v_{1}^{*})}_{i} - div {(v_{2}^{*})}_{i} + {(v_{3}^{*})}_{i} + {(v_{4}^{*})}_{i} - f_{i} = 0, in Ω},

and

\begin{matrix} A_{2}^{*} & = & {v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y^{*}]}^{2} \times {[Y_{1}^{*}]}^{2} : \\ - (- 1 - λ) div {(v_{1}^{*})}_{i} - λ div {(v_{2}^{*})}_{i} + (- 1 - λ) {(v_{3}^{*})}_{i} + λ {(v_{4}^{*})}_{i} = 0, in Ω} . \end{matrix}

(198)

Summarizing, we have got

\begin{matrix} inf_{(λ, u ϕ) \in (0, 1) \times V \times V_{0}} J_{3} (λ, u, ϕ) \\ \geq & sup_{v^{*} \in A^{*}} \{inf_{λ \in (0, 1)} \{- λ G^{*} (\frac{v_{1}^{*}}{λ}) - (1 - λ) G^{*} (\frac{v_{2}^{*}}{(1 - λ)}) \\ - F_{3}^{*} (v_{3}^{*}, λ) - F_{4}^{*} (v_{4}^{*}, λ) + \int_{\partial Ω} {(v_{1}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S + \int_{\partial Ω} {(v_{2}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S\}\} . \end{matrix}

(199)

Remark 33.1.

We highlight this last dual function in

v^{*}

is convex (in fact concave) on the convex set

A^{*}

34. A general convex primal dual formulation for an originally non-convex primal one

Let

Ω \subset R^{3}

be an open bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(200)

where

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

Y = Y^{*} = L^{2} (Ω) .

Define

F_{1} : V \to R

and

F_{2} : V \times Y^{*} \to R

F_{1} (u) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and

\begin{matrix} F_{2} (u, v_{0}^{*}) & = & - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{K}{2} \int_{Ω} u^{2} d x \\ + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(201)

Define also

F_{1}^{*} : Y^{*} \to R

and

F_{2}^{*} : Y^{*} \times Y^{*} \to R

\begin{matrix} F^{*} (v_{1}^{*}) & = & sup_{u \in V} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{1} (u)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} + f)}^{2}}{- γ \nabla^{2} + K} d x, \end{matrix}

(202)

and

\begin{matrix} F_{2}^{*} (v_{1}^{*}, v_{0}^{*}) & = & sup_{u \in V} {- {〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{2} (u, v_{0}^{*})} \\ = & - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{2 v_{0}^{*} - K} d x \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(203)

v_{0}^{*} \in B^{*}

, where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2},

for some appropriate

K > 0

to be specified.

At this point we define

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥_{\infty} \leq 5 / 4 K},

for appropriate

K_{3} > 0

to be specified, and

J_{1}^{*} : D^{*} \times B^{*} \to R

J_{1}^{*} (v_{1}^{*}, v_{0}^{*}) = - F_{1}^{*} (v_{1}^{*}) + F_{2}^{*} (v_{1}^{*}, v_{0}^{*}) .

Moreover, we define

J_{2}^{*} : D^{*} \times B^{*} \times V \to R

J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*}) = J_{1}^{*} (v_{1}^{*}, v_{0}^{*}) + \frac{K_{1}}{2} ∥ v_{1}^{*} - (- γ \nabla^{2} + K) {u ∥}_{2}^{2} + \frac{1}{5 K_{3}^{2} α} {∥ v_{1}^{*} - (- 2 v_{0}^{*} + K) u ∥}_{2}^{2} .

Observe that

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} = O (1 / K^{2}) + K_{1} + \frac{1}{5 K_{3}^{2} α},

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} = {(- γ \nabla^{2} + K)}^{2} + \frac{1}{5 K_{3}^{2} α} {(- 2 v_{0}^{*} + K)}^{2},

and

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}} = - (- γ \nabla^{2} + K) - \frac{1}{5 K_{3}^{2} α} (- 2 v_{0}^{*} + K) .

From such results, we may obtain

\begin{matrix} det {δ_{u, v_{1}^{*}}^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})} & = & \frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} \frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}})}^{2} \\ = & \frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{5 α K_{3}^{2}} + O (\frac{K_{1}}{K^{2}}) . \end{matrix}

(204)

From such results, in particular for

K_{1} ≫ max {K, K_{3}, 1, α, β, γ, 1 / α, 1 / γ, 1 / β},

K ≫ max {K_{3}, 1, α, β, γ, 1 / α, 1 / γ, 1 / β},

and

K_{3} \approx 3,

we have that

J_{2}^{*}

is convex in

(u, v_{1}^{*})

and concave in

v_{0}^{*}

(V_{1} \times D^{*}) \times B^{*} .

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 34.1.

Let

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in V_{1} \times D^{*} \times B^{*}

be such that

δ J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses,

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(205)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} = J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}),

may be done similarly as in the previous sections and will not be repeated.

Furthermore, since

δ J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0,

and

J_{2}^{*}

is convex in

(u, v_{1}^{*})

and concave in

v_{0}^{*}

V_{1} \times D^{*} \times B^{*}

, we have

J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{1} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}),

and

J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{1} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*}) .

From such results and the Saddle Point Theorem we may infer that

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(206)

The proof is complete.

□

35. Conclusion

In the first part of this article we have developed a relaxation proposal and duality principles suitable for a large class of models in physics and engineering.

In a second part we develop duality principles for the quasi-convex envelop of some vectorial models in the calculus of variations.

We highlight such dual variational formulations established are in general convex (in fact concave).

Finally, in the last sections, we develop mathematical models for some types of chemical reactions, including the hydrogen nuclear fusion and the water hydrolysis. Among such results, we highlight our proposal of modeling the Ginzburg-Landau theory in super-conductivity as a two-phase eigenvalue approach.

Data Availability Statement

Details on the software for numerical results avaialable upon request. e-mail: fabio.botelho@ufsc.br.

Conflicts of Interest

The author declares no conflict of interest concerning this article.

References

W.R. Bielski, A. Galka, J.J. Telega, The Complementary Energy Principle and Duality for Geometrically Nonlinear Elastic Shells. I. Simple case of moderate rotations around a tangent to the middle surface. Bulletin of the Polish Academy of Sciences, Technical Sciences, Vol. 38, No. 7-9, 1988.
W.R. Bielski and J.J. Telega,A Contribution to Contact Problems for a Class of Solids and Structures, Arch. Mech., 37, 4-5, pp. 303-320, Warszawa 1985.
J.J. Telega, On the complementary energy principle in non-linear elasticity. Part I: Von Karman plates and three dimensional solids, C.R. Acad. Sci. Paris, Serie II, 308, 1193-1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, ibid, pp. 1313-1317 (1989).
A.Galka and J.J.Telega, Duality and the complementary energy principle for a class of geometrically non-linear structures. Part I. Five parameter shell model; Part II. Anomalous dual variational priciples for compressed elastic beams, Arch. Mech. 47 (1995) 677-698, 699-724.
J.F. Toland, A duality principle for non-convex optimisation and the calculus of variations, Arch. Rat. Mech. Anal., 71, No. 1 (1979), 41-61.
R.A. Adams and J.F. Fournier, Sobolev Spaces, 2nd edn. (Elsevier, New York, 2003).
F. Botelho, Functional Analysis and Applied Optimization in Banach Spaces, Springer Switzerland, 2014.
F.S. Botelho, Functional Analysis, Calculus of Variations and Numerical Methods in Physics and Engineering, CRC Taylor and Francis, Florida, 2020.
F.S. Botelho, Variational Convex Analysis, Ph.D. thesis, Virginia Tech, Blacksburg, VA -USA, (2009).
F. Botelho, Topics on Functional Analysis, Calculus of Variations and Duality, Academic Publications, Sofia, (2011).
F. Botelho, Existence of solution for the Ginzburg-Landau system, a related optimal control problem and its computation by the generalized method of lines, Applied Mathematics and Computation, 218, 11976-11989, (2012).
R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, (1970).
F.S. Botelho, Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations, Mathematics 2023, 11(1), 63. [CrossRef]
J.F. Annet, Superconductivity, Superfluids and Condensates, 2nd edn. ( Oxford Master Series in Condensed Matter Physics, Oxford University Press, Reprint, 2010).
L.D. Landau and E.M. Lifschits, Course of Theoretical Physics, Vol. 5- Statistical Physics, part 1. (Butterworth-Heinemann, Elsevier, reprint 2008).
F.S. Botelho, Advanced Calculus and its Applications in Variational Quantum Mechanics and Relativity Theory, CRC Taylor and Francis, Florida, 2021.
P.Ciarlet, Mathematical Elasticity, Vol. II – Theory of Plates, North Holland Elsevier (1997).
H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, MPS-SIAM Series in Optimization, Philadelphia, 2006.
J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, second edition (Philadelphia, 2004).
N.N. Bogoliubov and D.V. Shirkov, Introduction of the Theory of Quantized Fields, Monographs and Texts in Physics and Astronomy, Vol. III, Intercience Publishers INC., New York, USA, 1959.
G.B. Folland, Quantum Field Theory, A Tourist Guide for Mathematicians, Mathematical Surveys and Monographs, Vol. 149, Americam Mathematical Society, AMS, Providence, Rhode Island, USA, 2008.

Figure 1. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 1. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 2. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 2. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 3. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 3. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 4. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 4. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 5. Density

t (x, y)

for the Case A.

Figure 5. Density

t (x, y)

for the Case A.

Figure 6. Density

t (x, y)

for the Case B.

Figure 6. Density

t (x, y)

for the Case B.

Figure 7. Solution

u (x) = v_{3}^{*} (x) / β

for the example 1.

Figure 7. Solution

u (x) = v_{3}^{*} (x) / β

for the example 1.

Figure 8. Solution

u (x) = v_{3}^{*} (x) / β

for the example 2.

Figure 8. Solution

u (x) = v_{3}^{*} (x) / β

for the example 2.

Figure 9. Solution

u_{0} (x)

for the example A.

Figure 9. Solution

u_{0} (x)

for the example A.

Figure 10. Solution

u_{0} (x)

for the example B.

Figure 10. Solution

u_{0} (x)

for the example B.

Figure 11. Solution

ϕ_{N} (x)

for the

ω = 1.8

Figure 11. Solution

ϕ_{N} (x)

for the

ω = 1.8

Figure 12. Solution

ϕ_{S} (x)

for the

ω = 1.8

Figure 12. Solution

ϕ_{S} (x)

for the

ω = 1.8

Figure 13. Solution

ϕ_{N} (x)

for the

ω = 1.0

Figure 13. Solution

ϕ_{N} (x)

for the

ω = 1.0

Figure 14. Solution

ϕ_{S} (x)

for the

ω = 1.0

Figure 14. Solution

ϕ_{S} (x)

for the

ω = 1.0

Figure 15. Solution

ϕ_{u} (x)

for

ω = 1

Figure 15. Solution

ϕ_{u} (x)

for

ω = 1

Figure 16. Solution

ϕ_{v} (x)

for

ω = 1

Figure 16. Solution

ϕ_{v} (x)

for

ω = 1

Figure 17. Solution

ϕ_{u} (x)

for

ω = 15

Figure 17. Solution

ϕ_{u} (x)

for

ω = 15

Figure 18. Solution

ϕ_{v} (x)

for

ω = 15

Figure 18. Solution

ϕ_{v} (x)

for

ω = 15

Figure 19. Solution

u (x)

for the example B.

Figure 19. Solution

u (x)

for the example B.

Figure 20. Optimal solution

w (x)

for a simply supported beam.

Figure 20. Optimal solution

w (x)

for a simply supported beam.

Figure 21. Optimal shape solution

t (x)

for a simply supported beam.

Figure 21. Optimal shape solution

t (x)

for a simply supported beam.

Figure 22. Optimal solution

w (x)

for a bi-clamped beam.

Figure 22. Optimal solution

w (x)

for a bi-clamped beam.

Figure 23. Optimal shape solution

t (x)

for a bi-clamped beam.

Figure 23. Optimal shape solution

t (x)

for a bi-clamped beam.

Figure 24. Optimal solution

w (x, y)

for a simply supported plate.

Figure 24. Optimal solution

w (x, y)

for a simply supported plate.

Figure 25. Optimal shape solution

t (x, y)

for a simply supported plate.

Figure 25. Optimal shape solution

t (x, y)

for a simply supported plate.

Figure 26. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 26. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 27. Optimal solution

u (x)

for the case

f (x) = cos (π x) / 2

Figure 27. Optimal solution

u (x)

for the case

f (x) = cos (π x) / 2

Figure 28. Optimal solution

u (x)

for the case

f (x) = 0

Figure 28. Optimal solution

u (x)

for the case

f (x) = 0

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Abstract

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Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

Similar models on the superconductivity physics may be found in [14,15,16].

Also, for the applications in physics in the final sections, we believe to have found a path to connect the quantum approach with a more classical one in a unified framework.

Indeed, we have presented a path to model a great variety of chemical reactions through such a connection between the atomic and classical worlds.

Finally, in this text we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated.

In order to clarify the notation, here we introduce the definition of topological dual space.

Definition 1.1

U^{*}

, through a bilinear form

{〈 \cdot, \cdot 〉}_{U} : U \times U^{*} \to R

(here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given

f : U \to R

linear and continuous, we assume the existence of a unique

u^{*} \in U^{*}

such that

\begin{matrix} f (u) = {〈 u, u^{*} 〉}_{U}, \forall u \in U . \end{matrix}

(1)

The norm of f , denoted by

{∥ f ∥}_{U^{*}}

, is defined as

\begin{matrix} {∥ f ∥}_{U^{*}} = sup_{u \in U} {| {〈 u, u^{*} 〉}_{U} {| : ∥ u ∥}_{U} \leq 1} \equiv ∥ u^{*} ∥_{U^{*}} . \end{matrix}

(2)

At this point we start to describe the primal and dual variational formulations.

2. A general duality principle non-convex optimization

In this section we present a duality principle applicable to a model in phase transition.

This case corresponds to the vectorial one in the calculus of variations.

Let

Ω \subset R^{n}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u_{1}, \dots, \nabla u_{N}) + G (u_{1}, \dots, u_{N}) - {〈 u_{i}, h_{i} 〉}_{L^{2}},

and where

F (\nabla u_{1}, \dots, \nabla u_{N}) = \int_{Ω} f (\nabla u_{1}, \dots, \nabla u_{N}) d x

f : R^{N \times n} \to R

is a three times Fréchet differentiable function not necessarily convex. Moreover,

V = {u = (u_{1}, \dots, u_{N}) \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω},

h = (h_{1}, \dots, h_{N}) \in L^{2} (Ω; R^{N}),

and

1 < p < + \infty .

We assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Define

V_{0} = W_{0}^{1, 2} (Ω; R^{N})

and

V_{0} (u) = {ϕ \in V_{0} : supp ϕ \subset B (u)},

where

B (u) = {x \in Ω : f^{* *} (\nabla u (x)) < f (\nabla u (x))} .

Moreover,

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{N \times n})

Y_{2} = Y_{2}^{*} = L^{2} (Ω; R^{N \times n})

Y_{3} = Y_{3}^{*} = L^{2} (Ω; R^{N}),

so that at this point we define,

F_{1} : V \times V_{0} \to R

G_{1} : V \to R

G_{2} : V \to R

G_{3} : V_{0} \to R

and

G_{4} : V \to R,

\begin{matrix} F_{1} (u, ϕ) & = & F (\nabla u_{1} + \nabla ϕ_{1}, \dots, \nabla u_{N} + \nabla ϕ_{N}) + \frac{K}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x \\ + \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x \end{matrix}

(3)

and

G_{1} (u_{1}, \dots, u_{n}) = G (u_{1}, \dots, u_{N}) + \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x - {〈 u_{i}, f_{i} 〉}_{L^{2}},

G_{2} (\nabla u_{1}, \dots, \nabla u_{N}) = \frac{K_{1}}{2} \int_{Ω} \nabla u_{j} \cdot \nabla u_{j} d x,

G_{3} (\nabla ϕ_{1}, \dots, \nabla ϕ_{N}) = \frac{K_{2}}{2} \int_{Ω} \nabla ϕ_{j} \cdot \nabla ϕ_{j} d x,

and

G_{4} (u_{1}, \dots, u_{N}) = \frac{K_{1}}{2} \int_{Ω} u_{j} u_{j} d x .

Define now

J_{1} : V \times V_{0} \to R

J_{1} (u, ϕ) = F (\nabla u + \nabla ϕ) + G (u) - {〈 u_{i}, h_{i} 〉}_{L^{2}} .

Observe that

\begin{matrix} J_{1} (u, ϕ) & = & F_{1} (u, ϕ) + G_{1} (u) - G_{2} (\nabla u) - G_{3} (\nabla ϕ) - G_{4} (u) \\ \leq & F_{1} (u, ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + sup_{v_{1} \in Y_{1}} {{〈 v_{1}, z_{1}^{*} 〉}_{L^{2}} - G_{2} (v_{1})} \\ + sup_{v_{2} \in Y_{2}} {{〈 v_{2}, z_{2}^{*} 〉}_{L^{2}} - G_{3} (v_{2})} \\ + sup_{u \in V} {{〈 u, z_{3}^{*} 〉}_{L^{2}} - G_{4} (u)} \\ = & F_{1} (u, ϕ) + G_{1} (u) - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}) \\ = & J_{1}^{*} (u, ϕ, z^{*}), \end{matrix}

(4)

\forall u \in V, ϕ \in V_{0} (u), z^{*} = (z_{1}^{*}, z_{2}^{*}, z_{3}^{*}) \in Y^{*} = Y_{1}^{*} \times Y_{2}^{*} \times Y_{3}^{*}

From the general results in [5], we may infer that

\begin{matrix} inf_{(u, ϕ) \in V \times V_{0} (u)} J (u, ϕ) & = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) . \end{matrix}

(5)

On the other hand

inf_{u \in V} J (u) \geq inf_{(u, ϕ) \in V \times V_{0} (u)} J_{1} (u, ϕ) .

From these last two results we may obtain

inf_{u \in V} J (u) \geq inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) .

Moreover, from standards results on convex analysis, we may have

\begin{matrix} inf_{u \in V} J_{1}^{*} (u, ϕ, z^{*}) & = & inf_{u \in V} {F_{1} (u, ϕ) + G_{1} (u) \\ - {〈 \nabla u, z_{1}^{*} 〉}_{L^{2}} - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} - {〈 u, z_{3}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})} \\ = & sup_{(v_{1}^{*}, v_{2}^{*}) \in C^{*}} {- F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} \\ + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*})}, \end{matrix}

(6)

where

C^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in Y_{1}^{*} \times Y_{3}^{*} : - div {(v_{1}^{*})}_{i} + {(v_{2}^{*})}_{i} = 0, \forall i \in {1, \dots, N}},

F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) = sup_{u \in V} {{〈 u, - div (z_{1}^{*} + v_{1}^{*}) 〉}_{L^{2}} - F_{1} (u, ϕ)},

and

G_{1}^{*} (v_{2}^{*} + z_{2}^{*}) = sup_{u \in V} {{〈 u, v_{2}^{*} + z_{2}^{*} 〉}_{L^{2}} - G_{1} (u)} .

Thus, defining

J_{2}^{*} (ϕ, z^{*}, v^{*}) = F_{1}^{*} (v_{1}^{*} + z_{1}^{*}, ϕ) - G_{1}^{*} (v_{2}^{*} + z_{3}^{*}) - {〈 \nabla ϕ, z_{2}^{*} 〉}_{L^{2}} + G_{2}^{*} (z_{1}^{*}) + G_{3}^{*} (z_{2}^{*}) + G_{4}^{*} (z_{3}^{*}),

we have got

\begin{matrix} inf_{u \in V} J (u) & \geq & inf_{(u, ϕ) \in V \times V_{0}} J_{1} (u, ϕ) \\ = & inf_{(u, ϕ, z^{*}) \in V \times V_{0} (u) \times Y^{*}} J_{1}^{*} (u, ϕ, z^{*}) \\ = & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0}} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} . \end{matrix}

(7)

Finally, observe that

\begin{matrix} inf_{u \in V} J (u) \\ \geq & inf_{z^{*} \in Y^{*}} \{inf_{ϕ \in V_{0} (u)} \{sup_{v^{*} \in C^{*}} J_{2}^{*} (ϕ, z^{*}, v^{*})\}\} \\ \geq & sup_{v^{*} \in C^{*}} \{inf_{(z^{*}, ϕ) \in Y^{*} \times V_{0} (u)} J_{2}^{*} (ϕ, z^{*}, v^{*})\} . \end{matrix}

(8)

This last variational formulation corresponds to a concave relaxed formulation in

v^{*}

concerning the original primal formulation.

3. Another duality principle for a simpler related model in phase transition with a respective numerical example

In this section we present another duality principle for a related model in phase transition.

Let

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

We intend to use duality theory to approximately solve such a global optimization problem.

Denoting

V_{0} = W_{0}^{1, 4} (Ω)

, at this point we define,

F : V \to R

and

F_{1} : V \times V_{0} \to R

F (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x,

and

F_{1} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x .

Observe that

F (u) \geq inf_{ϕ \in V_{0}} F_{1} (u, ϕ), \forall u \in V .

In order to restrict the action of

ϕ

on the region where the primal functional is non-convex, we redefine a not relabeled

V_{0} = \{ϕ \in W_{0}^{1, 4} (Ω) : {(ϕ^{'})}^{2} - 1 \leq 0, in Ω\}

and define also

F_{2} : V \times V_{0} \to R,

F_{3} : V \times V_{0} \to R

and

G : V \times V_{0} \to R

F_{2} (u, ϕ) = \frac{1}{2} \int_{Ω} {({(u^{'} + ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

\begin{matrix} F_{3} (u, ϕ) & = & F_{2} (u, ϕ) + \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(9)

and

\begin{matrix} G (u, ϕ) & = & \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x \\ + \frac{K_{1}}{2} \int_{Ω} {(ϕ^{'})}^{2} d x \end{matrix}

(10)

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

G^{*} : Y^{*} \times Y^{*} \to R

G^{*} (v^{*}, v_{0}^{*}) = sup_{(u, ϕ) \in V \times V_{0}} {{〈 u, v^{*} 〉}_{L^{2}} + {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} - G (u, ϕ)} .

Observe that

inf_{u \in U} J (u) \geq inf_{((u, ϕ), (v^{*}, v_{0}^{*})) \in V \times V_{0} \times {[Y^{*}]}^{2}} {G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ)} .

With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by

J_{1}^{*} : V \times V_{0} \times {[Y^{*}]}^{2} \to R

, where

J_{1}^{*} (u, ϕ, v^{*}, v_{0}^{*}) = G^{*} (v^{*}, v_{0}^{*}) - {〈 u, v^{*} 〉}_{L^{2}} - {〈 ϕ, v_{0}^{*} 〉}_{L^{2}} + F_{3} (u, ϕ) .

{(ϕ^{'})}^{2} - 1 \leq 0 in Ω

1.: Set $K \approx 0.1$ and $K_{1} = 120.0$ and $0 < ε ≪ 1 .$
2.: Choose $(u_{1}, ϕ_{1}) \in V \times V_{0},$ such that $∥ u_{1} ∥_{1, \infty} < 1$ and $∥ ϕ_{1} ∥_{1, \infty} < 1 .$
3.: Set $n = 1 .$
4.: Calculate $(v_{n}^{*}, {(v_{0}^{*})}_{n})$ solution of the system of equations:

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n}, ϕ_{n}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} = 0,$

that is

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v^{*}} - u_{n} = 0$

and

$\frac{\partial G^{*} (v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial v_{0}^{*}} - ϕ_{n} = 0$

so that

$v_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial u}$

and

${(v_{0}^{*})}_{n}^{*} = \frac{\partial G (u_{n}, ϕ_{n})}{\partial ϕ}$
5.: Calculate $(u_{n + 1}, ϕ_{n + 1})$ by solving the system of equations:

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial u} = 0$

and

$\frac{\partial J_{1}^{*} (u_{n + 1}, ϕ_{n + 1}, v_{n}^{*}, {(v_{0}^{*})}_{n})}{\partial ϕ} = 0$

that is

$- v_{n}^{*} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial u} = 0$

and

$- {(v_{0}^{*})}_{n} + \frac{\partial F_{3} (u_{n + 1}, ϕ_{n + 1})}{\partial ϕ} = 0$
6.: If $max {∥ u_{n} - u_{n + 1} ∥_{\infty}, ∥ ϕ_{n + 1} - ϕ_{n} ∥_{\infty}} \leq ε$ , then stop, else set $n : = n + 1$ and go to item 4.

At this point, we present the corresponding software in MAT-LAB, in finite differences and based on the one-dimensional version of the generalized method of lines.

Here the software.

***********************

1.: clear all

m8=300;

d=1/m8;

K=0.1;

K1=120;

for i=1:m8

$u o (i, 1) = i^{2} * d / 2;$

vo(i,1)=i*d/10;

yo(i,1)=sin(i*d*pi)/2;

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4.3})$ and $(k < 230000)$

k=k+1;

for i=1:m8-1

duo(i,1)=(uo(i+1,1)-uo(i,1))/d;

dvo(i,1)=(vo(i+1,1)-vo(i,1))/d;

end;

m9=zeros(2,2);

m9(1,1)=1;

i=1;

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2;$

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2};$

m50(:,:,i)=m80(:,:,i)*inv(m12);

z(:,i)=inv(m12)*y11(:,i)* $d^{2}$ ;

for i=2:m8-1

$f 1 = 6 * {(d u o (i, 1) + d v o (i, 1))}^{2} - 2$ ;

m80(1,1,i)=-f1-K;

m80(1,2,i)=-f1;

m80(2,1,i)=-f1;

m80(2,2,i)=-f1-K1;

$y 11 (1, i) = K * (u o (i + 1, 1) - 2 * u o (i, 1) + u o (i - 1, 1)) / d^{2} - y o (i, 1);$

$y 11 (2, i) = K 1 * (v o (i + 1, 1) - 2 * v o (i, 1) + v o (i - 1, 1)) / d^{2};$

$m 12 = 2 * m 80 (:, :, i) - m 9 * d^{2} - m 80 (:, :, i) * m 50 (:, :, i - 1);$

m50(:,:,i)=inv(m12)*m80(:,:,i);

$z (:, i) = i n v (m 12) * (y 11 (:, i) * d^{2} + m 80 (:, :, i) * z (:, i - 1));$

end;

U(1,m8)=1/2;

U(2,m8)=0.0;

for i=1:m8-1

U(:,m8-i)=m50(:,:,m8-i)*U(:,m8-i+1)+z(:,m8-i);

end;

for i=1:m8

u(i,1)=U(1,i);

v(i,1)=U(2,i);

end;

b12=max(abs(u-uo))

uo=u;

vo=v;

u(m8/2,1)

end;

for i=1:m8

y(i)=i*d;

end;

plot(y,uo)

**************************************

For the case in which

f (x) = 0

, we have obtained numerical results for

K = 0.1

and

K_{1} = 120

. For such a concerning solution

u_{0}

obtained, please see Figure 1. For the case in which

f (x) = sin (π x) / 2

, we have obtained numerical results also for

K = 0.1

and

K_{1} = 120

. For such a concerning solution

u_{0}

obtained, please see Figure 2.

Remark 3.1.

3.1. A general proposal for relaxation

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (\nabla u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}},

where

V = \{u \in W^{1, 4} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (Ω; R^{N}),

f_{1} \in L^{2} (Ω; R^{N})

G : V \to R

is convex and Fréchet differentiable, and

F (\nabla u) = \int_{Ω} f (\nabla u) d x,

where

f : R^{N \times n} \to R

is also Fréchet differentiable.

Assume there exists

\hat{N} \in N

such that

W_{h} \equiv \{y \in R^{N \times n} : f^{* *} (y) < f (y)\} = \cup_{j = 1}^{\hat{N}} W_{j}

where for each

j \in {1, \dots, \hat{N}}

W_{j} \subset R^{N \times n}

is an open connected set such that

\partial W_{j}

is regular. We also suppose

\bar{W_{j}} \cap \bar{W_{k}} = \emptyset, \forall j \neq k .

Define

{\hat{W}}_{j} = \{v_{j} \in W_{0}^{1, 4} (Ω; R^{N}); \nabla v_{j} (x) \in W_{j}, a . e . in Ω\}

and define also

W = \{v = (v_{1}, \dots, v_{\hat{N}}) : v_{j} \in {\hat{W}}_{j} \forall j \in {1, \dots, \hat{N}} and supp v_{j} \cap supp v_{k} = \emptyset, \forall j \neq k\} .

At this point we define

h_{5} (u (x), v (x)) = \{\begin{matrix} f (\nabla u (x) + \nabla v_{j} (x)), & if \nabla u (x) \in W_{j}, \\ f (\nabla u (x)), & if \nabla u (x) \notin W_{h}, \end{matrix}

(11)

and

H (u) = inf_{v \in W_{u}} \int_{Ω} h_{5} (u, v) d x,

where

W_{u} = {v \in W : \nabla u (x) + \nabla v_{j} (x) \in W_{j}, if \nabla u (x) \in W_{j}, a . e . in Ω, \forall j \in {1, \dots, \hat{N}}} .

Moreover, we propose the relaxed functional

J_{1} (u) = H (u) + G (u) - {〈 u, f_{1} 〉}_{L^{2}} .

Observe that clearly

inf_{u \in V} J_{1} (u) \leq inf_{u \in V} J (u) .

4. A convex dual variational formulation for a third similar model

In this section we present another duality principle for a third related model in phase transition.

Let

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and where

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2}

and

f \in L^{2} (Ω) .

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close to the infimum of J.

We intend to use the duality theory to solve such a global optimization problem in an appropriate sense to be specified.

At this point we define,

F : V \to R

and

G : V \to R

\begin{matrix} F (u) & = & \frac{1}{2} \int_{Ω} min {{(u^{'} - 1)}^{2}, {(u^{'} + 1)}^{2}} d x \\ = & \frac{1}{2} \int_{Ω} {(u^{'})}^{2} d x - \int_{Ω} | u^{'} | d x + 1 / 2 \\ \equiv & F_{1} (u^{'}), \end{matrix}

(12)

and

G (u) = \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functional

F_{1}^{*} : Y^{*} \to R

and

G^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v^{*}) & = & sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - F_{1} (v)} \\ = & \frac{1}{2} \int_{Ω} {(v^{*})}^{2} d x + \int_{Ω} | v^{*} | d x, \end{matrix}

(13)

and

\begin{matrix} G^{*} ({(v^{*})}^{'}) & = & sup_{u \in V} {- {〈 u^{'}, v^{*} 〉}_{L^{2}} - G (u)} \\ = & \frac{1}{2} \int_{Ω} {({(v^{*})}^{'} + f)}^{2} d x - \frac{1}{2} v^{*} (1) . \end{matrix}

(14)

Observe this is the scalar case of the calculus of variations, so that from the standard results on convex analysis, we have

inf_{u \in V} J (u) = max_{v^{*} \in Y^{*}} {- F_{1}^{*} (v^{*}) - G^{*} (- {(v^{*})}^{'})} .

Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some

{\hat{v}}^{*} \in Y^{*}

Moreover, the corresponding solution

u_{0} \in V

is obtained from the equation

u_{0} = \frac{\partial G ({({\hat{v}}^{*})}^{'})}{\partial {(v^{*})}^{'}} = {({\hat{v}}^{*})}^{'} + f .

Finally, the Euler-Lagrange equations for the dual problem stands for

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) + f (0) = 0, {(v^{*})}^{'} (1) + f (1) = 1 / 2, \end{matrix}

(15)

where

sign (v^{*} (x)) = 1

v^{*} (x) > 0,

sign (v^{*} (x)) = - 1,

v^{*} (x) < 0

and

- 1 \leq sign (v^{*} (x)) \leq 1,

v^{*} (x) = 0 .

We have computed the solutions

v^{*}

and corresponding solutions

u_{0} \in V

for the cases in which

f (x) = 0

and

f (x) = sin (π x) / 2 .

For the solution

u_{0} (x)

for the case in which

f (x) = 0

, please see Figure 3.

For the solution

u_{0} (x)

for the case in which

f (x) = sin (π x) / 2

, please see Figure 4.

Remark 4.1.

Observe that such solutions

u_{0}

obtained are not the global solutions for the related primal optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

4.1. The algorithm through which we have obtained the numerical results

In this subsection we present the software in MATLAB through which we have obtained the last numerical results.

This algorithm is for solving the concerning Euler-Lagrange equations for the dual problem, that is, for solving the equation

\{\begin{matrix} {(v^{*})}^{''} + f^{'} - v^{*} - sign (v^{*}) = 0, & in Ω, \\ {(v^{*})}^{'} (0) = 0, {(v^{*})}^{'} (1) = 1 / 2 . \end{matrix}

(16)

Here the concerning software in MATLAB. We emphasize to have used the smooth approximation

| v^{*} | \approx \sqrt{{(v^{*})}^{2} + e_{1}},

where a small value for

e_{1}

is specified in the next lines.

*************************************

1.: clear all
2.: $m_{8} = 800;$ (number of nodes)
3.: $d = 1 / m_{8};$
4.: $e_{1} = 0.00001;$
5.: $f o r i = 1 : m_{8}$

$y o (i, 1) = 0.01;$

$y_{1} (i, 1) = sin (π * i / m_{8}) / 2;$

$e n d;$
6.: $f o r i = 1 : m_{8} - 1$

$d y_{1} (i, 1) = (y_{1} (i + 1, 1) - y_{1} (i, 1)) / d;$

$e n d;$
7.: $f o r k = 1 : 3000$ (we have fixed the number of iterations)

$i = 1;$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 1 + d^{2} * h_{3} + d^{2};$

$m_{50} (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (d y_{1} (i, 1) * d^{2});$
8.: $f o r i = 2 : m_{8} - 1$

$h_{3} = 1 / \sqrt{v o {(i, 1)}^{2} + e_{1}};$

$m_{12} = 2 + h_{3} * d^{2} + d^{2} - m 50 (i - 1);$

$m 50 (i) = 1 / m_{12};$

$z (i) = m_{50} (i) * (z (i - 1) + d y_{1} (i, 1) * d^{2});$

$e n d;$
9.: $v (m_{8}, 1) = (d / 2 + z (m_{8} - 1)) / (1 - m_{50} (m_{8} - 1));$
10.: $f o r i = 1 : m_{8} - 1$

$v (m_{8} - i, 1) = m_{50} (m_{8} - i) * v (m_{8} - i + 1) + z (m_{8} - i);$

$e n d;$
11.: $v (m_{8} / 2, 1)$
12.: $v o = v;$

$e n d;$
13.: $f o r i = 1 : m_{8} - 1$

$u (i, 1) = (v (i + 1, 1) - v (i, 1)) / d + y_{1} (i, 1);$

$e n d;$
14.: $f o r i = 1 : m_{8} - 1$

$x (i) = i * d;$

$e n d;$

$p l o t (x, u (:, 1))$

********************************

5. An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation

In this section we develop an approximate primal dual formulation suitable for a large class of variational models.

Here, the applications are for the Kirchhoff-Love plate model, which may be found in Ciarlet, [17].

At this point we start to describe the primal variational formulation.

Let

Ω \subset R^{2}

be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of

Ω

, which is assumed to be regular (Lipschitzian), is denoted by

\partial Ω

. The vectorial basis related to the cartesian system

{x_{1}, x_{2}, x_{3}}

is denoted by

(a_{α}, a_{3})

, where

α = 1, 2

(in general Greek indices stand for 1 or 2), and where

a_{3}

is the vector normal to

Ω

, whereas

a_{1}

and

a_{2}

are orthogonal vectors parallel to

Ω .

Also,

n

is the outward normal to the plate surface.

The displacements will be denoted by

\hat{u} = {{\hat{u}}_{α}, {\hat{u}}_{3}} = {\hat{u}}_{α} a_{α} + {\hat{u}}_{3} a_{3} .

The Kirchhoff-Love relations are

\begin{matrix} {\hat{u}}_{α} (x_{1}, x_{2}, x_{3}) = u_{α} (x_{1}, x_{2}) - x_{3} w {(x_{1}, x_{2})}_{, α} \\ and {\hat{u}}_{3} (x_{1}, x_{2}, x_{3}) = w (x_{1}, x_{2}) . \end{matrix}

(17)

Here

- h / 2 \leq x_{3} \leq h / 2

so that we have

u = (u_{α}, w) \in U

where

\begin{matrix} U & = & \{u = (u_{α}, w) \in W^{1, 2} (Ω; R^{2}) \times W^{2, 2} (Ω), \\ u_{α} = w = \frac{\partial w}{\partial n} = 0 on \partial Ω\} \\ = & W_{0}^{1, 2} (Ω; R^{2}) \times W_{0}^{2, 2} (Ω) . \end{matrix}

It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

We also define the operator

Λ : U \to Y \times Y

, where

Y = Y^{*} = L^{2} (Ω; R^{2 \times 2})

, by

Λ (u) = {γ (u), κ (u)},

γ_{α β} (u) = \frac{u_{α, β} + u_{β, α}}{2} + \frac{w_{, α} w_{, β}}{2},

κ_{α β} (u) = - w_{, α β} .

The constitutive relations are given by

N_{α β} (u) = H_{α β λ μ} γ_{λ μ} (u),

(18)

M_{α β} (u) = h_{α β λ μ} κ_{λ μ} (u),

(19)

where:

{H_{α β λ μ}}

and

\{h_{α β λ μ} = \frac{h^{2}}{12} H_{α β λ μ}\}

, are symmetric positive definite fourth order tensors. From now on, we denote

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}^{- 1}

and

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

Furthermore

{N_{α β}}

denote the membrane force tensor and

{M_{α β}}

the moment one. The plate stored energy, represented by

(G \circ Λ) : U \to R

is expressed by

(G \circ Λ) (u) = \frac{1}{2} \int_{Ω} N_{α β} (u) γ_{α β} (u) d x + \frac{1}{2} \int_{Ω} M_{α β} (u) κ_{α β} (u) d x

(20)

and the external work, represented by

F : U \to R

, is given by

F (u) = {〈 w, P 〉}_{L^{2}} + {〈 u_{α}, P_{α} 〉}_{L^{2}},

(21)

where

P, P_{1}, P_{2} \in L^{2} (Ω)

are external loads in the directions

a_{3}

a_{1}

and

a_{2}

respectively. The potential energy, denoted by

J : U \to R

is expressed by:

J (u) = (G \circ Λ) (u) - F (u)

Define now

J_{3} : \tilde{U} \to R

J_{3} (u) = J (u) + J_{5} (w) .

where

J_{5} (w) = 10 \int_{Ω} \frac{a^{K b w}}{ln (a) K^{3 / 2}} d x + 10 \int_{Ω} \frac{a^{- K (b w - 1 / 100)}}{ln (a) K^{3 / 2}} d x .

In such a case for

a = 2.71

K = 185

b = P / | P |

Ω

and

\tilde{U} = {{u \in U : ∥ w ∥}_{\infty} \leq 0.01 and P w \geq 0 a . e . in Ω},

we get

\begin{matrix} \frac{\partial J_{3} (u)}{\partial w} & = & \frac{\partial J (u)}{\partial w} + \frac{\partial J_{5} (u)}{\partial w} \\ \approx & \frac{\partial J (u)}{\partial w} + O (\pm 3.0), \end{matrix}

(22)

and

\begin{matrix} \frac{\partial^{2} J_{3} (u)}{\partial w^{2}} & = & \frac{\partial^{2} J (u)}{\partial w^{2}} + \frac{\partial^{2} J_{5} (u)}{\partial w^{2}} \\ \approx & \frac{\partial^{2} J (u)}{\partial w^{2}} + O (850) . \end{matrix}

(23)

This new functional

J_{3}

has a relevant improvement in the convexity conditions concerning the previous functional J.

Indeed, we have obtained a gain in positiveness for the second variation

\frac{\partial^{2} J (u)}{\partial w^{2}},

which has increased of order

O (700 - 1000) .

Moreover the difference between the approximate and exact equation

\frac{\partial J (u)}{\partial w} = 0

is of order

O (\pm 3.0)

which corresponds to a small perturbation in the original equation for a load of

P = 1500 N / m^{2},

for example. Summarizing, the exact equation may be approximately solved in an appropriate sense.

5.1. A duality principle for the concerning quasi-convex envelope

In this section, denoting

V_{1} = {ϕ = ϕ (x, y) \in W^{1, 2} (Ω \times Ω; R^{2}) : ϕ = 0 on Ω \times \partial Ω},

we define the functional

J_{1} : U \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} . \end{matrix}

(24)

where

G_{1} ({w_{, α β}}) = \frac{1}{2} \int_{Ω} h_{α β λ μ} w_{, α β} w_{, λ μ} d x

and,

\begin{matrix} G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{α β λ μ} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} (x, y) + \frac{1}{2} w_{, α} w_{, β}) \\ \times (\frac{1}{2} (u_{λ, μ} + u_{λ, μ}) + ϕ_{λ, y_{μ}} (x, y) + \frac{1}{2} w_{, λ} w_{, μ}) d x d y \end{matrix}

We define also

J_{2} ({u_{α}}, ϕ) = inf_{w \in W_{0}^{2, 2} (Ω)} J_{1} (u, ϕ),

and

J_{3} ({u_{α}}) = inf_{ϕ \in V_{1}} J_{2} ({u_{α}}, ϕ) .

It is a well known result from the modern Calculus of Variations theory (please, see [18] for details) that

inf_{u \in U} J (u) = inf_{{u_{α}} \in W_{0}^{1, 2} (Ω; R^{2})} J_{3} ({u_{α}}) .

At this point we denote

Y_{1} = Y_{1}^{*} = Y_{3} = Y_{3}^{*} \equiv L^{2} (Ω \times Ω; R^{4})

and

Y_{2} = Y_{2}^{*} \equiv L^{2} (Ω \times Ω; R^{2}) .

Observe that

\begin{matrix} J (u) \\ = & G_{1} ({w_{, α β}}) + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - {〈 w, P 〉}_{L^{2}} - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ = & G_{1} ({w_{, α β}}) - {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + {〈 w_{, α β}, M_{α β} 〉}_{L^{2}} \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}} \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x), Q_{α} (x, y) d x d y + G_{2} (\{\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}\}) \\ - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} w_{, α} w_{, β}), v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} \\ \geq & inf_{v_{3} \in Y_{3}} {- {〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} + G_{1} ({(v_{3})}_{α β})} \\ + inf_{w \in W_{0}^{2, 2} (Ω)} \{{〈 w_{, α β}, M_{α β} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} w_{, α} (x) Q_{α} (x, y) d x d y - {〈 w, P 〉}_{L^{2}}\} \\ + inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{α β} v_{α β}^{*} d x d y + G_{2} ({v_{α β}})\} \\ + inf_{(v_{2}, {u_{α}}) \in Y_{2} \times W_{0}^{1, 2} (Ω; R^{2})} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{1}{2} (u_{α, β} + u_{β, α}) + ϕ_{α, y_{β}} + \frac{1}{2} {(v_{2})}_{α} (x, y) {(v_{2})}_{β} (x, y)) \\ \times v_{α β}^{*} (x, y) d x d y - {〈 u_{α}, P_{α} 〉}_{L^{2}} + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{α} (x, y) Q_{α} (x, y) d x d y\} \\ \geq & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

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\forall u \in U, (M, Q) \in C^{*}

v = {v_{α β}} \in A^{*}

where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap B^{*},

A_{1}^{*} = {{v_{α β}^{*}} \in Y_{1}^{*} : {(v_{α β}^{*})}_{, y_{β}} = 0, in Ω},

A_{2}^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} v_{α β}^{*} d y)}_{, x_{β}} + P_{α} = 0, in Ω\},

B^{*} = \{{v_{α β}^{*}} \in Y_{1}^{*} : \{v_{α β}^{*} (x, y)\} is positive definite in Ω \times Ω\} .

and

C^{*} = \{(M, Q) \in Y_{3}^{*} \times Y_{2}^{*} : M_{α β, α β} - {(\int_{Ω} Q_{α} d y)}_{, x_{α}} - P = 0, in Ω\} .

Also

\{\bar{v_{α β}^{*}}\} = {\{v_{α β}^{*}\}}^{- 1},

and

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}

in an appropriate tensor sense.

Here it is worth highlighting we have denoted,

\begin{matrix} G_{1}^{*} (M) & = & sup_{v_{3} \in Y_{3}} {{〈 {(v_{3})}_{α β}, M_{α β} 〉}_{L^{2}} - G_{1} (v_{3})} \\ = & \frac{1}{2} \int_{Ω} {\bar{h}}_{α β λ μ} M_{α β} M_{λ μ} d x, \end{matrix}

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where we recall that

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

in an appropriate tensorial sense.

Summarizing, defining

J^{*} : C^{*} \times A^{*} \to R

\begin{matrix} J^{*} ((M, Q), v^{*}) & = & - G_{1}^{*} (M) - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} (\bar{v_{α β}^{*}}) Q_{α} Q_{β} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{α β λ μ} v_{α β}^{*} v_{λ μ}^{*} d x d y, \end{matrix}

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we have got

inf_{u \in U} J (u) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) .

Remark 5.1.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize such results are extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if

((M_{0}, Q_{0}), v_{0}^{*}) \in C^{*} \times A^{*}

is such that

δ J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(w_{0})}_{, α β} = {\bar{h}}_{α β λ μ} {(M_{0})}_{λ μ},

and

{(v_{0}^{*})}_{α β} = H_{α β λ} (\frac{{(u_{0})}_{λ, μ} + {(u_{0})}_{μ, λ}}{2} + \frac{{(ϕ_{0})}_{λ, y_{μ}} + {(ϕ_{0})}_{μ, y_{λ}}}{2} + \frac{1}{2} {(v_{2_{0}})}_{λ} {(v_{2_{0}})}_{μ}),

{(v_{0}^{*})}_{α β, y_{β}} = 0,

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{((M, Q), v^{*}) \in C^{*} \times A^{*}} J^{*} ((M, Q), v^{*}) \\ = & J^{*} ((M_{0}, Q_{0}), v_{0}^{*}) \\ = & inf_{u \in V} J (u) . \end{matrix}

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Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 0 .

Here we highlight such last results are valid just for this finite-dimensional model version.

6. A duality principle for a related relaxed formulation concerning the vectorial approach in the calculus of variations

In this section we develop a duality principle for a related vectorial model in the calculus of variations.

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ .

For

1 < p < + \infty

, consider a functional

J : V \to R

where

J (u) = G (\nabla u) + F (u) - {〈 u, f 〉}_{L^{2}},

where

V = \{u \in W^{1, p} (Ω; R^{N}) : u = u_{0} on \partial Ω\},

u_{0} \in C^{1} (\bar{Ω}; R^{N})

and

f \in L^{2} (Ω; R^{N}) .

We assume

G : Y \to R

and

F : V \to R

are Fréchet differentiable and F is also convex.

Also

G (\nabla u) = \int_{Ω} g (\nabla u) d x,

where

g : R^{N \times n} \to R

it is supposed to be Fréchet differentiable. Here we have denoted

Y = L^{p} (Ω; R^{N \times n})

We define also

J_{1} : V \times Y_{1} \to R

J_{1} (u, ϕ) = G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}},

where

Y_{1} = W^{1, p} (Ω \times Ω; R^{N})

and

G_{1} (\nabla u + \nabla_{y} ϕ) = \frac{1}{| Ω |} \int_{Ω} \int_{Ω} g (\nabla u (x) + \nabla_{y} ϕ (x, y)) d x d y .

Moreover, we define the relaxed functional

J_{2} : V \to R

J_{2} (u) = inf_{ϕ \in V_{0}} J_{1} (u, ϕ),

where

V_{0} = {ϕ \in Y_{1} : ϕ (x, y) = 0, on Ω \times \partial Ω} .

Now observe that

\begin{matrix} J_{1} (u, ϕ) & = & G_{1} (\nabla u + \nabla_{y} ϕ) + F (u) - {〈 u, f 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + G_{1} (\nabla u + \nabla_{y} ϕ) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{2}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x + G_{1} (v)\} \\ + inf_{(v, ϕ) \in V \times V_{0}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x + F (u) - {〈 u, f 〉}_{L^{2}}\} \\ = & - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) \\ + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ, \end{matrix}

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\forall (u, ϕ) \in V \times V_{0}, v^{*} \in A^{*}

, where

A^{*} = {v^{*} \in Y_{2}^{*} : {div}_{y} v^{*} (x, y) = 0, in Ω} .

Here we have denoted

G_{1}^{*} (v^{*}) = sup_{v \in Y_{2}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot v (x, y) d y d x - G_{1} (v)\},

where

Y_{2} = L^{p} (Ω \times Ω; R^{N \times n}),

Y_{2}^{*} = L^{q} (Ω \times Ω; R^{N \times n}),

and where

\frac{1}{p} + \frac{1}{q} = 1 .

Furthermore, for

v^{*} \in A^{*}

, we have

\begin{matrix} F^{*} ({div}_{x} (\frac{1}{| Ω |} \int_{Ω} v^{*} (x, y) d y) + f) - \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ \\ = & sup_{(v, ϕ) \in V \times V_{0}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v^{*} (x, y) \cdot (\nabla u + \nabla_{y} ϕ (x, y)) d y d x - F (u) + {〈 u, f 〉}_{L^{2}}\}, \end{matrix}

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Therefore, denoting

J_{3}^{*} : Y_{2}^{*} \to R

J_{3}^{*} (v^{*}) = - G_{1}^{*} (v^{*}) - F^{*} ({div}_{x} (\int_{Ω} v^{*} (x, y) d y) + f) + \frac{1}{| Ω |} \int_{\partial Ω} (\int_{Ω} v^{*} (x, y) d y) \otimes n u_{0} d Γ,

we have got

inf_{u \in V} J_{2} (u) \geq sup_{v^{*} \in A^{*}} J_{3}^{*} (v^{*}) .

Finally, we highlight such a dual functional

J_{3}^{*}

is convex (in fact concave).

6.1. An example in finite elasticity

In this section we develop an application of results obtained in the last section to a model in non-linear elasticity.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Concerning a standard model in non-linear elasticity, consider a functional

J : V \to R

where

\begin{matrix} J (u) \\ = & \frac{1}{2} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{1}{2} u_{m, i} u_{m, j}) (\frac{u_{k, l} + u_{j, i}}{2} + \frac{1}{2} u_{m, k} u_{m, l}) d x \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}} \end{matrix}

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where

f \in L^{2} (Ω; R^{3})

and

V = W_{0}^{1, 2} (Ω; R^{3}) .

Here

{H_{i j k l}}

is a fourth-order and positive definite symmetric tensor (in an appropriate standard sense). Moreover,

u = (u_{1}, u_{2}, u_{3}) \in V

is a field of displacements resulting from the f load field action on the volume comprised by

Ω

At this point, we define the functional

J_{1} : V \times V_{1} \to R

, where

\begin{matrix} J_{1} (u, ϕ) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

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where

V_{1} = {ϕ \in W^{1, 2} (Ω \times Ω; R^{3}) : ϕ = 0 on Ω \times \partial Ω} .

We define also the quasi-convex envelop of J, denoted by

Q_{J} : V \to R

, as

Q_{J} (u) = inf_{ϕ \in V_{1}} J_{1} (u, ϕ) .

It is a well known result from the modern calculus of variations theory (please see [18] for details), that

inf_{u \in V} J (u) = inf_{u \in V} Q_{J} (u) .

Observe now that, denoting

Y_{1} = Y_{1}^{*} = L^{2} (Ω \times Ω; R^{9})

Y_{2} = Y_{2}^{*} = L^{2} (Ω \times Ω; R^{3}),

and

\begin{matrix} G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ = & \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} H_{i j k l} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ \times (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}})) d x d y \end{matrix}

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we have that

\begin{matrix} J_{1} (u, ϕ) \\ = & G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ = & - \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y \\ + G_{1} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) \\ + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2} + \frac{1}{2} (u_{m, i} + ϕ_{m, y_{i}}) (u_{m, j} + ϕ_{m, y_{j}})) σ_{i j} d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}} \\ \geq & inf_{v \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} v_{i j} σ_{i j} d x d y - G_{1} ({v_{i j}})\} \\ + inf_{v_{2} \in Y_{1}} \{- \frac{1}{| Ω |} \int_{Ω} \int_{Ω} {(v_{2})}_{i j} Q_{i j} d x d y + \frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} \frac{1}{2} ({(v_{2})}_{m i} {(v_{2})}_{m j})) d x d y\} \\ + inf_{(u, ϕ) \in V \times V_{1}} \{\frac{1}{| Ω |} \int_{Ω} \int_{Ω} (σ_{i j} + Q_{i j}) (\frac{u_{i, j} + u_{j, i}}{2} + \frac{ϕ_{i, y_{j}} + ϕ_{j, y_{i}}}{2}) d x d y - {〈 u_{i}, f_{i} 〉}_{L^{2}}\} \\ \geq & - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y \\ - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y, \end{matrix}

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\forall (u, ϕ) \in V \times V_{1}, (σ, Q) \in A^{*}

, where

A^{*} = A_{1}^{*} \cap A_{2}^{*} \cap A_{3}^{*}

A_{1}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : σ_{i j, y_{j}} + Q_{i j, y_{j}} = 0, in Ω \times Ω} .

A_{2}^{*} = \{(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : \frac{1}{| Ω |} {(\int_{Ω} (σ_{i j}) d y)}_{x_{j}} + \frac{1}{| Ω |} {(\int_{Ω} (Q_{i j}) d y)}_{x_{j}} + f_{i} = 0, in Ω\},

A_{3}^{*} = {(σ, Q) \in Y_{1}^{*} \times Y_{1}^{*} : {σ_{i j}} is positive definite in Ω \times Ω} .

Hence, denoting

J^{*} (σ, Q) = - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x d y - \frac{1}{2 | Ω |} \int_{Ω} \int_{Ω} \bar{σ_{i j}} Q_{m i} Q_{m k} d x d y,

we have obtained

inf_{u \in V} J (u) \geq sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) .

Remark 6.1.

This last dual functional is concave and such a concerning inequality corresponds a duality principle for the relaxed primal formulation.

We emphasize again such results are also extensions and in some sense complement the original duality principles in the works of Telega and Bielski, [1,2,3].

Moreover, if

(σ_{0}, Q_{0}) \in A^{*}

is such that

δ J^{*} (σ_{0}, Q_{0}) = 0,

it is a well known result from the Legendre transform proprieties that the corresponding

(u_{0}, ϕ_{0}) \in V \times V_{1}

such that

{(σ_{0})}_{i j} = H_{i j k l} (\frac{u_{k, l} + u_{l, k}}{2} + \frac{ϕ_{k, y_{l}} + ϕ_{l, y_{k}}}{2} + \frac{1}{2} (u_{m, k} + ϕ_{m, y_{k}}) (u_{m, l} + ϕ_{m, y_{l}}))

and

{(Q_{0})}_{i j} = {(σ_{0})}_{i m} {(v_{2_{0}})}_{m j},

is also such that

δ J_{1} (u_{0}, ϕ_{0}) = 0

and

J_{1} (u_{0}, ϕ_{0}) = J^{*} (σ_{0}, Q_{0}) .

From this and

inf_{u \in V} J (u) = inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \geq sup_{(σ, Q) A^{*}} J^{*} (σ, Q),

we obtain

\begin{matrix} J_{1} (u_{0}, ϕ_{0}) & = & inf_{(u, ϕ) \in V \times V_{1}} J_{1} (u, ϕ) \\ = & sup_{(σ, Q) \in A^{*}} J^{*} (σ, Q) \\ = & J^{*} (σ_{0}, Q_{0}) \\ = & inf_{u \in V} J (u) . \end{matrix}

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Also, from the modern calculus of variations theory, there exists a sequence

{u_{n}} \subset V

such that

u_{n} ⇀ u_{0}, weakly in V,

and

J (u_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

From this and the Ekeland variational principle, there exists

{v_{n}} \subset V

such that

∥ u_{n} - v_{n} ∥_{V} \leq 1 / n,

J (v_{n}) \leq inf_{u \in V} J (u) + 1 / n,

and

∥ δ J (v_{n}) ∥_{V^{*}} \leq 1 / n, \forall n \in N,

so that

v_{n} ⇀ u_{0}, weakly in V,

and

J (v_{n}) \to J_{1} (u_{0}, ϕ_{0}) = inf_{u \in V} J (u) .

Assume now we are dealing with a finite dimensional version of such a model, in a finite elements of finite differences context, for example.

In such a case we have

v_{n} \to u_{0}, strongly in R^{N}

for an appropriate

N \in N .

From continuity we obtain

δ J (v_{n}) \to δ J (u_{0}) = 0,

J (v_{n}) \to J (u_{0}) .

Summarizing, we have got

J (u_{0}) = inf_{u \in V} J (u),

δ J (u_{0}) = 0 .

Here we highlight such last results are valid just for this finite-dimensional model version.

7. An exact convex dual variational formulation for a non-convex primal one

In this section we develop a convex dual variational formulation suitable to compute a critical point for the corresponding primal one.

Let

Ω \subset R^{2}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = F (u_{x}, u_{y}) - {〈 u, f 〉}_{L^{2}},

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Here we denote

Y = Y^{*} = L^{2} (Ω)

and

Y_{1} = Y_{1}^{*} = L^{2} (Ω) \times L^{2} (Ω) .

Defining

V_{1} = {{u \in V : ∥ u ∥}_{1, \infty} \leq K_{1}}

for some appropriate

K_{1} > 0

, suppose also F is twice Fréchet differentiable and

det \{\frac{\partial^{2} F (u_{x}, u_{y})}{\partial v_{1} \partial v_{2}}\} \neq 0,

\forall u \in V_{1} .

Define now

F_{1} : V \to R

and

F_{2} : V \to R

F_{1} (u_{x}, u_{y}) = F (u_{x}, u_{y}) + \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

and

F_{2} (u_{x}, u_{y}) = \frac{ε}{2} \int_{Ω} u_{x}^{2} d x + \frac{ε}{2} \int_{Ω} u_{y}^{2} d x,

where here we denote

d x = d x_{1} d x_{2} .

Moreover, we define the respective Legendre transform functionals

F_{1}^{*}

and

F_{2}^{*}

F_{1}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{1} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{1} (v_{1}, v_{2})}{\partial v_{2}},

and

F_{2}^{*} (v^{*}) = {〈 v_{1}, v_{1}^{*} + f_{1} 〉}_{L^{2}} + {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - F_{2} (v_{1}, v_{2}),

where

v_{1}, v_{2} \in Y

are such that

v_{1}^{*} + f_{1} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{1}},

v_{2}^{*} = \frac{\partial F_{2} (v_{1}, v_{2})}{\partial v_{2}} .

Here

f_{1}

is any function such that

{(f_{1})}_{x} = f, in Ω .

Furthermore, we define

\begin{matrix} J^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + F_{2}^{*} (v^{*}) \\ = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x . \end{matrix}

(36)

Observe that through the target conditions

v_{1}^{*} + f_{1} = ε u_{x},

v_{2}^{*} = ε u_{y},

we may obtain the compatibility condition

{(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0 .

Define now

A^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}) \in B_{r} (0, 0) \subset Y_{1}^{*} : {(v_{1}^{*} + f_{1})}_{y} - {(v_{2}^{*})}_{x} = 0, in Ω},

for some appropriate

r > 0

such that

J^{*}

is convex in

B_{r} (0, 0) .

Consider the problem of minimizing

J^{*}

subject to

v^{*} \in A^{*} .

Assuming

r > 0

is large enough so that the restriction in r is not active, at this point we define the associated Lagrangian

J_{1}^{*} (v^{*}, φ) = J^{*} (v^{*}) + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}},

where

φ

is an appropriate Lagrange multiplier.

Therefore

\begin{matrix} J_{1}^{*} (v^{*}) & = & - F_{1}^{*} (v^{*}) + \frac{1}{2 ε} \int_{Ω} {(v_{1}^{*} + f_{1})}^{2} d x + \frac{1}{2 ε} \int_{Ω} {(v_{2}^{*})}^{2} d x \\ + {〈 φ, {(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} 〉}_{L^{2}} . \end{matrix}

(37)

The optimal point in question will be a solution of the corresponding Euler-Lagrange equations for

J_{1}^{*} .

From the variation of

J_{1}^{*}

v_{1}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + \frac{v_{1}^{*} + f}{ε} - \frac{\partial φ}{\partial y} = 0 .

(38)

From the variation of

J_{1}^{*}

v_{2}^{*}

we obtain

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + \frac{v_{2}^{*}}{ε} + \frac{\partial φ}{\partial x} = 0 .

(39)

From the variation of

J_{1}^{*}

φ

we have

{(v_{1}^{*} + f)}_{y} - {(v_{2}^{*})}_{x} = 0 .

From this last equation, we may obtain

u \in V

such that

v_{1}^{*} + f = ε u_{x},

and

v_{2}^{*} = ε u_{y} .

From this and the previous extremal equations indicated we have

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}} + u_{x} - \frac{\partial φ}{\partial y} = 0,

and

- \frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}} + u_{y} + \frac{\partial φ}{\partial x} = 0 .

so that

v_{1}^{*} + f = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}},

and

v_{2}^{*} = \frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}} .

From this and equation (38) and (39) we have

\begin{matrix} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{1}^{*}})}_{x} - ε {(\frac{\partial F_{1}^{*} (v^{*})}{\partial v_{2}^{*}})}_{y} \\ + {(v_{1}^{*} + f_{1})}_{x} + {(v_{2}^{*})}_{y} \\ = & - ε u_{x x} - ε u_{y y} + {(v_{1}^{*})}_{x} + {(v_{2}^{*})}_{y} + f = 0 . \end{matrix}

(40)

Replacing the expressions of

v_{1}^{*}

and

v_{2}^{*}

into this last equation, we have

- ε u_{x x} - ε u_{y y} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0,

so that

{(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{1}})}_{x} + {(\frac{\partial F (u_{x} - φ_{y}, u_{y} + φ_{x})}{\partial v_{2}})}_{y} + f = 0, in Ω .

(41)

Observe that if

\nabla^{2} φ = 0

then there exists

\hat{u}

such that u and

φ

are also such that

u_{x} - φ_{y} = {\hat{u}}_{x}

and

u_{y} + φ_{x} = {\hat{u}}_{y} .

The boundary conditions for

φ

must be such that

\hat{u} \in W_{0}^{1, 2} .

From this and equation (41) we obtain

δ J (\hat{u}) = 0 .

Summarizing, we may obtain a solution

\hat{u} \in W_{0}^{1, 2}

of equation

δ J (\hat{u}) = 0

by minimizing

J^{*}

A^{*}

Finally, observe that clearly

J^{*}

is convex in an appropriate large ball

B_{r} (0, 0)

for some appropriate

r > 0

8. Another primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(42)

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(43)

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K}

for some appropriate

K > 0

to be specified.

Observe that, denoting

φ = - γ \nabla^{2} u + 2 v_{0}^{*} u - f

we have

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} = \frac{1}{α} + 4 K_{1} u^{2}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} = γ \nabla^{2} - 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}} = K_{1} (2 φ + 2 (- γ \nabla^{2} u + 2 v_{0}^{*} u)) - 2 u

so that

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*})}{\partial u \partial v_{0}^{*}})}^{2} \\ = & \frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} - \frac{γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}}{α} \\ - 4 K_{1}^{2} φ^{2} - 8 K_{1} φ (- γ \nabla^{2} + 2 v_{0}^{*}) u + 8 K_{1} φ u \\ + 4 K_{1} (- γ \nabla^{2} u + 2 v_{0}^{*} u) u . \end{matrix}

(44)

Observe now that a critical point

φ = 0

and

(- γ \nabla^{2} u + 2 v_{0}^{*} u) u = f u \geq 0

Ω

Therefore, for an appropriate large

K_{1} > 0

, also at a critical point, we have

\begin{matrix} det {δ^{2} J_{1}^{*} (u, v_{0}^{*})} \\ = & 4 K_{1} f u - \frac{δ^{2} J (u)}{α} + K_{1} \frac{{(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{α} > 0 . \end{matrix}

(45)

Remark 8.1.

From this last equation we may observe that

J_{1}^{*}

has a large region of convexity about any critical point

(u_{0}, {\hat{v}}_{0}^{*})

, that is, there exists a large

r > 0

such that

J_{1}^{*}

is convex on

B_{r} (u_{0}, {\hat{v}}_{0}^{*}) .

With such results in mind, we may easily prove the following theorem.

Theorem 8.2.

Assume

K_{1} ≫ max {1, K, K_{3}}

and suppose

(u_{0}, {\hat{v}}_{0}^{*}) \in V_{1} \times B^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, there exists

r > 0

such that

J_{1}^{*}

is convex in

E^{*} = B_{r} (u_{0}, {\hat{v}}_{0}^{*}) \cap (V_{1} \times B^{*})

δ J (u_{0}) = 0,

and

- J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{0}^{*}) \in E^{*}} J_{1}^{*} (u, v_{0}^{*}) .

9. A third primal dual formulation for a related model

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(46)

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

f \in L^{2} (Ω) .

Denoting

Y = Y^{*} = L^{2} (Ω)

, define now

J_{1}^{*} : V \times Y^{*} \times Y^{*} \to R

\begin{matrix} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{1}{2} \int_{Ω} K u^{2} d x \\ - {〈 u, v_{1}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{(- 2 v_{0}^{*} + K)} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x + {〈 u, f 〉}_{L^{2}} \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x, \end{matrix}

(47)

where

ε > 0

is a small real constant.

Define also

A^{+} = {u \in V : u f \geq 0, a . e . in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

V_{1} = V_{2} \cap A^{+}

for some appropriate

K_{3} > 0

to be specified.

Moreover define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K_{4}}

and

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥ \leq K_{5}},

for some appropriate real constants

K_{4}, K_{5} > 0

to be specified.

Remark 9.1.

Define now

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2}

For an appropriate function (or, in a more general fashion, an appropriate bounded operator)

M_{1}

define

B_{1}^{*} = {v_{0}^{*} \in B^{*} : 2 v_{0}^{*} + M_{1} \geq ε_{1}},

for some small parameter

ε_{1} > 0 .

Moreover, define

E^{*} = {u \in V_{1} : \sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |} .

Since for

(u, v_{0}^{*}) \in V_{1} \times B_{1}^{*}

we have

u f \geq 0, in Ω

, so that for

u_{1}, u_{2} \in V_{1}

we have

sign (u_{1}) = sign (u_{2}) in Ω,

we may infer that

E^{*}

is a convex set.

Moreover if

(u, v_{0}^{*}) \in E^{*} \times B_{1}^{*}

, then

\sqrt{4 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} |}

so that

4 α u^{2} \geq M_{1} + γ \nabla^{2}

and

2 v_{0}^{*} + M_{1} \geq ε_{1}

so that

H_{1} (u, v_{0}^{*}) = - γ \nabla^{2} + 2 v_{0}^{*} + 4 α u^{2} \geq ε_{1} .

Such a result we will be used many times in the next sections.

Observe that, defining

φ = v_{0}^{*} - α (u^{2} - β)

we may obtain

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u^{2}} = - γ \nabla^{2} + K + \frac{α}{α + ε} 4 u^{2} - 2 φ \frac{α}{α + ε}

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial {(v_{1}^{*})}^{2}} = \frac{1}{- 2 v_{0}^{*} + K}

and

\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}} = - 1

so that

\begin{matrix} det \{\frac{\partial^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{\partial u \partial v_{1}^{*}}\} \\ = & \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}})}^{2} \\ = & \frac{- γ \nabla^{2} + 2 v_{0}^{*} + 4 \frac{α^{2}}{α + ε} u^{2} - 2 \frac{α}{α + ε} φ}{- 2 v_{0}^{*} + K} \\ \equiv & H (u, v_{0}^{*}) . \end{matrix}

(48)

However, at a critical point, we have

φ = 0

so that, for a fixed

v_{0}^{*} \in B^{*}

we define the non-active but convex restriction

{(C_{1})}_{v_{0}^{*}}^{*} = {u \in V_{1} : {(φ)}^{2} \leq ε},

for a small parameter

ε > 0 .

From such results, assuming

K ≫ max {K_{3}, K_{4}, K_{5}},

and

0 < ε ≪ ε_{1} ≪ 1

, we have that

H (u, v_{0}^{*}) > 0,

for

v_{0}^{*} \in B_{1}^{*}

and

u \in E^{*} \cap {(C_{1})}_{v_{0}^{*}}^{*} .

With such results in mind, we may easily prove the following theorem.

Theorem 9.2.

Suppose

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in (E^{*} \cap {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}) \times D^{*} \times B_{1}^{*}

is such that

δ J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses, we have that

δ J (u_{0}) = 0

and

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(49)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*})

may be easily made similarly as in the previous sections.

Moreover, observe that for

K > 0

sufficiently large, we have

\frac{\partial^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0, \forall v_{0}^{*} \in B^{*}

so that this and the other hypotheses, we have also

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*})

and

J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*}) .

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that

\begin{matrix} J (u_{0}) & = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(50)

Moreover, observe that

\begin{matrix} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}) \\ \leq & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x \\ + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - \frac{K}{2} \int_{Ω} u^{2} d x \\ - \frac{1}{2 α} \int_{Ω} {({\hat{v}}_{0}^{*})}^{2} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {({\hat{v}}_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \leq & sup_{v_{0}^{*} \in Y^{*}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + 〈 u^{2}, v_{0}^{*} 〉 \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x \\ + \frac{1}{2 (α + ε)} \int_{Ω} {(v_{0}^{*} - α (u^{2} - β))}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \forall u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} . \end{matrix}

(51)

Summarizing, we have got

J (u_{0}) = J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \leq inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) .

From such results, we may infer that

\begin{matrix} J (u_{0}) & = & inf_{u \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*}} J (u) \\ = & J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in {(C_{1})}_{{\hat{v}}_{0}^{*}}^{*} \times D^{*}} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})\} . \end{matrix}

(52)

The proof is complete. □

10. An algorithm for a related model in shape optimization

The next two subsections have been previously published by Fabio Silva Botelho and Alexandre Molter in [8], Chapter 21.

10.1. Introduction

Consider an elastic solid which the volume corresponds to an open, bounded, connected set, denoted by

Ω \subset R^{3}

with a regular (Lipschitzian) boundary denoted by

\partial Ω = Γ_{0} \cup Γ_{t}

where

Γ_{0} \cap Γ_{t} = \emptyset .

Consider also the problem of minimizing the functional

\hat{J} : U \times B \to R

where

\hat{J} (u, t) = \frac{1}{2} {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

subject to

\{\begin{matrix} {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} = 0 in Ω, \\ H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3} . \end{matrix}

(53)

Here

n = (n_{1}, n_{2}, n_{3})

denotes the outward normal to

\partial Ω

and

U = {u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3}) : u = (0, 0, 0) = 0 on Γ_{0}},

B = \{t : Ω \to [0, 1] measurable : \int_{Ω} t (x) d x = t_{1} | Ω |\},

where

0 < t_{1} < 1

and

| Ω |

denotes the Lebesgue measure of

Ω .

Moreover

u = (u_{1}, u_{2}, u_{3}) \in W^{1, 2} (Ω; R^{3})

is the field of displacements relating the cartesian system

(0, x_{1}, x_{2}, x_{3})

, resulting from the action of the external loads

f \in L^{2} (Ω; R^{3})

and

\hat{f} \in L^{2} (Γ_{t}; R^{3}) .

We also define the stress tensor

{σ_{i j}} \in Y^{*} = Y = L^{2} (Ω; R^{3 \times 3}),

σ_{i j} (u) = H_{i j k l} (t) e_{k l} (u),

and the strain tensor

e : U \to L^{2} (Ω; R^{3 \times 3})

e_{i j} (u) = \frac{1}{2} (u_{i, j} + u_{j, i}), \forall i, j \in {1, 2, 3} .

Finally,

{H_{i j k l} (t)} = {t H_{i j k l}^{0} + (1 - t) H_{i j k l}^{1}},

where

H^{0}

corresponds to a strong material and

H^{1}

to a very soft material, intending to simulate voids along the solid structure.

The variable t is the design one, which the optimal distribution values along the structure are intended to minimize its inner work with a volume restriction indicated through the set B.

Moreover, details on the Sobolev spaces addressed may be found in [6]. In addition, the primal variational development of the topology optimization problem has been described in [7].

10.2. Mathematical formulation of the topology optimization problem

Our mathematical topology optimization problem is summarized by the following theorem.

Theorem 10.1.

Consider the statements and assumptions indicated in the last section, in particular those refereing to Ω and the functional

\hat{J} : U \times B \to R .

Define

J_{1} : U \times B \to R

J_{1} (u, t) = - G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})},

where

G (e (u), t) = \frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x,

and where

d x = d x_{1} d x_{2} d x_{3} .

Define also

J^{*} : U \to R

\begin{matrix} J^{*} (u) & = & inf_{t \in B} {J_{1} (u, t)} \\ = & inf_{t \in B} {- G (e (u), t) + {〈 u_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 u_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}} . \end{matrix}

(54)

Assume there exists

c_{0}, c_{1} > 0

such that

H_{i j k l}^{0} z_{i j} z_{k l} > c_{0} z_{i j} z_{i j}

and

H_{i j k l}^{1} z_{i j} z_{k l} > c_{1} z_{i j} z_{i j}, \forall z = {z_{i j}} \in R^{3 \times 3}, such that z \neq 0 .

Finally, define

J : U \times B \to R \cup {+ \infty}

J (u, t) = \hat{J} (u, t) + I n d (u, t),

where

I n d (u, t) = \{\begin{matrix} 0, & if (u, t) \in A^{*}, \\ + \infty, & otherwise, \end{matrix}

(55)

where

A^{*} = A_{1} \cap A_{2},

A_{1} = {(u, t) \in U \times B : {(σ_{i j} (u))}_{, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

A_{2} = {(u, t) \in U \times B : σ_{i j} (u) n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Under such hypotheses, there exists

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} J (u_{0}, t_{0}) & = & inf_{(u, t) \in U \times B} J (u, t) \\ = & sup_{\hat{u} \in U} J^{*} (\hat{u}) \\ = & J^{*} (u_{0}) \\ = & \hat{J} (u_{0}, t_{0}) \\ = & inf_{(t, σ) \in B \times C^{*}} G^{*} (σ, t) \\ = & G^{*} (σ (u_{0}), t_{0}), \end{matrix}

(56)

where

\begin{matrix} G^{*} (σ, t) & = & sup_{v \in Y} {{〈 v_{i j}, σ_{i j} 〉}_{L^{2} (Ω)} - G (v, t)} \\ = & \frac{1}{2} \int_{Ω} {\bar{H}}_{i j k l} (t) σ_{i j} σ_{k l} d x, \end{matrix}

(57)

{{\bar{H}}_{i j k l} (t)} = {H_{i j k l} (t)}^{- 1}

and

C^{*} = C_{1} \cap C_{2},

where

C_{1} = {σ \in Y^{*} : σ_{i j, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}}

and

C_{2} = {σ \in Y^{*} : σ_{i j} n_{j} - {\hat{f}}_{i} = 0, on Γ_{t}, \forall i \in {1, 2, 3}} .

Proof.

Observe that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{inf_{u \in U} J (u, t)\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, {(H_{i j k l} (t) e_{k l} (u))}_{, j} + f_{i} 〉}_{L^{2} (Ω)} \\ - {〈 {\hat{u}}_{i}, H_{i j k l} (t) e_{k l} (u) n_{j} - {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{inf_{u \in U} \{\frac{1}{2} \int_{Ω} H_{i j k l} (t) e_{i j} (u) e_{k l} (u) d x \\ - \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\}\} \\ = & inf_{t \in B} \{sup_{\hat{u} \in U} \{- \int_{Ω} H_{i j k l} (t) e_{i j} (\hat{u}) e_{k l} (\hat{u}) d x \\ {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {\hat{u}}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})}\}\} \\ = & inf_{t \in B} \{inf_{σ \in C^{*}} G^{*} (σ, t)\} . \end{matrix}

(58)

Also, from this and the min-max theorem, there exist

(u_{0}, t_{0}) \in U \times B

such that

\begin{matrix} inf_{(u, t) \in U \times B} J (u, t) & = & inf_{t \in B} \{sup_{\hat{u} \in U} J_{1} (u, t)\} \\ = & sup_{u \in U} \{inf_{t \in B} J_{1} (u, t)\} \\ = & J_{1} (u_{0}, t_{0}) \\ = & inf_{t \in B} J_{1} (u_{0}, t) \\ = & J^{*} (u_{0}) . \end{matrix}

(59)

Finally, from the extremal necessary condition

\frac{\partial J_{1} (u_{0}, t_{0})}{\partial u} = 0

we obtain

{(H_{i j k l} (t_{0}) e_{k l} (u_{0}))}_{, j} + f_{i} = 0 in Ω,

and

H_{i j k l} (t_{0}) e_{k l} (u_{0}) n_{j} - {\hat{f}}_{i} = 0 on Γ_{t}, \forall i \in {1, 2, 3},

so that

G (e (u_{0})) = \frac{1}{2} {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + \frac{1}{2} {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} .

Hence

(u_{0}, t_{0}) \in A^{*}

so that

I n d (u_{0}, t_{0}) = 0

and

σ (u_{0}) \in C^{*} .

Moreover

\begin{matrix} J^{*} (u_{0}) & = & - G (e (u_{0})) + {〈 {(u_{0})}_{i}, f_{i} 〉}_{L^{2} (Ω)} + {〈 {(u_{0})}_{i}, {\hat{f}}_{i} 〉}_{L^{2} (Γ_{t})} \\ = & G (e (u_{0})) \\ = & G (e (u_{0})) + I n d (u_{0}, t_{0}) \\ = & J (u_{0}, t_{0}) \\ = & G^{*} (σ (u_{0}), t_{0}) . \end{matrix}

(60)

This completes the proof. □

10.3. About a concerning algorithm and related numerical method

For numerically solve this optimization problem in question, we present the following algorithm

1.: Set $t_{1} = 0.5 in Ω$ and $n = 1$ .
2.: Calculate $u_{n} \in U$ such that

$J_{1} (u_{n}, t_{n}) = sup_{u \in U} J_{1} (u, t_{n}) .$
3.: Calculate $t_{n + 1} \in B$ such that

$J_{1} (u_{n}, t_{n + 1}) = inf_{t \in B} J_{1} (u_{n}, t) .$
4.: If $∥ t_{n + 1} - t_{n} ∥_{\infty} < 10^{- 4}$ or $n > 100$ then stop, else set $n : = n + 1$ and go to item 2.

We have developed a software in finite differences for solving such a problem.

Here the software.

**************************************

1.: clear all

global P m8 d w u v Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3

m8=27;

m9=24;

c3=0.95;

d=1.0/m8;

d1=0.5/m9;

Ea= $210 * 10^{5}$ ; (stronger material)

Eb=1000; (softer material simulating voids)

w=0.30;

P=-42000000;

z1=(m8-1)*(m9-1);

A3=zeros(z1,z1);

for i=1:z1

A3(1,i)=1.0;

end;

b=zeros(z1,1);

uo=0.000001*ones(z1,1);

u1=ones(z1,1);

b(1,1)=c3*z1;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=c3;

end; end;

for i=1:z1

x1(i)=c3*z1;

end;

for i=1:2*m8*m9

xo(i)=0.000;

end;

xw=xo;

xv=Lo;

for k2=1:24

c3=0.98*c3;

b(1,1)=c3*z1;

k2

b14=1.0;

k3=0;

while $(b 14 > 10^{- 3.5})$ and $(k 3 < 5)$

k3=k3+1;

b12=1.0;

k=0;

while $(b 12 > 10^{- 4.0})$ and $(k < 120)$

k=k+1;

k2

k3

k

X=fminunc(’funbeam’,xo);

xo=X;

b12=max(abs(xw-xo));

xw=X;

end;

for i=1:m9-1

for j=1:m8-1

$E 1 = L o {(i, j)}^{2} * (E a - E b);$

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

$S x = E 1 * (e x + w * e y) / (1 - w^{2});$

$S y = E 1 * (w * e x + e y) / (1 - w^{2});$

Sxy=E1/(2*(1+w))*exy;

dc3(i,j)=-(Sx*ex+Sy*ey+2*Sxy*exy);

end;

end;

for i=1:m9-1

for j=1:m8-1

f(j+(i-1)*(m8-1))=dc3(i,j);

end;

end;

for k1=1:1

k1

X1=linprog(f, $[]$ , $[]$ ,A3,b,uo,u1,x1);

x1=X1;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=X1(j+(m8-1)*(i-1));

end;

end;

b14=max(max(abs(Lo-xv)))

xv=Lo;

colormap(gray); imagesc(-Lo); axis equal; axis tight; axis off;pause(1e-6)

end;

end;

****************************************************

Here the auxiliary Function ’funbeam’

function S=funbeam(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2

for i=1:m9

for j=1:m8

u(i,j)=x(j+(m8)*(i-1));

v(i,j)=x(m8*m9+(i-1)*m8+j);

end;

for i=1:m9

end;

u(m9-1,1)=0;

v(m9-1,1)=0;

u(m9-1,m8-1)=0;

v(m9-1,m8-1)=0;

for i=1:m9-1

for j=1:m8-1

du1(i,j)=(u(i,j+1)-u(i,j))/d;

du2(i,j)=(u(i+1,j)-u(i,j))/d1;

dv1(i,j)=(v(i,j+1)-v(i,j))/d;

dv2(i,j)=(v(i+1,j)-v(i,j))/d1;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E 1 = L o {(i, j)}^{3} * E a + (1 - L o {(i, j)}^{3}) * E b;

ex=du1(i,j);

ey=dv2(i,j);

exy=1/2*(dv1(i,j)+du2(i,j));

S x = E 1 * (e x + w * e y) / (1 - w^{2});

S y = E 1 * (w * e x + e y) / (1 - w^{2});

Sxy=E1/(2*(1+w))*exy;

S=S+1/2*(Sx*ex+Sy*ey+2*Sxy*exy);

end;

S=S*d*d1-P*v(2,(m8)/3)*d*d1;

***********************************************

For a two dimensional beam of dimensions

1 m \times 0.5 m

and

t_{1} = 0.63

we have obtained the following results:

1.: Case A: For the optimal shape for a clamped beam at left (cantilever) and load $P = - 4 10^{6} N j$ at $(x, y) = (1, 0.25)$ , please Figure 5.
2.: Case B :For the optimal shape for a simply supported beam at $(0, 0)$ and $(1, 0)$ and load $P = - 4 10^{6} N j$ at $(x, y) = (1 / 3, 0.5)$ , please Figure 6.

In the first case the mesh was $28 \times 24$ . In the second one the mesh was $27 \times 24$

11. A duality principle for a general vectorial case in the calculus of variations

In this section we develop a duality principle for a general vectorial case in variational optimization.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

. Let

J : V \to R

be a functional where

J (u) = G (\nabla u_{1}, \dots, \nabla u_{N}) - {〈 u, f 〉}_{L^{2}},

where

V = W_{0}^{1, 2} (Ω; R^{N})

and

f = (f_{1}, \dots, f_{N}) \in L^{2} (Ω; R^{N}) .

Here we have denoted

u = (u_{1}, \dots, u_{N}) \in V

and

{〈 u, f 〉}_{L^{2}} = {〈 u_{i}, f_{i} 〉}_{L^{2}},

so that we may also denote

J (u) = G (\nabla u) - {〈 u, f 〉}_{L^{2}} .

Assume

G (\nabla u) = \int_{Ω} g (\nabla u) d x

where

g : R^{3 N} \to R

is a differentiable function such that

g (y) \to + \infty

| y | \to \infty

. Moreover, suppose there exists

α \in R

such that

α = inf_{u \in V} J (u) .

It is well known that

\begin{matrix} α & = & inf_{u \in V} J (u) \\ = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

(61)

Under some mild hypotheses, from convexity, we have that

\begin{matrix} inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} \\ = & sup_{v^{*} \in A^{*}} {- {(G \circ \nabla)}^{*} (- d i v v^{*})} = - {(G \circ \nabla)}^{*} (f), \end{matrix}

(62)

where

A^{*} = {v^{*} \in Y = Y^{*} = L^{2} (Ω; R^{3 N}) : d i v v^{*} + f = 0} .

Now observe that the restriction

v = \nabla u

for some

u \in V

is equivalent to the restriction

curl v_{i} = 0, in Ω

where

v = {v_{i}} = {v_{i j}}_{j = 1}^{3}

\forall i \in {1, \dots, N},

with appropriate boundary conditions, so that with an appropriate Lagrange multiplier

ϕ = {ϕ_{i}}

, we obtain

\begin{matrix} {(G \circ \nabla)}^{*} (- d i v v^{*}) & = & sup_{u \in V} {{〈 u, - d i v v^{*} 〉}_{L^{2}} - G (\nabla u)} \\ = & sup_{u \in V} {{〈 \nabla u, v^{*} 〉}_{L^{2}} - G (\nabla u)} \\ \leq & inf_{ϕ \in Y^{*}} \{sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G (v) + {〈 ϕ, curl v 〉}_{L^{2}}\} \\ = & inf_{ϕ \in Y^{*}} G^{*} (v^{*} + curl ϕ) . \end{matrix}

(63)

where we have denoted

curl v = {curl v_{i}}

and

curl ϕ = {curl ϕ_{i}} .

Joining the pieces, we have got

\begin{matrix} inf_{u \in V} J (u) & = & inf_{u \in V} {G (\nabla u) - {〈 u, f 〉}_{L^{2}}} \\ \geq & sup_{(v^{*}, ϕ) \in A^{*} \times Y^{*}} {- G^{*} (v^{*} + curl ϕ)}, \end{matrix}

(64)

where we recall that

Y = Y^{*} = L^{2} (Ω; R^{3 N}) .

We emphasize such a dual formulation in

(v^{*}, ϕ)

is convex (in fact concave).

12. A note on the Galerkin Functional

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \end{matrix}

(65)

Here

V = W_{0}^{1, 2} (Ω)

γ > 0, α > 0, β > 0 .

We denote also

Y = Y^{*} = L^{2} (Ω) .

At this point we define

A^{+} = {u \in V : u f \geq 0, in Ω},

V_{2} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

for some appropriate real constant

K_{3} > 0

and

V_{1} = A^{+} \cap V_{2} .

Observe that

J^{'} (u) = - γ \nabla^{2} u + α u^{3} - β - f,

so that we define the Galerkin functional

J_{1} : V \to R

J_{1} (u) = \frac{1}{2} {∥ J^{'} (u) ∥}_{2}^{2} = \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x .

From this, we get

\begin{matrix} \frac{\partial^{2} J_{1} (u)}{\partial u^{2}} & = & (- γ \nabla u + α u^{3} - β u - f) 6 α u \\ + {(- γ \nabla^{2} + 3 α u^{2} - β)}^{2} . \end{matrix}

(66)

Define now

φ_{2} = {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} .

At this point, for an appropriate small real constant

ε_{1} > 0

and bounded constant operator

M_{1} > ε_{1}

, we set the intended non-active restriction

\sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |},

and define

B_{1} = {u \in V_{1} : \sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |}} .

Observe that since for

u \in V_{1}

we have

u f \geq 0

Ω

so that if

u_{1}, u_{2} \in V_{1}

then

sign (u_{1}) = sign (u_{2}), in Ω,

we may infer that

B_{1}

is a convex set.

Furthermore, if

u \in B_{1}

, then

\sqrt{3 α} | u | \geq \sqrt{| M_{1} + γ \nabla^{2} + β |},

so that

3 α u^{2} \geq M_{1} + γ \nabla^{2} + β,

and hence

δ^{2} J (u) = - γ \nabla^{2} + 3 α u^{2} - β \geq M_{1} > ε_{1} > 0 .

For a small parameter

ε > 0

we define the intended non-active restriction

φ_{2} \leq ε, in Ω,

and define

B_{2} = {u \in V_{1} : φ_{2} \leq ε, in Ω} .

Observe that for

α > 0

and

β > 0

sufficiently large

φ_{2}

is convex in

V_{1}

(positive definite Hessian) so that

B_{2}

is a convex set. Assuming

0 < ε ≪ ε_{1} ≪ 1

, define

B_{3} = B_{1} \cap B_{2}

, which is a convex set.

Summarizing, if

u \in B_{3}

, then

δ^{2} J_{1} (u) \geq 0 .

With such results in mind, we define the following convex optimization problem for finding a critical point of J.

Minimize

J_{1} (u) = \frac{1}{2} {∥ J^{'} (u) ∥}_{2}^{2} = \frac{1}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x,

subject to

u \in B_{3} .

Observe that a critical point

u_{0} \in B_{3}

J_{1}

, from such a concerning convexity of

J_{1}

on the convex set

B_{1}

, is also such that

J (u_{0}) = min_{u \in B_{3}} J_{1} (u) .

Finally, we may also define the convex optimization problem of minimizing

\begin{matrix} J_{3} (u) & = & K_{1} J_{1} (u) + J (u) \\ = & \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + α u^{3} - β u - f)}^{2} d x \\ + \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}}, \end{matrix}

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subject to

u \in B_{3} .

Here

K_{1} > 0

is a large real constant.

Such a functional

J_{3}

is also convex on

B_{3}

so that a critical point

u_{0} \in B_{3}

of J is also a critical point of

J_{3}

, and thus

J_{3} (u_{0}) = min_{u \in B_{3}} J_{3} (u) .

13. A note on the Legendre-Galerkin functional

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{4} \int_{Ω} u^{4} d x \\ - \frac{β}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \end{matrix}

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Here

V = W_{0}^{1, 2} (Ω)

γ > 0, α > 0, β > 0 .

We denote also

Y = Y^{*} = L^{2} (Ω)

and

F_{1} : V \to R

F_{2} : V \to R

and

F_{3} : V \to R

F_{1} (u) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x,

F_{2} (u) = \frac{α}{4} \int_{Ω} u^{4} d x,

F_{3} (u) = \frac{β}{2} \int_{Ω} u^{2} d x .

Moreover, we define

F_{1}^{*}, F_{2}^{*}, F_{3}^{*} : Y^{*} \to R

\begin{matrix} F_{1}^{*} (v_{1}^{*}) & = & sup_{u \in V} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{1} (u)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{- γ \nabla^{2}} d x, \end{matrix}

(69)

\begin{matrix} F_{2}^{*} (v_{2}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u)} \\ = & \frac{3}{4} \int_{Ω} \frac{{(v_{2}^{*})}^{4 / 3}}{α^{1 / 3}} d x, \end{matrix}

(70)

\begin{matrix} F_{3}^{*} (v_{3}^{*}) & = & sup_{u \in V} {{〈 u, v_{3}^{*} 〉}_{L^{2}} - F_{3} (u)} \\ = & \frac{1}{2 β} \int_{Ω} {(v_{3}^{*})}^{2} d x . \end{matrix}

(71)

Observe now that these three last suprema are attained through the equations,

v_{1}^{*} = \frac{\partial F_{1} (u)}{\partial u} = - γ \nabla^{2} u,

v_{2}^{*} = \frac{\partial F_{2} (u)}{\partial u} = α u^{3}

v_{3}^{*} = \frac{\partial F_{3} (u)}{\partial u} = β u .

From such results, at a critical point, we obtain the following compatibility conditions

u = \frac{v_{1}^{*}}{- γ \nabla^{2}} = {(\frac{v_{2}^{*}}{β})}^{1 / 3} = \frac{v_{3}^{*}}{β} .

From such relations we have

\frac{v_{1}^{*}}{- γ \nabla^{2}} = \frac{v_{3}^{*}}{β},

and

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3},

so that

v_{1}^{*} = - γ \nabla^{2} (\frac{v_{3}^{*}}{β}),

and

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3} .

Moreover, we define the functional

F_{4}^{*} : Y^{*} \to R,

F_{4}^{*} (v^{*}) = sup_{u \in V} {{〈 u, v_{1}^{*} + v_{2}^{*} - v_{3}^{*} 〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}}} .

Therefore

F_{4}^{*} (v^{*}) = \{\begin{matrix} 0, & if v_{1}^{*} + v_{2}^{*} - v_{3}^{*} - f = 0, in Ω, \\ + \infty, & otherwise . \end{matrix}

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Hence, a critical point of J corresponds to the solution of the following system of equations

v_{1}^{*} = - γ \nabla^{2} (\frac{v_{3}^{*}}{β}),

v_{2}^{*} = α {(\frac{v_{3}^{*}}{β})}^{3},

and

v_{1}^{*} + v_{2}^{*} - v_{3}^{*} - f = 0, in Ω .

From this last equation we may obtain

v_{1}^{*} = - v_{2}^{*} + v_{3}^{*} + f,

so that the final equations to be solved are

- v_{2}^{*} + v_{3}^{*} + f + γ \nabla^{2} (\frac{v_{3}^{*}}{β}) = 0

and

v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3} = 0, in Ω,

with the boundary conditions

u = \frac{v_{3}^{*}}{β} = 0, on \partial Ω .

With such results in mind, we define the Legendre-Galerkin functional

J^{*} : {[Y^{*}]}^{2} \to R

, where

\begin{matrix} J^{*} (v^{*}) & = & \frac{1}{2} \int_{Ω} {(- v_{2}^{*} + v_{3}^{*} + f + \frac{γ \nabla^{2} v_{3}^{*}}{β})}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3})}^{2} d x . \end{matrix}

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At this point, defining

φ = v_{2}^{*} - α {(\frac{v_{3}^{*}}{β})}^{3},

we obtain

\frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{2}^{*})}^{2}} = 2;

\frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{3}^{*})}^{2}} = {(- 1 - \frac{γ \nabla^{2}}{β})}^{2} + \frac{9 α^{2} {(v_{3}^{*})}^{4}}{β^{6}} + O (φ),

\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}} = \frac{- 3 α {(v_{3}^{*})}^{2}}{β^{3}} + (- 1 - \frac{γ \nabla^{2}}{β}) .

From such results we may infer that

\begin{matrix} det (\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}}) & = & \frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{2}^{*})}^{2}} \frac{\partial^{2} J^{*} (v^{*})}{\partial {(v_{3}^{*})}^{2}} - {(\frac{\partial^{2} J^{*} (v^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}})}^{2} \\ = & {(- 1 - \frac{γ \nabla^{2}}{β} + 3 α \frac{{(v_{3}^{*})}^{2}}{β^{3}})}^{2} + O (φ) \end{matrix}

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Observe that a critical point

φ = 0

so that

δ^{2} J^{*} (v^{*}) > 0

at a neighborhood of any critical point.

At this point we define

A^{+} = \{v^{*} = (v_{2}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{2} : \frac{v_{3}^{*}}{β} f \geq 0, in Ω\},

D^{*} = {v^{*} = (v_{2}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{2} : ∥ v^{*} ∥_{\infty} \leq K},

for an appropriate real constant

K > 0

Define now

E^{*} = A^{+} \cap D^{*}

C_{1}^{*} = {v^{*} = (v_{2}^{*}, v_{3}^{*}) \in E^{*} : φ^{2} \leq ε, in Ω},

for a small real constant

ε > 0,

C_{2}^{*} = \{v^{*} = (v_{2}^{*}, v_{3}^{*}) \in E^{*} : (- 1 - \frac{γ \nabla^{2}}{β} + 3 α \frac{{(v_{3}^{*})}^{2}}{β^{3}}) \geq ε_{1}\},

and

C^{*} = C_{1}^{*} \cap C_{2}^{*} .

Similarly as done in the previous section, we may prove that

C^{*}

is a convex set.

Furthermore, for

0 < ε ≪ ε_{1} ≪ 1,

we have that

J^{*}

is convex on

C^{*}

Summarizing, we may define the following convex optimization problem to obtain a critical point of the primal functional J,

Minimize J^{*} (v_{2}^{*}, v_{3}^{*}) subject to v^{*} = (v_{2}^{*}, v_{3}^{*}) \in C^{*} .

We call

J^{*}

the Legendre-Galerkin functional associated to J.

13.1. Numerical examples

We have obtained numerical solutions for two one-dimensional examples.

1.: For $γ = 1.0,$ $α = 3.0,$ $β = 30.0,$ $f \equiv 10, in Ω = [0, 1] .$

For the respective solution please see Figure 7.
2.: For $γ = 0.01,$ $α = 3.0,$ $β = 30.0,$ $f \equiv 10, in Ω = [0, 1] .$

For the respective solution please see Figure 8.

14. A general concave dual variational formulation for global optimization

Let

Ω \subset R^{3}

be an open, bounded and connected set a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

where

J (u) = G (u) - {〈 u, f 〉}_{L^{2}}, \forall u \in V .

Here

V = W_{0}^{1, 2} (Ω)

f \in L^{2} (Ω)

and we also denote

Y = Y^{*} = L^{2} (Ω) .

Assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Furthermore, suppose G is three times Fréchet differentiable and there exists

K > 0

such that

\frac{\partial^{2} G (u)}{\partial u^{2}} + K > 0, \forall u \in V .

Define now

J_{1} : V \times Y \to R

where,

J_{1} (u, v) = G_{1} (u, v) + F (u),

where

G_{1} (u, v) = G (v) - \frac{ε}{2} \int_{Ω} v^{2} d x + \frac{K}{2} \int_{Ω} {(v - u)}^{2} d x,

and

F (u) = \frac{ε}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Moreover, we define the polar functionals

G_{1}^{*} : Y^{*} \times V \to R

and

F^{*} : Y^{*} \to R

, where

\begin{matrix} G_{1}^{*} (v^{*}, u) & = & sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G_{1} (u, v)} \\ = & - G_{K_{ε}}^{*} (v^{*} + K u) + \frac{K}{2} \int_{Ω} u^{2} d x, \end{matrix}

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G_{K_{ε}}^{*} (v^{*} + K u) = sup_{v \in Y} \{{〈 v, v^{*} 〉}_{L^{2}} - G (v) - \frac{K}{2} \int_{Ω} v^{2} d x + \frac{ε}{2} \int_{Ω} v^{2} d x\},

and

\begin{matrix} F^{*} (- v^{*}) & = & sup_{u \in V} {- {〈 u, v^{*} 〉}_{L^{2}} - F (u)} \\ = & \frac{1}{2 ε} \int_{Ω} {(v^{*} - f)}^{2} d x . \end{matrix}

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At this point we define the functional

J_{2}^{*} : Y^{*} \times V \to R

J_{2}^{*} (v^{*}, u) = - G_{K_{ε}}^{*} (v^{*} + K u) + \frac{K}{2} \int_{Ω} u^{2} d x - F^{*} (- v^{*}) .

With such results in mind we define

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

and

D^{*} = {v^{*} \in Y^{*} : ∥ v^{*} ∥_{\infty} \leq K_{4}},

for appropriated real constants

K_{3} > 0

and

K_{4} > 0 .

Moreover, we define also the penalized functional

J_{3}^{*} : Y^{*} \times V \to R

where

J_{3}^{*} (v^{*}, u) = J_{2}^{*} (v^{*}, u) - \frac{K_{1}}{2} \int_{Ω} {(v^{*} - \frac{\partial G (u)}{\partial u} + ε u)}^{2} d x .

Finally, we remark that for

ε > 0

sufficiently small and

K_{1} > 0

sufficiently large,

J_{3}^{*}

is concave in

D^{*} \times V_{1}

around a concerning critical point. We recall that a critical point

v^{*} - \frac{\partial G (u)}{\partial u} + ε u = 0, in Ω .

15. A related restricted problem in phase transition

In this section we develop a convex (in fact concave) dual variational for a model similar to those found in phase transition problems.

Let

Ω = [0, 1] \subset R .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{1}{2} \int_{Ω} min {{(u^{'} + 1)}^{2}, {(u^{'} - 1)}^{2}} d x \\ + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \\ = & \frac{1}{2} \int_{Ω} {(u^{'})}^{2} d x - \int_{Ω} | u^{'} | d x + 1 / 2 \\ + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} . \end{matrix}

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Here

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2} .

We also denote

V_{1} = W_{0}^{1, 2} (Ω),

and

Y = Y^{*} = L^{2} (Ω) .

Furthermore, we define the functionals G and

F : V \times V_{1} \to R

G (u^{'}, v^{'}) = \frac{1}{2} \int_{Ω} {(u^{'} + v^{'})}^{2} d x - \int_{Ω} | u^{'} + v^{'} | d x + 1 / 2,

and

F (u, v) = \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} .

Moreover we define

J_{1} : V \times V_{1} \to R

J_{1} (u, v) = G (u^{'}, v^{'}) + F (u, v),

and consider the problem of minimizing

J_{1}

on the set

A = {(u, v) \in V \times V_{1} : {(v^{'})}^{2} \leq K_{2}, in Ω} .

Already including the Lagrange multiplier

ϕ

concerning such restrictions, we define

J_{2} (u, v, ϕ) = J_{1} (u, v) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} .

Observe now that

\begin{matrix} J_{2} (u, v, ϕ) & = & J_{1} (u, v) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ = & G (u^{'}, v^{'}) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ + F (u, v) \\ = & - {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} - {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + G (u^{'}, v^{'}) \\ + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} \\ {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + F (u, v) \\ \geq & inf_{(v_{1}, v_{2}) \in Y \times Y} \{- {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} + G_{1} (v_{1}, v_{2}, ϕ) \\ + \frac{1}{2} {〈 ϕ^{2}, {(v_{2})}^{2} - K_{2} 〉}_{L^{2}}\} \\ + inf_{(u, v) \in V \times V_{1}} {{〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + F (u, v)} \\ = & - G_{1}^{*} (v_{1}^{*}, v_{2}^{*}, ϕ) - {\tilde{F}}^{*} (v_{1}^{*}, v_{2}^{*}), \forall (u, v) \in V \times V_{1}, (v_{1}^{*}, v_{2}^{*}, ϕ) \in {[Y^{*}]}^{3}, \end{matrix}

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where

G_{1} (u^{'}, v^{'}, ϕ) = G (u^{'}, v^{'}) + \frac{1}{2} {〈 ϕ^{2}, {(v^{'})}^{2} - K_{2} 〉}_{L^{2}} .

Also,

\begin{matrix} G_{1}^{*} (v_{1}^{*}, v_{2}^{*}, ϕ) & = & sup_{(v_{1}, v_{2}) \in Y \times Y} {{〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - G_{1} (v_{1}, v_{2}, ϕ)} \\ = & \frac{1}{2} \int_{Ω} {(v_{1}^{*})}^{2} d x \\ + \int_{Ω} | v_{1}^{*} | d x + \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{ϕ^{2}} \\ + \frac{K_{2}}{2} \int_{Ω} ϕ^{2} d x, \end{matrix}

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where

{\tilde{F}}^{*} (v^{*}) = \{\begin{matrix} \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - v_{1}^{*} (1) u (1), & if {(v_{2}^{*})}^{'} = 0, in Ω, \\ + \infty, & otherwise . \end{matrix}

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From this we may infer that

v_{2}^{*} = c, in Ω,

for some

c \in R .

Summarizing, denoting

v^{*} = (v_{1}^{*}, v_{2}^{*}) = (v_{1}^{*}, c)

, and

J^{*} (v^{*}, ϕ) = - G_{1}^{*} (v^{*}, ϕ) - {\tilde{F}}^{*} (v^{*})

we have got

inf_{(u, v) \in A} J_{1} (u, v) \geq sup_{(v^{*}, ϕ) \in Y^{*} \times R \times Y^{*}} J^{*} (v^{*} ϕ) .

We have developed numerical results by maximizing the dual functional

J^{*}

for two examples, namely.

1.: Example A: In this case, we consider $f (x) = cos (π x) / 2$ , $K_{2} = 10^{- 4}$ .

For the optimal

$u_{0} = {(v_{1}^{*})}^{'} + f,$

please see Figure 9.
2.: Example B: In this case, we consider $f (x) = cos (π x) / 2$ , $K_{2} = 30$ .

For the optimal

$u_{0} = {(v_{1}^{*})}^{'} + f,$

please see Figure 10.

16. One more dual variational formulation

In this section we develop one more dual variational formulation for a related model.

Let

Ω = [0, 1] \subset R

and consider the functional

J : V \to R

defined by

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2} .

We define also the relaxed functional

J_{1} : V \times V_{0} \to R

, already including a concerning restriction and corresponding non-negative Lagrange multiplier

Λ^{2}

, where

J_{1} (u, v, Λ) = \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} .

where

V_{0} = {v \in W_{0}^{1, 4} (Ω) : {(v^{'})}^{2} - K \leq 0 in Ω} .

Observe that

\begin{matrix} \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} \\ = & - {〈 v_{0}^{*}, {(u^{'} + v^{'})}^{2} - 1 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x \\ + {〈 v_{0}^{*}, {(u^{'} + v^{'})}^{2} - 1 〉}_{L^{2}} + {〈 Λ^{2}, {(v^{'})}^{2} - K 〉}_{L^{2}} - {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} - {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} \\ + {〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}} \\ \geq & inf_{w \in Y} \{- {〈 v_{0}^{*}, w 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} {(w)}^{2} d x\} \\ inf_{(v_{1}, v_{2}) \in Y \times Y} \{{〈 v_{0}^{*}, {(v_{1} + v_{2})}^{2} - 1 〉}_{L^{2}} + {〈 Λ^{2}, {(v_{2})}^{2} - K 〉}_{L^{2}} - {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}}\} \\ + inf_{(u, v) \in V \times V_{0}} \{{〈 u^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 v^{'}, v_{2}^{*} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ = & - \frac{1}{2} \int_{Ω} {(v_{0}^{*})}^{2} d x - \int_{Ω} v_{0}^{*} d x \\ - \frac{1}{4} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{v_{0}^{*}} d x - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{2 Λ^{2}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - \frac{1}{2} \int_{Ω} K Λ^{2} d x + v_{1}^{*} (1) u (1) . \end{matrix}

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Here, we highlight

v_{2}^{*} = c \in R in Ω

, for some real constant c.

Hence, denoting

\begin{matrix} J_{1}^{*} (v^{*}, Λ) & = & - \frac{1}{2} \int_{Ω} {(v_{0}^{*})}^{2} d x - \int_{Ω} v_{0}^{*} d x \\ - \frac{1}{4} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{v_{0}^{*}} d x - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} - v_{2}^{*})}^{2}}{2 Λ^{2}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + f)}^{2} d x - \frac{1}{2} \int_{Ω} K Λ^{2} d x + v_{1}^{*} (1) u (1) \end{matrix}

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and

J_{2} (u, v) = \frac{1}{2} \int_{Ω} {({(u^{'} + v^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

we have obtained

inf_{(u, v) \in V \times V_{0}} J_{2} (u, v)} \geq sup_{(v^{*}, Λ) \in A^{*} \times [Y^{*}] \times R \times Y^{*}} J_{1}^{*} (v^{*}, Λ) .

Finally, for

A^{*} = {v_{0}^{*} \in Y^{*} : v_{0}^{*} \geq ε in Ω}

we emphasize

J_{1}^{*}

is concave on

A^{*} \times [Y^{*}] \times R \times Y^{*}

Here

ε > 0

is a small regularizing real constant.

Remark 16.1.

The constraint

{(v^{'})}^{2} - K \leq 0, in Ω

is included to restrict the action of v on the region where the primal functional is non-convex, through an appropriate constant

K > 0 .

17. A model in superconductivity through an eigenvalue approach

In this section we intend to model superconductivity through a two phase eigenvalue approach.

Let

Ω = [0, 5] \subset R

be a straight wire corresponding to a one-dimensional super-conducting sample.

Consider the functional

J : V \times V \times R \to R

where

\begin{matrix} J (u, v, E) & = & \frac{γ_{1}}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α_{1}}{2} \int_{Ω} {| u |}^{4} d x \\ - \frac{ω^{2}}{2} \int_{Ω} {| u |}^{2} d x \\ + \frac{γ_{2}}{2} \int_{Ω} \nabla v \cdot \nabla v d x + \frac{α_{2}}{2} \int_{Ω} {| v |}^{4} d x \\ - \frac{ω_{1}^{2}}{2 K_{3}^{2}} \int_{Ω} {| v |}^{2} d x \\ - \frac{E}{2} (\int_{Ω} {(| u |}^{2} + {| v |}^{2}) d x - m_{T}) . \end{matrix}

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Here, in atomic units,

m_{T}

is the total electronic charge,

V = W_{0}^{1, 2} (Ω)

and we set

α_{1} = 10^{4}

corresponding to higher self-interacting energy which is related to a normal phase. We also set

α_{2} = 10^{- 1}

corresponding to a lower self-interacting energy which is related to a super-conducting phase and respective super-currents.

Moreover, we set

γ_{1} = γ_{2} = 1,

and initially

ω = 1.8

which is gradually decreased to

ω = 1.0

Furthermore, we define

| ϕ_{N} |^{2} = \frac{{| u |}^{2}}{{| u |}^{2} + {| v |}^{2}}

and

| ϕ_{S} |^{2} = \frac{{| v |}^{2}}{{| u |}^{2} + {| v |}^{2}}

where

ϕ_{N}

corresponds to a normal phase and

ϕ_{S}

to a super-conducting one.

At this point we observe that the temperature

T = T (x, t)

is proportional the frequency

ω / (2 π)

of vibration for the normal phase.

We start the process with

ω = 1.8

which in atomic units corresponds to a higher temperature and gradually decreases it to the value

ω = 1.0

Between

ω = 1.2

and

ω = 1.0

the system changes from an almost total normal phase to an almost total super-conducting phase, as expected.

We highlight that the temperature is proportional to the vibrational kinetics energy

E_{1} (t) = \frac{1}{2} \int_{Ω} {| u |}^{2} \frac{\partial r_{N} (x, t)}{\partial t} \cdot \frac{\partial r_{N} (x, t)}{\partial t} d x

so that for

r_{N} (x, t) = e^{i ω t} w_{5} (x)

and for a suitable vectorial function

w_{5}

, we have

T \propto E_{1} \propto ω^{2}

so that we may model the decreasing of temperature T through the decreasing of

ω^{2}

For

ω = 1.8

, for the corresponding normal phase

ϕ_{N}

and super-conducting phase

ϕ_{S}

, please se Figure 11 and Figure 12, respectively.

For

ω = 1.0

, for the corresponding normal phase

ϕ_{N}

and super-conducting phase

ϕ_{S}

, please se Figure 13 and Figure 14, respectively.

Finally, we have set

ω_{1} / K_{3} \approx 1

which for large

ω_{1}

corresponds to the super-currents.

18. A simplified qualitative many body model for the hydrogen nuclear fusion

In this section we develop a qualitative simple model for the hydrogen nuclear fusion.

Let

Ω = {[0, L]}^{3} \subset R^{3}

be a box in which is confined a gas comprised by an amount of ionized deuterium and tritium isotopes of hydrogen.

Though a suitable increasing in temperature, we intend to develop the following nuclear reaction

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom comprises a proton and a neutron and the ionized Tritium atom comprises a proton and two neutrons.

At this point we denote by

m_{D}, m_{T}, m_{H_{e}} and m_{N}

the masses of the ionized Deuterium, Tritium and Helium atoms, and the single neutron, respectively.

Therefore, we have the following mass relation

m_{D} + m_{T} = m_{H_{e}} + m_{N} .

To simplify our analysis, in such a chemical reaction, denoting the total masses of ionized Deuterium, Tritium, Helium and single Neutrons by

{(m_{D})}_{T}, {(m_{T})}_{T}, {(m_{H_{e}})}_{T}

and

{(m_{N})}_{T}

we assume there is a real constant

c > 0

such that

{(m_{D})}_{T} = c m_{D}, {(m_{T})}_{T} = c m_{T}, {(m_{H_{e}})}_{T} = c m_{H_{e}}, {(m_{N})}_{T} = c m_{N} .

With such statements and definitions in mind, we define the following functional J, where

J (ϕ, r) = J (ϕ_{D}, ϕ_{T}, ϕ_{H_{e}}, ϕ_{N}, r) = G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r),

where, in a simplified many body context,

| ϕ_{D} {(x, y) |}^{2} = | ϕ_{p}^{D} {(y) |}^{2} + | ϕ_{N}^{D} {(x, y) |}^{2} {| ϕ_{p}^{D} (y) |}^{2} \frac{1}{m_{p}},

| ϕ_{T} {(x, y) |}^{2} = | ϕ_{p}^{T} {(y) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y) |}^{2}) | ϕ_{p}^{T} {(y) |}^{2} \frac{1}{m_{p}},

| ϕ_{H_{e}} {(x, y) |}^{2} = | ϕ_{2 P}^{H_{e}} {(y) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y) |}^{2}) | ϕ_{2 P}^{H_{e}} {(y) |}^{2} \frac{1}{2 m_{p}},

ϕ_{N} = ϕ_{N} (x) .

Here

x, y \in Ω \subset R^{3}

refers to the particle densities.

Furthermore, we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0,

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0,

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y \\ + \frac{γ_{N}^{D}}{2} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y \\ \frac{γ_{p}^{T}}{2} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y \\ + \frac{γ_{N}}{2} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x, \end{matrix}

(84)

and,

\begin{matrix} F (ϕ) & = & \frac{α_{D}}{2} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{T}}{2} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{D T}}{2} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{H_{e}}}{2} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{N} {(x - ξ) |}^{2} {| ϕ_{N} (ξ) |}^{2}}{| x - ξ |} d x d ξ \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}) |^{2} {| ϕ_{N} (ξ_{j}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} \end{matrix}

(85)

and the kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y, \end{matrix}

(86)

where we also assume

r_{D} \approx e^{i ω t} w_{5} (x, y),

r_{T} \approx e^{i ω t} w_{6} (x, y),

so that considering such a vibrational motion, the temperature T is proportional to

ω^{2}

, that is

T \propto ω^{2} .

Therefore, an increasing in T corresponds to a proportional increasing in

ω^{2} .

Summarizing, we have supposed

E_{c} (ϕ, r) \approx \frac{1}{2} ω^{2} \int_{Ω} | ϕ_{D} |^{2} + | ϕ_{T} |^{2} d x C_{1} + \frac{1}{2} ω_{1}^{2} \int_{Ω} {| ϕ_{N} |}^{2} d x C_{2},

so that we represent the increasing in T through an increasing in

ω^{2} .

Moreover, we denote by

m_{N}

the mass of a single neutron and by

m_{p}

the mass of a single proton.

Thus, denoting also by

λ_{1}, λ_{2}

the proportion of non-reacted and reacted masses respectively, we have the following constraints.

1.: $\int_{Ω} {| ϕ_{N}^{D} (x, y) |}^{2} d x = m_{N},$
2.: $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y) |}^{2} d x = m_{N},$
3.: $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y) |}^{2} d x = m_{N},$
4.: $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
5.: $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
6.: $\int_{Ω} {| ϕ_{p}^{D} (y) |}^{2} d y = λ_{1} c m_{p},$
7.: $\int_{Ω} {| ϕ_{p}^{T} (y) |}^{2} d y = λ_{1} c m_{p},$
8.: $\int_{Ω} {| ϕ_{2 P}^{H_{e}} (y) |}^{2} d y = λ_{2} (2 c m_{p}),$

Similar constraints are valid corresponding to the charge of a single proton.

We have also the following complementing constraints,

1.: $\int_{Ω} {| ϕ_{D} |}^{2} d x d y = λ_{1} {(m_{D})}_{T},$
2.: $\int_{Ω} {| ϕ_{T} |}^{2} d x d y = λ_{1} {(m_{T})}_{T},$
3.: $\int_{Ω} {| ϕ_{H_{e}} |}^{2} d x d y = λ_{2} {(m_{H_{e}})}_{T},$
4.: $\int_{Ω} {| ϕ_{N} |}^{2} d x d y = λ_{2} {(m_{N})}_{T},$
5.: $λ_{1} + λ_{2} = 1 .$

With such results and statements in mind and simplifying the interacting terms, we re-define the functional J now denoting it by

J_{1}

, here already including the Lagrange multipliers concerning the constraints, where

\begin{matrix} J_{1} (ϕ, ω, E, λ) & = & \frac{γ_{p}^{D}}{2} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y \\ + \frac{γ_{N}^{D}}{2} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y \\ \frac{γ_{p}^{T}}{2} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y \\ + \frac{γ_{N}}{2} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x \\ + \frac{α_{D}}{2} \int_{Ω} | ϕ_{D} |^{4} d x + \frac{α_{T}}{2} \int_{Ω} {| ϕ_{T} |}^{4} d x \\ + \frac{α_{H_{e}}}{2} \int_{Ω} | ϕ_{H_{e}} |^{4} d x + \frac{α_{N}}{2} \int_{Ω} {| ϕ_{N} |}^{4} d x \\ - ω^{2} \int_{Ω} (| ϕ_{D} |^{2} + | ϕ_{T} |^{2}) d x \\ - ω_{1}^{2} \int_{Ω} {| ϕ_{N} |}^{2} d x + J_{A u x}, \end{matrix}

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where the functional

J_{A u x}

stands for

\begin{matrix} J_{A u x} & = & - \int_{Ω} {(E_{N}^{D})}_{5} (y) (\int_{Ω} {| ϕ_{N}^{D} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y) |}^{2} d x - m_{N}) d y \\ - \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y) |}^{2} d x - m_{N}) d y \\ - {(E_{D})}_{2} (\int_{Ω} {| ϕ_{p}^{D} (y) |}^{2} d y - λ_{1} c m_{p}) \\ - {(E_{T})}_{3} (\int_{Ω} {| ϕ_{p}^{T} (y) |}^{2} d y - λ_{1} c m_{p}) \\ - {(E_{H_{e}})}_{3} (\int_{Ω} {| ϕ_{2 P}^{H_{e}} (x, y) |}^{2} d y - λ_{2} 2 c m_{p}) \\ - E_{5} (\int_{Ω} {| ϕ_{D} |}^{2} d x d y - λ_{1} {(m_{D})}_{T}) \\ - E_{6} (\int_{Ω} {| ϕ_{T} |}^{2} d x d y - λ_{1} {(m_{T})}_{T}) \\ - E_{7} (\int_{Ω} {| ϕ_{H_{e}} |}^{2} d x d y - λ_{2} {(m_{H_{e}})}_{T}) \\ - E_{8} (\int_{Ω} {| ϕ_{N} |}^{2} d x d y - λ_{2} {(m_{N})}_{T}) \\ - E_{9} (λ_{1} + λ_{2} - 1) . \end{matrix}

(88)

Remark 18.1.

In order to obtain consistent results it is necessary to set

(α_{N}, α_{H_{e}}) ≫ (α_{D}, α_{T}) .

In such a case, a higher temperature corresponding to a large

ω^{2}

, though such a nuclear reaction, will result in a small

λ_{1}

and a higher kinetics energy for the neutron field, corresponding to a large

ω_{1}^{2}

and

λ_{2}

closer to 1.

19. A more detailed mathematical description of the hydrogen nuclear fusion

In this section we develop in more details another model for the hydrogen nuclear fusion.

Remark 19.1.

Denoting by

i \in C

the imaginary unit, in this and in the subsequent sections, for the time-dependent case we generically define the gradient of a scalar function

u (x, t)

with domain in

R^{4}

, denoted by

\nabla u (x, t)

, as

\nabla u (x, t) = (i u_{t} (x, t), u_{x_{1}} (x, t), u_{x_{2}} (x, t), u_{x_{3}} (x, t)),

so that

\nabla u \cdot \nabla u = - u_{t}^{2} + \sum_{j = 1}^{3} u_{x_{j}}^{2} .

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Here such a set

Ω

Symbolically such a reaction stands for

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval

[0, t_{f}],

at this point we define the following density functions:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + | ϕ_{N}^{D} {(x, y, t) |}^{2} {| ϕ_{p}^{D} (y, t) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{T} {(x, y, t) |}^{2} = | ϕ_{p}^{T} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t) |}^{2}) | ϕ_{p}^{T} {(y, t) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t) .$

Furthermore, we define also the related densities

1.: $ρ_{D} (y, t) = \int_{Ω} {| ϕ_{D} (x, y, t) |}^{2} d x,$
2.: $ρ_{T} (y, t) = \int_{Ω} {| ϕ_{T} (x, y, t) |}^{2} d x,$

$ρ_{H_{e}} (y, t) = \int_{Ω} {| ϕ_{H_{e}} (x, y, t) |}^{2} d x,$

$ρ_{N} (x, t) = {| ϕ_{N} (x, t) |}^{2},$

$ρ_{e} (y, t) = \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x .$

For the chemical reaction in question we consider that one unit of mass of fractional proportion

α_{D}

of ionized Deuterium and

α_{T}

of ionized Tritium results in one unit of mass of fractional proportion

α_{H_{e}}

of ionized Helium and

α_{N}

of neutrons.

Symbolic, this stands for

1 = α_{D} + α_{T} = α_{H_{e}} + α_{N} .

Concerning the control volume

Ω

in question and related surface control

\partial Ω,

we assume such a volume has an initial (fot

t = 0

) amount of ionized Deuterium of

{(m_{D})}_{0}

and an initial amount of ionized Tritium of

{(m_{T})}_{0} .

The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control

\partial Ω

, we assume there is a part

Ω_{1} \subset \partial Ω

for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part

\partial Ω_{2} \subset \partial Ω

such that

\partial Ω_{1} \cap \partial Ω_{2} = \emptyset

for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

\partial Ω_{2}

is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
6.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
7.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
8.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
9.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

Here

n

denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, we define the functional

J (ϕ, ρ, r, u, E, A, B)

where

J = G (\nabla u) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3},

and where we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0, γ_{e} > 0

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0, α_{e, e} > 0, α_{H_{e}, e} < 0

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y d t \\ + \frac{γ_{N}^{D}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y d t \\ \frac{γ_{p}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y d t \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y d t \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y d t \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y d t \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y d t \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y d t \\ + \frac{γ_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{e}) \cdot (\nabla ϕ_{e}) d x d y d t, \end{matrix}

(89)

and

\begin{matrix} F (ϕ) & = & \frac{α_{D}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{T}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{D T}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{N} {(x - ξ, t) |}^{2} {| ϕ_{N} (ξ) |}^{2}}{| x - ξ, t |} d x d ξ d t \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{N} (ξ_{j}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}, e}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{e, e}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{e} (x - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \end{matrix}

(90)

and the internal kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{e} |}^{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} d x d y d t, \end{matrix}

(91)

Here it is worth highlighting we have approximated the initially discrete set of indices s of particles as a continuous positive real variable s.

Moreover,

F_{1} = \frac{1}{4 π} \int_{0}^{t_{f}} {∥ curl A - B_{0} ∥}_{2} d t,

\begin{matrix} F_{2} & = & \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d x d y d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{i n d} \cdot K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d x d y d t, \end{matrix}

(92)

where

K_{p}

and

K_{e}

are appropriate real constants related to the respective charges.

Here

u = (u_{1}, u_{2}, u_{3})

is the fluid velocity field and

r_{D}, r_{T}, r_{H_{e}}, r_{N}, r_{e}

are fields of displacements for the corresponding atom fields.

Also

A

denotes the magnetic potential,

B_{0}

an external magnetic field and

B

is the total magnetic field.

Moreover,

E_{i n d}

is an induced electric field.

Finally,

\begin{matrix} F_{3} & = & \frac{C_{D}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{D} \cdot \nabla_{(x, y)} r_{D} d x d y d t + \frac{C_{T}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{T} \cdot \nabla_{(x, y)} r_{T} d x d y d t \\ \frac{C_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{H_{e}} \cdot \nabla_{(x, y)} r_{H_{e}} d x d y d t + \frac{C_{N}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{N} \cdot \nabla_{(x, y)} r_{N} d x d y d t \\ \frac{C_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla_{(x, y)} r_{e} \cdot \nabla_{(x, y)} r_{e} d x d y d t, \end{matrix}

(93)

for appropriate real positive constants

C_{D} C_{T}, C_{H_{e}}, C_{N}, C_{e} .

Such a functional J is subject to the following constraints:

1.

The momentum conservation equation for the fluid motion

ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) = ρ f_{k} - \frac{\partial P}{\partial x_{k}} + τ_{k j, j} + {(F_{E})}_{k} + {(F_{M})}_{k},

\forall k \in {1, 2, 3} .

Here

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e}

is the total density and P is the fluid pressure field.

Furthermore,

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} δ_{i j} \sum_{k = 1}^{3} \frac{\partial u_{k}}{\partial x_{k}}),

\forall i, j \in {1, 2, 3},

F_{E} = {{(F_{E})}_{k}} = (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} |}^{2} d x) E,

and

\begin{matrix} F_{M} & = & {{(F_{M})}_{k}} \\ = & (K_{p} (| ϕ_{p}^{D} |^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ | ϕ_{p}^{T} |^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + | ϕ_{2 p}^{H_{e}} |^{2} (u + \frac{\partial r_{H_{e}}}{\partial t})) \\ + K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t})) \times B . \end{matrix}

(94)

2.

Mass conservation equation:

\frac{\partial ρ}{\partial t} + div (ρ u) = 0 .

3.

Energy equation

ρ \frac{D e}{D t} + P (div u) = \frac{\partial Q}{\partial t} - div q,

where we assume the Fourier law

q = - K \nabla T,

where

T = T (x, t)

is the scalar field of temperature.

Also,

\begin{matrix} e & = & \frac{ρ}{2} u \cdot u + \frac{ρ_{D}}{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} \\ + \frac{ρ_{T}}{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} \\ + \frac{ρ_{H_{e}}}{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} \\ + \frac{ρ_{N}}{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} \\ + \frac{ρ_{e}}{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} \end{matrix}

(95)

and

\frac{D e}{D t} = \frac{\partial e}{\partial t} + u_{j} \frac{\partial e}{\partial x_{j}} .

4.

P = F_{7} (ρ, T),

for an appropriate scalar function

F_{7}

5.

Mass relations

(a): $m_{D} (t) = \int_{Ω} ρ_{D} (x, t) d x,$
(b): $m_{T} (t) = \int_{Ω} ρ_{T} (x, t) d x,$
(c): $m_{H e} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
(d): $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
(e): $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x,$
(f): ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
(g): ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
(h): $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$

where,

(a): ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$
(b): $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
(c): $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
(d): $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
(e): ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$
(f): $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

6.

Other mass constraints

(a): $\int_{Ω} {| ϕ_{N}^{D} (x, y, t) |}^{2} d x = m_{N},$
(b): $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t) |}^{2} d x = m_{N},$
(c): $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t) |}^{2} d x = m_{N},$
(d): $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t) |}^{2} d x = m_{N},$
(e): $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t) |}^{2} d x = m_{N} .$

7.

For the induced electric field, we must have

\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x)) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0}) = 0, \end{matrix}

(96)

where

{\hat{K}}_{p}

and

{\hat{K}}_{e}

are appropriate real constants related to the respective charges.

8.

A Maxwell equation:

div B = 0,

where

B = B_{0} - curl A .

9.

Another Maxwell equation:

div E = 4 π (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x),

where the total electric field

E

stands for

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, ξ) d x d ξ,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, ξ)}{\partial x_{k}} d ξ\} .

At this point we generically denote

{〈 h_{1}, h_{2} 〉}_{L^{2}} = \int_{0}^{t_{f}} \int_{Ω} h_{1} h_{2} d x d y d t .

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional

J_{3}

stands for

\begin{matrix} J_{3} = J_{3} (ϕ, u, r, P, A, B, E, Λ, E) \\ = & G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} \\ + {〈Λ_{k}, ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) - ρ f_{k} + \frac{\partial P}{\partial x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{\partial ρ}{\partial t} + div (ρ u)〉}_{L^{2}} + J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}}, \end{matrix}

(97)

where,

\begin{matrix} J_{A u x_{1}} & = & {〈Λ_{5}, ρ \frac{D e}{D t} + P (div u) - \frac{\partial Q}{\partial t} + div q〉}_{L^{2}} \\ + {〈Λ_{6}, P - F_{7} (ρ, T)〉}_{L^{2}}, \end{matrix}

(98)

\begin{matrix} J_{A u x_{2}} & = & {〈Λ_{7}, m_{D} (t) - \int_{Ω} ρ_{D} (x, t) d x〉}_{L^{2}} \\ + {〈Λ_{8}, m_{T} (t) - \int_{Ω} ρ_{T} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{9}, m_{H_{e}} (t) - \int_{Ω} ρ_{H_{e}} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{10}, m_{N} (t) - \int_{Ω} ρ_{N} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{11}, m_{e} (t) - \int_{Ω} ρ_{e} (x, t) d x〉}_{L^{2}} \\ \int_{0}^{t_{f}} E_{1} 2 (t) {(α_{N} m_{H_{e}})}_{T} (t) - α_{H_{e}} {(m_{N})}_{T} (t)) d t, \end{matrix}

(99)

\begin{matrix} J_{A u x_{3}} & = & - \int_{0}^{t_{f}} \int_{Ω} {(E_{N}^{D})}_{5} (y, t) (\int_{Ω} {| ϕ_{N}^{D} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y, t) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y, t) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y, t) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y, t) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t) |}^{2} d x - m_{N}) d y d t, \end{matrix}

(100)

\begin{matrix} J_{A u x_{4}} & = & 〈Λ_{12}, curl E_{i n d} \\ + \frac{1}{c} curl ({\hat{K}}_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x)) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0})〉}_{L^{2}} \\ + {〈Λ_{13}, div B〉}_{L^{2}} \\ + {〈Λ_{14}, div E - 4 π (K_{p} (| ϕ_{p}^{D} |^{2} + | ϕ_{p}^{T} |^{2} + | ϕ_{2 p}^{H_{e}} |^{2}) + K_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x)〉}_{L^{2}} . \end{matrix}

(101)

Here we recall the following definitions and relations:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + | ϕ_{N}^{D} {(x, y, t) |}^{2} {| ϕ_{p}^{D} (y, t) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{D} {(x, y, t) |}^{2} = | ϕ_{p}^{D} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{D} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{D} {(x, y, t) |}^{2}) | ϕ_{p}^{D} {(y, t) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t) .$

1.: $ρ_{D} (y, t) = \int_{Ω} {| ϕ_{D} (x, y, t) |}^{2} d x,$
2.: $ρ_{T} (y, t) = \int_{Ω} {| ϕ_{T} (x, y, t) |}^{2} d x,$

$ρ_{H_{e}} (y, t) = \int_{Ω} {| ϕ_{H_{e}} (x, y, t) |}^{2} d x,$

$ρ_{N} (x, t) = {| ϕ_{N} (x, t) |}^{2},$

$ρ_{e} (y, t) = \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} d x .$

Also,

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e},

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

Finally,

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, ξ) d x d ξ,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, ξ)}{\partial x_{k}} d ξ\} .

and,

B = B_{0} - curl A .

20. A final mathematical description of the hydrogen nuclear fusion

In this section we develop in even more details another model for the hydrogen nuclear fusion.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Here such a set

Ω

Symbolically such a reaction stands for

{Deuterium}^{+} + {Tritium}^{+} \to {Helium}^{+ +} + Neutron (energetic) .

We recall that the ionized Deuterium atom is comprised by a proton and a neutron and the ionized Tritium atom is comprised by a proton and two neutrons.

Moreover, the ionized Helium atom is comprised by two protons and two neutrons.

We highlight the model here presented includes electric and magnetic fields and the corresponding potential ones.

Denoting by t the time on the interval

[0, t_{f}],

at this point we define the following density functions:

1.: For a single Deuterium atom indexed by s:

$| ϕ_{D} {(x, y, t, s) |}^{2} = | ϕ_{p}^{D} {(y, t, s) |}^{2} + | ϕ_{N}^{D} {(x, y, t, s) |}^{2} {| ϕ_{p}^{D} (y, t, s) |}^{2} \frac{1}{m_{p}},$
2.: For a single Tritium atom indexed by s:

$| ϕ_{T} {(x, y, t, s) |}^{2} = | ϕ_{p}^{T} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t, s) |}^{2}) | ϕ_{p}^{T} {(y, t, s) |}^{2} \frac{1}{m_{p}},$
3.: For a single Helium atom indexed by s:

$| ϕ_{H_{e}} {(x, y, t, s) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t, s) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field:

$ϕ_{N} = ϕ_{N} (x, t, s),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

Furthermore, we define also the related densities

1.: $ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
2.: $ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

For the chemical reaction in question we consider that one unit of mass of fractional proportion

α_{D}

of ionized Deuterium and

α_{T}

of ionized Tritium results in one unit of mass of fractional proportion

α_{H_{e}}

of ionized Helium and

α_{N}

of neutrons.

Symbolically, this stands for

1 = α_{D} + α_{T} = α_{H_{e}} + α_{N} .

Concerning the control volume

Ω

in question and related surface control

\partial Ω,

we assume such a volume has an initial (fot

t = 0

) amount of ionized Deuterium of

{(m_{D})}_{0}

and an initial amount of ionized Tritium of

{(m_{T})}_{0} .

The initial amount of ionized Helium and single neutrons are supposed to be zero.

On the other hand, about the surface control

\partial Ω

, we assume there is a part

Ω_{1} \subset \partial Ω

for which is allowed the entrance and exit of Deuterium and Tritium ionized atoms.

We assume also there is another part

\partial Ω_{2} \subset \partial Ω

such that

\partial Ω_{1} \cap \partial Ω_{2} = \emptyset

for which is allowed only the exit of ionized Helium atoms and neutrons, but not their entrance.

\partial Ω_{2}

is allowed the exit only (not the entrance) of ionized Deuterium and Tritium atoms.

Indeed, we assume the following relations for the masses:

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

(102)

Here

n

denotes the outward normal vectorial fields to the concerning surfaces.

Having clarified such masses relations, denoting by

N_{D} (t) N_{T} (t), N_{H_{e}} (t), N_{N} (t), N_{e} (t)

the respective indexed number of particles at time t, we define the functional

J (ϕ, ρ, r, u, E, A, B, {N_{D}, N_{T}, N_{H_{e}}, N_{N}, N_{e}})

where

J = G (\nabla u) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} + F_{4},

and where we assume

γ_{p}^{D} > 0, γ_{p}^{T} > 0, γ_{N}^{D} > 0, γ_{N_{1}}^{T} > 0, γ_{N_{2}}^{T} > 0,

γ_{2 p}^{H_{e}} > 0, γ_{N_{1}}^{H_{e}} > 0,

γ_{N_{2}}^{H_{e}} > 0, γ_{N} > 0, γ_{e} > 0

and

α_{D} > 0, α_{T} > 0, α_{H_{e}} > 0, α_{N} > 0

α_{D T} > 0, α_{H_{e} N} > 0, α_{e, e} > 0, α_{H_{e}, e} < 0

so that

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}^{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} (\nabla ϕ_{p}^{D}) \cdot (\nabla ϕ_{p}^{D}) d y d s d t \\ + \frac{γ_{N}^{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} (\nabla ϕ_{N}^{D}) \cdot (\nabla ϕ_{N}^{D}) d x d y d s d t \\ \frac{γ_{p}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{p}^{T}) \cdot (\nabla ϕ_{p}^{T}) d y d s d t \\ + \frac{γ_{N_{1}}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{N_{1}}^{T}) \cdot (\nabla ϕ_{N_{1}}^{T}) d x d y d s d t \\ + \frac{γ_{N_{2}}^{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} (\nabla ϕ_{N_{2}}^{T}) \cdot (\nabla ϕ_{N_{2}}^{T}) d x d y d s d t \\ + \frac{γ_{2 p}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{2 p}^{H_{e}}) \cdot (\nabla ϕ_{2 p}^{H_{e}}) d y d s d t \\ + \frac{γ_{N_{1}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{N_{1}}^{H_{e}}) \cdot (\nabla ϕ_{N_{1}}^{H_{e}}) d x d y d s d t \\ + \frac{γ_{N_{2}}^{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} (\nabla ϕ_{N_{2}}^{H_{e}}) \cdot (\nabla ϕ_{N_{2}}^{H_{e}}) d x d y d s d t \\ + \frac{γ_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} (\nabla ϕ_{N}) \cdot (\nabla ϕ_{N}) d x d s d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} (\nabla ϕ_{e}) \cdot (\nabla ϕ_{e}) d x d y d s d t, \end{matrix}

(103)

and

\begin{matrix} F (ϕ) = \\ \frac{α_{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{0}^{N_{D} (t)} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{D} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{0}^{N_{T} (t)} \int_{Ω} \frac{| ϕ_{T} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{D T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{0}^{N_{T} (t)} \int_{Ω} \frac{| ϕ_{D} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{T} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d t \\ + \frac{α_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{H_{e}} (ξ_{1}, ξ_{2}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}, s_{1}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{0}^{N_{N} (t)} \int_{Ω} \frac{| ϕ_{N} (x - ξ, t, s - s_{1}) |^{2} {| ϕ_{N} (ξ, t, s_{1}) |}^{2}}{| x - ξ |} d x d ξ d s d s_{1} d t \\ + \sum_{j = 1}^{2} \frac{α_{H_{e} N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{D} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x_{1} - ξ_{1}, y - ξ_{2}, t) |^{2} {| ϕ_{N} (ξ_{j}, t) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{H_{e}, e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{0}^{N_{e} (t)} \int_{Ω} \frac{| ϕ_{H_{e}} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \\ + \frac{α_{e, e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{0}^{N_{e} (t)} \int_{Ω} \frac{| ϕ_{e} (x - ξ_{1}, y - ξ_{2}, t, s - s_{1}) |^{2} {| ϕ_{e} (ξ_{1}, ξ_{2}, t, s_{1}) |}^{2}}{| (x, y) - (ξ_{1}, ξ_{2}) |} d x d y d ξ_{1} d ξ_{2} d s d s_{1} d t \end{matrix}

(104)

and the internal kinetics energy is expressed by

\begin{matrix} E_{c} (ϕ, r) & = & \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} |}^{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} |}^{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} {| ϕ_{N} |}^{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} d x d y d s d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} |}^{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} d x d y d s d t, \end{matrix}

(105)

Moreover,

F_{1} = \frac{1}{4 π} \int_{0}^{t_{f}} {∥ curl A - B_{0} ∥}_{2} d t,

\begin{matrix} F_{2} & = & \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} E_{i n d} \cdot K_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d x d y d s d t \\ + \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} E_{i n d} \cdot K_{e} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d x d y d s d t, \end{matrix}

(106)

where

K_{p}

and

K_{e}

are appropriate real constants related to the respective charges.

Here

u = (u_{1}, u_{2}, u_{3})

is the fluid velocity field and

r_{D}, r_{T}, r_{H_{e}}, r_{N}, r_{e}

are fields of displacements for the corresponding particle fields.

Also

A

denotes the magnetic potential,

B_{0}

an external magnetic field and

B

is the total magnetic field.

Moreover,

E_{i n d}

is an induced electric field.

Also,

\begin{matrix} F_{3} & = & \frac{C_{D}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{D} (t)} \int_{Ω} \nabla_{(x, y)} r_{D} \cdot \nabla_{(x, y)} r_{D} d x d y d s d t \\ + \frac{C_{T}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{T} (t)} \int_{Ω} \nabla_{(x, y)} r_{T} \cdot \nabla_{(x, y)} r_{T} d x d y d s d t \\ + \frac{C_{H_{e}}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{H_{e}} (t)} \int_{Ω} \nabla_{(x, y)} r_{H_{e}} \cdot \nabla_{(x, y)} r_{H_{e}} d x d y d s d t \\ + \frac{C_{N}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{N} (t)} \int_{Ω} \nabla_{(x, y)} r_{N} \cdot \nabla_{(x, y)} r_{N} d x d y d s d t \\ \frac{C_{e}}{2} \int_{0}^{t_{f}} \int_{0}^{N_{e} (t)} \int_{Ω} \nabla_{(x, y)} r_{e} \cdot \nabla_{(x, y)} r_{e} d x d y d s d t, \end{matrix}

(107)

for appropriate real positive constants

C_{D} C_{T}, C_{H_{e}}, C_{N}, C_{e} .

Finally,

\begin{matrix} F_{4} & = & \frac{ε_{D}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{D} (t)}{\partial t})}^{2} d t + \frac{ε_{T}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{D} (t)}{\partial t})}^{2} d t \\ + \frac{ε_{N}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{N} (t)}{\partial t})}^{2} d t + \frac{ε_{H_{e}}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{H_{e}} (t)}{\partial t})}^{2} d t \\ + \frac{ε_{e}}{2} \int_{0}^{t_{f}} {(\frac{\partial N_{e} (t)}{\partial t})}^{2} d t, \end{matrix}

(108)

where

ε_{D}, ε_{T}, ε_{N}, ε_{H_{e}}, ε_{e}

are small real positive constants.

Such a functional J is subject to the following constraints:

1.

The momentum conservation equation for the fluid motion

ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) = ρ f_{k} - \frac{\partial P}{\partial x_{k}} + τ_{k j, j} + {(F_{E})}_{k} + {(F_{M})}_{k},

\forall k \in {1, 2, 3} .

Here

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e}

is the total density and P is the fluid pressure field.

Furthermore,

τ_{i j} = μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}} - \frac{2}{3} δ_{i j} \sum_{k = 1}^{3} \frac{\partial u_{k}}{\partial x_{k}}),

\forall i, j \in {1, 2, 3},

\begin{matrix} F_{E} = {{(F_{E})}_{k}} = \\ (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} d s) E, \end{matrix}

and

\begin{matrix} F_{M} & = & {{(F_{M})}_{k}} \\ = & (K_{p} (\int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s) \\ + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} (u + \frac{\partial r_{e}}{\partial t}) d s) \times B . \end{matrix}

(109)

2.

Mass conservation equation:

\frac{\partial ρ}{\partial t} + div (ρ u) = 0 .

3.

Energy equation

ρ \frac{D e}{D t} + P (div u) = \frac{\partial Q}{\partial t} - div q,

where we assume the Fourier law

q = - K \nabla T,

where

T = T (x, t)

is the scalar field of temperature.

Also,

\begin{matrix} e & = & \frac{ρ}{2} u \cdot u + \frac{ρ_{D}}{2} \frac{\partial r_{D}}{\partial t} \cdot \frac{\partial r_{D}}{\partial t} \\ + \frac{ρ_{T}}{2} \frac{\partial r_{T}}{\partial t} \cdot \frac{\partial r_{T}}{\partial t} \\ + \frac{ρ_{H_{e}}}{2} \frac{\partial r_{H_{e}}}{\partial t} \cdot \frac{\partial r_{H_{e}}}{\partial t} \\ + \frac{ρ_{N}}{2} \frac{\partial r_{N}}{\partial t} \cdot \frac{\partial r_{N}}{\partial t} \\ + \frac{ρ_{e}}{2} \frac{\partial r_{e}}{\partial t} \cdot \frac{\partial r_{e}}{\partial t} \end{matrix}

(110)

and

\frac{D e}{D t} = \frac{\partial e}{\partial t} + u_{j} \frac{\partial e}{\partial x_{j}} .

4.

P = F_{7} (ρ, T),

for an appropriate scalar function

F_{7}

5.

Mass relations

(a): $m_{D} (t) = \int_{Ω} ρ_{D} (x, t) d x,$
(b): $m_{T} (t) = \int_{Ω} ρ_{T} (x, t) d x,$
(c): $m_{H e} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
(d): $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
(e): $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x,$

where,

(a): ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$
(b): $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
(c): $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
(d): $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
(e): ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
(f): ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
(g): $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
(h): ${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$
(i): $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

(111)

6.

Other mass constraints

(a): $\int_{Ω} {| ϕ_{N}^{D} (x, y, t, s) |}^{2} d x = m_{N},$
(b): $\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t, s) |}^{2} d x = m_{N},$
(c): $\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t, s) |}^{2} d x = m_{N},$
(d): $\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t, s) |}^{2} d x = m_{N},$
(e): $\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t, s) |}^{2} d x = m_{N},$
(f): $\int_{Ω} {| ϕ_{p}^{D} (x, t, s) |}^{2} d x = m_{p},$
(g): $\int_{Ω} {| ϕ_{p}^{T} (x, t, s) |}^{2} d x = m_{p},$
(h): $\int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t, s) |}^{2} d x = 2 m_{p},$

7.

m_{D} (t) = m_{p} N_{D} (t) + m_{N} N_{D} (t)

m_{T} (t) = m_{p} N_{T} (t) + m_{N} N_{T} (t),

m_{H_{e}} (t) = 2 m_{p} N_{H_{e}} (t) + 2 m_{N} N_{H_{e}} (t),

m_{e} (t) = m_{e} N_{D} (t) + m_{e} N_{T} (t) + 2 m_{e} N_{H_{e}} (t) .

8.

For the induced electric field, we must have

\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} \int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t)}{\partial t} d x) d s) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0}) = 0, \end{matrix}

(112)

where

{\hat{K}}_{p}

and

{\hat{K}}_{e}

are appropriate real constants related to the respective charges.

9.

A Maxwell equation:

div B = 0,

where

B = B_{0} - curl A .

10.

Another Maxwell equation:

\begin{matrix} div E & = & 4 π (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) \\ + K_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s), \end{matrix}

(113)

where the total electric field

E

stands for

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, t ξ, s) d x d ξ d s,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, t, ξ, s)}{\partial x_{k}} d ξ d s\} .

At this point we generically denote

{〈 h_{1}, h_{2} 〉}_{L^{2}} = \int_{0}^{t_{f}} \int_{Ω} h_{1} h_{2} d x d y d t .

Thus, already including the Lagrange multipliers concerning the restrictions indicated, the extended functional

J_{3}

stands for

\begin{matrix} J_{3} = J_{3} (ϕ, u, r, P, A, B, E, Λ, E, {N_{D}, N_{T}, N_{H_{e}}, N_{N}, N_{e}}) \\ = & G (\nabla ϕ) + F (ϕ) + E_{c} (ϕ, r) + F_{1} + F_{2} + F_{3} + F_{4} \\ + {〈Λ_{k}, ρ (\frac{\partial u_{k}}{\partial t} + u_{j} \frac{\partial u_{k}}{\partial x_{j}}) - ρ f_{k} + \frac{\partial P}{\partial x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{\partial ρ}{\partial t} + div (ρ u)〉}_{L^{2}} + J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}} + J_{A u x_{5}}, \end{matrix}

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where,

\begin{matrix} J_{A u x_{1}} & = & {〈Λ_{5}, ρ \frac{D e}{D t} + P (div u) - \frac{\partial Q}{\partial t} + div q〉}_{L^{2}} \\ + {〈Λ_{6}, P - F_{7} (ρ, T)〉}_{L^{2}}, \end{matrix}

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\begin{matrix} J_{A u x_{2}} & = & {〈Λ_{7}, m_{D} (t) - \int_{Ω} ρ_{D} (x, t) d x〉}_{L^{2}} \\ + {〈Λ_{8}, m_{T} (t) - \int_{Ω} ρ_{T} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{9}, m_{H_{e}} (t) - \int_{Ω} ρ_{H_{e}} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{10}, m_{N} (t) - \int_{Ω} ρ_{N} (x, t) d x〉}_{L^{2}} \\ {〈Λ_{11}, m_{e} (t) - \int_{Ω} ρ_{e} (x, t) d x〉}_{L^{2}} \\ \int_{0}^{t_{f}} E_{12} (t) {(α_{N} m_{H_{e}})}_{T} (t) - α_{H_{e}} {(m_{N})}_{T} (t)) d t, \end{matrix}

(116)

\begin{matrix} J_{A u x_{3}} & = & - \int_{0}^{t_{f}} \int_{Ω} {(E_{N}^{D})}_{5} (y, t, s) (\int_{Ω} {| ϕ_{N}^{D} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{T})}_{6} (y, t, s) (\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{T})}_{7} (y, t, s) (\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{1}}^{H_{e}})}_{8} (y, t, s) (\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y, t, s) |}^{2} d x - m_{N}) d y d t \\ - \int_{0}^{t_{f}} \int_{Ω} {(E_{N_{2}}^{H_{e}})}_{9} (y, t, s) (\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t, s) |}^{2} d x - m_{N}) d y d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{p}^{D}) (t, s) (\int_{Ω} {| ϕ_{p}^{D} (y, t, s) |}^{2} d y - m_{p}) d s d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{p}^{T}) (t, s) (\int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y - m_{p}) d s d t, \\ - \int_{0}^{t_{f}} \int_{Ω} (E_{2 p}^{H_{e}}) (t, s) (\int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y - 2 m_{p}) d s d t, \end{matrix}

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\begin{matrix} J_{A u x_{4}} & = & 〈Λ_{12}, curl E_{i n d} \\ + \frac{1}{c} curl ({\hat{K}}_{p} \int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{\partial r_{D}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{\partial r_{T}}{\partial t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{\partial r_{H_{e}}}{\partial t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{\partial r_{e} (x, y, t, s)}{\partial t} d x) d s) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{\partial}{\partial t} (curl A - B_{0})〉}_{L^{2}} \\ + {〈Λ_{13}, div B〉}_{L^{2}} \\ + 〈Λ_{14}, div E - 4 π (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) \\ {+ K_{e} \int_{Ω} {| ϕ_{e} |}^{2} d x d s)〉}_{L^{2}} . \end{matrix}

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\begin{matrix} J_{A u x_{5}} & = & {〈 Λ_{15}, m_{D} (t) - (m_{p} N_{D} (t) + m_{N} N_{D} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{16}, m_{T} (t) - (m_{p} N_{T} (t) + m_{N} N_{T} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{17}, m_{H_{e}} (t) - (2 m_{p} N_{H_{e}} (t) + 2 m_{N} N_{H_{e}} (t)) 〉}_{L^{2}} \\ + {〈 Λ_{18}, m_{e} (t) - (m_{e} N_{D} (t) + m_{e} N_{T} (t) + 2 m_{e} N_{H_{e}} (t)) 〉}_{L^{2}} . \end{matrix}

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Here we recall the following definitions and relations:

1.: For the Deuterium field

$| ϕ_{D} {(x, y, t, s) |}^{2} = | ϕ_{p}^{D} {(y, t, s) |}^{2} + | ϕ_{N}^{D} {(x, y, t, s) |}^{2} {| ϕ_{p}^{D} (y, t, s) |}^{2} \frac{1}{m_{p}},$
2.: For the Tritium field

$| ϕ_{T} {(x, y, t, s) |}^{2} = | ϕ_{p}^{T} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t, s) |}^{2}) | ϕ_{p}^{D} {(y, t, s) |}^{2} \frac{1}{m_{p}},$
3.: For the Helium field

$| ϕ_{H_{e}} {(x, y, t, s) |}^{2} = | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{H_{e}} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{H_{e}} {(x, y, t, s) |}^{2}) | ϕ_{2 p}^{H_{e}} {(y, t, s) |}^{2} \frac{1}{2 m_{p}},$
4.: For the Neutron field

$ϕ_{N} = ϕ_{N} (x, t, s),$
5.: For the electronic field resulting from the ionization

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

1.: $ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
2.: $ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

Also,

ρ = ρ_{D} + ρ_{T} + ρ_{H_{e}} + ρ_{N} + ρ_{e},

1.: ${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
2.: $m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
3.: $m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
4.: $m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
5.: $(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
6.: $(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{\partial Ω_{1} \cup \partial Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
7.: ${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
8.: ${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{\partial Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
9.: $\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} {(m_{H_{e}})}_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
10.: ${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{\partial Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d Γ d τ,$
11.: $m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
12.: $\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

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Finally,

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{Ω} f_{5} (ϕ, x, t, ξ, s) d x d ξ d s,

we have also

E_{ρ} = \{\int_{Ω} \frac{\partial f_{5} (ϕ, x, t, ξ, s)}{\partial x_{k}} d ξ d s\} .

and,

B = B_{0} - curl A .

21. A qualitative modeling for a general phase transition process

In this section we develop a general qualitative modeling for a phase transition process.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Such a set

Ω

is supposed to a be a fixed volume in which an amount of mass of a substance A with a density function u will develop phase a transition for another phase with corresponding density function

v .

The total mass

m_{T}

is suppose to be kept constant throughout such a process.

We model such transition in phase through a functional

J : V \times V \to R

where

\begin{matrix} J (u, v) & = & \frac{γ_{1}}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α_{1}}{2} \int_{Ω} u^{4} d x \\ \frac{γ_{2}}{2} \int_{Ω} \nabla v \cdot \nabla v d x + \frac{α_{2}}{2} \int_{Ω} v^{4} d x \\ - \frac{1}{2} \int_{Ω} ω^{2} (u^{2} + v^{2}) d x - \frac{E}{2} (\int_{Ω} (u^{2} + v^{2}) d x - m_{T}) . \end{matrix}

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Here

γ_{1} > 0, γ_{2} > 0, α_{1} > 0, α_{2} > 0

and

V = W^{1, 2} (Ω) .

The phases corresponding to u and v are connected through a Lagrange multiplier E, which represents the chemical potential of the chemical process in question.

We assume the temperature is directly proportional to the internal kinetics

E_{C}

energy where

E_{C} = \frac{1}{2} \int_{Ω} u^{2} \frac{\partial r_{u}}{\partial t} \cdot \frac{\partial r_{u}}{\partial t} d x .

For a internal vibrational motion, we assume approximately

r_{u} \approx e^{i ω t} w_{5} (x),

for an appropriate frequency

ω

and vectorial function

w_{5} .

Thus, the temperature

T = T (x, t)

is indeed proportional to

ω^{2}

, that is, symbolically, we may write

T \propto E_{1} \propto ω^{2} .

Therefore, we start with the system with a phase corresponding to

u \approx 1

and

v \approx 0

ω = 1

. Gradually increasing the temperature to a corresponding

ω = 15

, we obtain a transition to a phase corresponding to

u \approx 0

and

v \approx 1

At this point, we also define the index normalized corresponding densities

ϕ_{u} = \frac{u^{2}}{u^{2} + v^{2}}

and

ϕ_{v} = \frac{v^{2}}{u^{2} + v^{2}} .

Finally, we have obtained some numerical results for the following parameters:

Ω = [0, 1] \subset R,

γ_{1} = γ_{2} = 1

α = 0.1

α_{2} = 10^{3} .

1.: We start with $ω = 1$ corresponding to $ϕ_{u} \approx 1$ and $ϕ_{v} \approx 0$ in $Ω$ .

For the corresponding solutions $ϕ_{u}$ and $ϕ_{v}$ , please see Figure 15 and Figure 16, respectively.
2.: We end the process with $ω = 15$ corresponding to $ϕ_{u} \approx 0$ and $ϕ_{v} \approx 1$ in $Ω$ .

For the corresponding solutions $ϕ_{u}$ and $ϕ_{v}$ , please see Figure 17 and Figure 18, respectively.

22. A mathematical description of a hydrogen molecule in a quantum mechanics context

In this section we develop a mathematical description for a hydrogen molecule.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω

Observe that a single hydrogen molecule comprises two hydrogen atoms physically linked through their electrons.

We recall that each hydrogen atom comprises one proton, one neutron and one electron.

Since the electric charge interaction effects are much higher than those related to the respective masses, in a first analysis we neglect the single neutron densities.

Denoting

(x, y, z) \in Ω \times Ω \times Ω

and time

t \in [0, t_{f}]

, generically, for a particle

p_{j k l}

at the atom

A_{k l}

in the molecule

M_{l}

, we define the following general density:

| ϕ_{{(p_{j k l})}_{T}} {(x, y, z, t) |}^{2} = \frac{| ϕ_{p_{j k l}} {(x, y, z, t) |}^{2} | ϕ_{A_{k l}} {(y, z, t) |}^{2} {| ϕ_{M_{l}} (z, t) |}^{2}}{m_{A_{j k}} m_{M_{l}}} .

Here we have the particle density

| ϕ_{p_{j k l}} {(x, y, z, t) |}^{2}

in the atom

A_{k l}

with density

| ϕ_{A_{k l}} {(y, z, t) |}^{2}

, at the molecule

M_{l}

with a global density

| ϕ_{M_{l}} {(z, t) |}^{2}

Here we have also denoted,

m_{p_{j k l}}

the particle mass,

m_{A_{k l}}

the mass of atom

A_{k l}

and

m_{M_{l}}

the mass of molecule

M_{l}

, so that we set the following constraints:

1.: $\int_{Ω} {| ϕ_{p_{j k l}} (x, y, z, t) |}^{2} d x = m_{p_{j k l}},$
2.: $\int_{Ω} {| ϕ_{A_{k l}} (y, z, t) |}^{2} d y = m_{A_{k l}},$
3.: $\int_{Ω} {| ϕ_{M_{l}} (z, t) |}^{2} d z = m_{M_{l}} .$

At this point we denote for the atoms

A_{1}

A_{2}

of a hydrogen molecule:

1.: $m_{e_{j}} = m_{e}$ : mass of electron $e_{j}$ in the atom $A_{j}$ , where $j \in {1, 2} .$
2.: $m_{p_{j}} = m_{p}$ : mass of proton $p_{j}$ in the atom $A_{j}$ , where $j \in {1, 2} .$

Therefore, considering the respective indexed densities for the particles in question, we define the total hydrogen molecule density, denoted by

| ϕ_{H_{2}} {(x, y, z, t) |}^{2}

\begin{matrix} | ϕ_{H_{2}} {(x, y, z, t) |}^{2} & = & \frac{| ϕ_{p_{1}} {(x, y, z, t) |}^{2} | ϕ_{A_{1}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{1}} m_{M}} \\ + \frac{| ϕ_{e_{1}} {(x, y, z, t) |}^{2} | ϕ_{A_{1}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{1}} m_{M}} \\ + \frac{| ϕ_{p_{2}} {(x, y, z, t) |}^{2} | ϕ_{A_{2}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{2}} m_{M}} \\ + \frac{| ϕ_{e_{2}} {(x, y, z, t) |}^{2} | ϕ_{A_{2}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{2}} m_{M}} . \end{matrix}

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Such system is subject to the following constraints:

From the proton $p_{1}$ in the atom $A_{1}$ :

$\int_{Ω} {| ϕ_{p_{1}} (x, y, z, t) |}^{2} d x = m_{p},$
For the proton $p_{2}$ in the atom $A_{2}$ :

$\int_{Ω} {| ϕ_{p_{2}} (x, y, z, t) |}^{2} d x = m_{p},$
For the atom $A_{1}$ :

$\int_{Ω} {| ϕ_{A_{1}} (y, z, t) |}^{2} d y = m_{A_{1}},$
For the atom $A_{2}$ :

$\int_{Ω} {| ϕ_{A_{2}} (y, z, t) |}^{2} d y = m_{A_{2}},$
For the electrons $e_{1}$ and $e_{2}$ , concerning the physical electronic link between the atoms:

$\int_{Ω} | ϕ_{e_{1}} {(x, y, z, t) |}^{2} d x + \int_{Ω} {| ϕ_{e_{2}} (x, y, z, t) |}^{2} d x = 2 m_{e} .$
For the total molecular density:

$\int_{Ω} {| ϕ_{M} (z, t) |}^{2} d z = m_{M} .$

Therefore, already including the Lagrange multipliers, the corresponding variational formulation for such a system stands for

J : V \to R

, where

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + J_{A u x} (ϕ, E) .

Here we denote

| {(ϕ_{p_{j}})}_{T} |^{2} = \frac{| ϕ_{p_{j}} {(x, y, z, t) |}^{2} | ϕ_{A_{j}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{j}} m_{M}},

| {(ϕ_{e_{j}})}_{T} |^{2} = \frac{| ϕ_{e_{j}} {(x, y, z, t) |}^{2} | ϕ_{A_{j}} {(y, z, t) |}^{2} {| ϕ_{M} (z, t) |}^{2}}{m_{A_{j}} m_{M}}, \forall j \in {1, 2}

we assume

γ_{(p_{j})} > 0, γ_{e_{j}} > 0, γ_{A_{j}} > 0, γ_{M} > 0, α_{{(p_{j})}_{T}} > 0, α_{{(e_{j})}_{T}} > 0 α_{{(p_{j} e_{k})}_{T}} < 0, \forall j, k \in {1, 2},

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p_{j}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{p_{j}}) \cdot (\nabla ϕ_{p_{j}}) d x d y d z d t \\ + \frac{γ_{e_{j}}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{e_{j}}) \cdot (\nabla ϕ_{e_{j}}) d x d y d z d t \\ \frac{γ_{A_{j}}}{2} \int_{Ω} (\nabla ϕ_{A_{j}}) \cdot (\nabla ϕ_{A_{j}}) d y d z d t \\ + \frac{γ_{M}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{M}) \cdot (\nabla ϕ_{M}) d z d t \end{matrix}

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and

\begin{matrix} F (ϕ) = \\ \frac{α_{{(p_{j})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(p_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(p_{j})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z; d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{α_{{(e_{j})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(e_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(e_{j})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{α_{{(p_{j} e_{k})}_{T}}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{{(p_{j})}_{T}} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{{(e_{k})}_{T}} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \end{matrix}

Finally,

\begin{matrix} J_{A u x} (ϕ, E) & = & \int_{0}^{t_{f}} \int_{Ω} {(E_{p})}_{j} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} (E_{e}) (y, z, t) (\int_{Ω} (| ϕ_{e_{1}} {(x, y, z, t) |}^{2} + | ϕ_{e_{2}} (x, y, z, t) |^{2}) d x - 2 m_{e}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} {(E_{A})}_{j} (z, t) (\int_{Ω} {| ϕ_{A_{j}} (y, z, t) |}^{2} d y - m_{A_{j}}) d z d t \\ \int_{0}^{t_{f}} (E_{M}) (t) (\int_{Ω} {| ϕ_{M} (z, t) |}^{2} d z - m_{M}) d t . \end{matrix}

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Remark 22.1.

We highlight the two electrons which link the atoms are at same level of energy

E_{e}

. Morever, each atom has its energy level

E_{A_{j}}

and the molecule as a whole has also its energy level

E_{M} .

23. A mathematical model for the water hydrolysis

In this section we develop a modeling for a chemical reaction known as the water hydrolysis.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

In such a volume

Ω

containing a total mass

m_{T}

of water initially at the temperature 25 C with pressure 1 atm, we intend to model the following reaction

H_{2} O ⇌ O H^{-} + H^{+}

which as previously mentioned is the well known water hydrolysis.

We highlight

H_{2} O

stand for a water molecule which subject to an appropriate electric potential is decomposed into a ionized

O H^{-}

molecule and ionized

H^{+}

atom.

It is also well known that the water symbol

H_{2} O

corresponds to a molecule with two hydrogen (H) atoms and one oxygen (O) atom.

Moreover, the oxygen atom O has 8 protons, 8 neutrons and 8 electrons whereas the hydrogen atom H has one proton, one neutron and one electron.

Remark 23.1.

Here we have assumed that a unit mass of

H_{2} O

reacts into a fractional mass

α_{B}

O H^{-}

and a fractional mass

α_{C}

H^{+} .

Symbolically, we have:

1 = α_{B} + α_{C} .

To clarify the notation we set the conventions:

$H_{2} O$ molecule generically corresponds to wave function $ϕ_{1}$ .
$O H^{-}$ molecule corresponds to wave function $ϕ_{2} .$
$H^{+}$ hydrogen atom corresponds to wave function $ϕ_{3} .$

At this point we define the following densities:

For the $H_{2} O$ water density (for charges), denoted by $| ϕ_{1} |^{2}$ , we have

$\begin{matrix} | ϕ_{1} {(x, y, z, t) |}^{2} & = & K_{p} \sum_{j = 1}^{2} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A_{j}}^{H} {(m_{1})}_{M}} \\ + K_{e} \sum_{j = 1}^{2} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m_{1})}_{A_{j}}^{H} {(m_{1})}_{M}} \\ + K_{p} \sum_{j = 1}^{8} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{1})}_{M}} \\ + K_{e} \sum_{j = 1}^{8} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{1}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{1})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{1})}_{M}} \end{matrix}$

(125)

where ${(m_{1})}_{M}$ is the mass of a single water molecule and generically $| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |^{2}$ refers to the hydrogen proton $p_{j}$ at the hydrogen atom $A_{j}$ concerning the $H_{2} O$ molecular density and so on.
For the $O H^{-}$ density, denoted by $| ϕ_{2} |^{2}$ , we have

$\begin{matrix} | ϕ_{2} {(x, y, z, t) |}^{2} & = & K_{p} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{H})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{H} {(m_{2})}_{M}} \\ + K_{e} {| {(ϕ_{2}^{H})}_{e_{1}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{H})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{H} {(m_{2})}_{M}} \\ + K_{e} {| {(ϕ_{2}^{O H^{-}})}_{e_{2}} (x, z, t) |}^{2} \frac{| {(ϕ_{2})}_{M} (z, t) |^{2}}{{(m_{2})}_{M}} \\ + K_{p} \sum_{j = 1}^{8} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{2})}_{M}} \\ + K_{e} \sum_{j = 1}^{8} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} \frac{| {(ϕ_{2}^{O})}_{A} (y, z, t) |^{2} {| {(ϕ_{2})}_{M} (z, t) |}^{2}}{{(m)}_{A}^{O} {(m_{2})}_{M}}, \end{matrix}$

(126)

where ${(m_{2})}_{M}$ is the mass of a single molecule of $O H^{-}$ .
For the ionized hydrogen atom have

$| ϕ_{3} {(x, y, t) |}^{2} = K_{p} {| {(ϕ_{3}^{H})}_{p} (x, y, t) |}^{2} \frac{| {(ϕ_{3}^{H})}_{A} (y, t) |^{2}}{{(m_{3})}_{A}} .$

where we have denoted

{(m_{3})}_{A}

is the mass of a single atom of

H^{+}

Here

K_{p} > 0

and

K_{e} < 0

are appropriate real constants concerning a proton and an electron charge, respectively.

The system is subject to the following constraints:

$\int_{Ω} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} d x = m_{p},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{e_{1}} (x, y, z, t) |}^{2} d x = m_{e},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{e_{2}} (x, y, z, t) |}^{2} d x = m_{e},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 8},$
$\int_{Ω} {| {(ϕ_{3}^{H})}_{p} (x, z, t) |}^{2} d x = m_{p},$
$\int_{Ω} {| {(ϕ_{1}^{H})}_{A_{j}} (y, z, t) |}^{2} d y = m_{A}^{H}, \forall j \in {1, 2},$
$\int_{Ω} {| {(ϕ_{1}^{O})}_{A} (y, z, t) |}^{2} d y = m_{A}^{O},$
$\int_{Ω} {| {(ϕ_{2}^{H})}_{A} (y, z, t) |}^{2} d y = m_{A}^{H},$
$\int_{Ω} {| {(ϕ_{2}^{O})}_{A} (y, z, t) |}^{2} d y = m_{A}^{O},$
$\int_{Ω} {| {(ϕ_{3}^{H})}_{A} (y, z, t) |}^{2} d y = m_{A}^{H},$
$\int_{Ω} (| {(ϕ_{1})}_{M} (z, t) |^{2} + | {(ϕ_{2})}_{M} (z, t) |^{2} + | {(ϕ_{3})}_{M} (z, t) |^{2}) d z = m_{T},$
$\int_{Ω} (α_{C} | {(ϕ_{2})}_{M} (z, t) |^{2} - α_{B} | {(ϕ_{3})}_{M} (z, t) |^{2}) d z = 0 .$

Already including the Lagrange multipliers for the constraints, the variational formulation for such system. denoted by the functional

J (ϕ, E)

stands for

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + F_{1} (ϕ) - J_{A u x} (ϕ, E),

where

\begin{matrix} G (\nabla ϕ) & = & \frac{γ_{p}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{O})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{O})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{p} \cdot \nabla {(ϕ_{2}^{H})}_{p} d x d y d z d t \\ + \frac{γ_{e}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{e_{1}} \cdot \nabla {(ϕ_{2}^{H})}_{e_{1}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O H^{-}})}_{e_{2}} \cdot \nabla {(ϕ_{1}^{O H^{-}})}_{e_{2}} d x d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{p_{j}} \cdot \nabla {(ϕ_{2}^{O})}_{p_{j}} d x d y d z d t \\ + \frac{γ_{e}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{e_{j}} \cdot \nabla {(ϕ_{2}^{O})}_{e_{j}} d x d y d z d t \\ + \frac{γ_{p}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{p} \cdot \nabla {(ϕ_{2}^{O})}_{p} d x d y d t \\ + \frac{γ_{A^{H}}}{2} \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{H})}_{A_{j}} \cdot \nabla {(ϕ_{1}^{H})}_{A_{j}} d y d z d t \\ + \frac{γ_{A^{O}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{O})}_{A} \cdot \nabla {(ϕ_{1}^{O})}_{A} d y d z d t \\ + \frac{γ_{A^{H}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{H})}_{A} \cdot \nabla {(ϕ_{2}^{H})}_{A} d y d z d t \\ + \frac{γ_{A^{O}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{O})}_{A} \cdot \nabla {(ϕ_{2}^{O})}_{A} d y d z d t \\ + \frac{γ_{M_{1}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1})}_{M} \cdot \nabla {(ϕ_{1})}_{M} d z d t \\ + \frac{γ_{M_{2}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2})}_{M} \cdot \nabla {(ϕ_{2})}_{M} d z d t \\ \frac{γ_{A_{3}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{3})}_{A} \cdot \nabla {(ϕ_{3})}_{A} d y d t . \end{matrix}

Here

γ_{p} > 0

γ_{e} > 0

γ_{A}^{H} > 0,

γ_{A}^{O} > 0,

γ_{M_{1}} > 0

γ_{M_{2}} > 0

γ_{A_{3}} > 0 .

Moreover,

\begin{matrix} F (ϕ) \\ = & \frac{α_{1}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{1} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{1} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \\ + \frac{α_{2}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{2} (ξ_{1}, ξ_{2}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \\ + \frac{α_{3}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{3} (x - ξ_{1}, z - ξ_{3}, t) |^{2} {| ϕ_{3} (ξ_{1}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{3} d t \\ + \frac{α_{23}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} {| ϕ_{3} (ξ_{1}, ξ_{3}, t) |}^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d x_{1} d x_{2} d x_{3} d t \end{matrix}

where

α_{1} > 0

α_{2} > 0

α_{3} > 0

and

α_{23} > 0 .

Furthermore,

\begin{matrix} F_{1} (ϕ) & = & \int_{0}^{t_{f}} \int_{Ω} V (x, y, z, t) (| ϕ_{1} |^{2} + | ϕ_{2} |^{2} + | ϕ_{3} |^{2}) d x d y d z d t, \end{matrix}

(127)

where

V = V (x, y, z, t)

is an electric potential originated from an external electric field

E

applied on

Ω .

Finally,

\begin{matrix} J_{A u x} (ϕ, E) \\ = & \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{p_{j}}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{H})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{e_{j}}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{H})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{p_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{1})}_{e_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{p}^{H} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{H})}_{p} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{p_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{O})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 8}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{2})}_{e_{j}}^{O} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{O})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{3})}_{p}^{H} (y, t) (\int_{Ω} {| {(ϕ_{3}^{H})}_{p} (x, y, t) |}^{2} d x - m_{p}) d y d t \\ + \sum_{j = 1}^{2} \int_{0}^{t_{f}} \int_{Ω} {(E_{4})}_{A_{j}}^{H} (z, t) (\int_{Ω} (| {(ϕ_{1})}_{A_{j}}^{H} (y, z, t) |^{2} d y - m_{A_{j}}^{H}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} \int_{Ω} {(E_{4})}_{A}^{O} (z, t) (\int_{Ω} (| {(ϕ_{1})}_{A}^{O} (y, z, t) |^{2} d y - m_{A}^{O}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{5})}_{A}^{H} (z, t) (\int_{Ω} (| {(ϕ_{2})}_{A}^{H} (y, z, t) |^{2} d y - m_{A}^{H}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{5})}_{A}^{O} (z, t) (\int_{Ω} (| {(ϕ_{2})}_{A}^{O} (y, z, t) |^{2} d y - m_{A}^{O}) d z d t \\ \int_{0}^{t_{f}} {(E_{6})}_{A}^{H} (t) (\int_{Ω} (| {(ϕ_{3})}_{A}^{H} (y, t) |^{2} d y - m_{A}^{H}) d t \\ + \int_{0}^{t_{f}} (E_{7}) (t) (\int_{Ω} (| {(ϕ_{1})}_{M} (z, t) |^{2} + | {(ϕ_{2})}_{M} (z, t) |^{2} + | {(ϕ_{3})}_{M} (z, t) |^{2}) d z - m_{T}) d t \\ + \int_{0}^{t_{f}} (E_{8}) (t) (\int_{Ω} (α_{C} | {(ϕ_{2})}_{M} (z, t) |^{2} - α_{B} | {(ϕ_{3})}_{M} (z, t) |^{2}) d z) d t . \end{matrix}

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24. A mathematical model for the Austenite and Martensite phase transition

In this section we consider a phase transition of a solid solution of

γ - F e

(

γ - i r o n

) and carbon with a

0.75 / 100

proportion of carbon, known as austenite, initially at a temperature above and close to

723 C

and rapidly cooled to a temperature of about

25 C

, developing a phase transition which generates a solid solution of

α - F e

(

α - i r o n

) and carbon known as martensite.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular boundary denoted by

\partial Ω

which contains an amount of austenite at

723 C

and which, as previously mentioned, is rapidly cooled to a temperature

25 C

on a time interval

[0, t_{f}],

resulting a phase known as martensite.

We recall the

γ - F e

of austenite phase presents a multi-faced cubic crystalline structure in a micro-structure with carbon atoms.

On the other hand,

α - F_{e}

structure of the martensite phase has a

C C C

cubic centralized crystalline structure in a micro-structure with carbon atoms.

At this point, we also recall that the

F_{e}

(iron) atom has 26 protons, 26 electrons and 30 neutrons.

On the other hand a

C a r b o n_{12}

atom has 6 protons and this same number of electrons and neutrons.

Here we define the density function

ϕ_{1}

, representing the Austenite phase, where:

\begin{matrix} | ϕ_{1} {(x, y, z, t) |}^{2} & = & \sum_{j = 1}^{26} | ϕ_{p_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{26} | ϕ_{e_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{30} | ϕ_{N_{j}}^{γ - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{γ})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} . \end{matrix}

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Similarly, we define the density function for the Martensite phase, which is denoted by

ϕ_{2}

, where:

\begin{matrix} | ϕ_{2} {(x, y, z, t) |}^{2} & = & \sum_{j = 1}^{26} | ϕ_{p_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{γ - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{α} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{26} | ϕ_{e_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{α - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{γ} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{30} | ϕ_{N_{j}}^{α - F_{e}} {(x, y, z, t) |}^{2} | ϕ_{A}^{α - F_{e}} {(y, z, t) |}^{2} {| ϕ_{1}^{α} (z, t) |}^{2} \frac{1}{{(m_{A}^{α})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |^{2} | {(ϕ_{1}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{1}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |^{2} | {(ϕ_{2}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{2}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} \\ + \sum_{j = 1}^{6} | {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |^{2} | {(ϕ_{2}^{C})}_{A} (y, z, t) |^{2} {| ϕ_{2}^{C} (z, t) |}^{2} \frac{1}{{(m_{A}^{C})}^{2}} . \end{matrix}

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For the

C F C

γ - F_{e}

(γ - i r o n)

corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

C_{γ} = {(ε_{1}, 0, 0), (0, ε_{2}, 0), (0, 0, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

{(C_{γ})}_{1} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

and

{(C_{γ})}_{2} = {(ε_{1}, ε_{2}, 0), (ε_{1}, 0, ε_{3}), (0, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

we must have

ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in C_{γ},

where

δ_{z} \in R^{+}

is a small real parameter related to

γ - F_{e}

crystalline structure dimensions.

We must have also,

ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in {(C_{γ})}_{1}

and,

{(ϕ_{1}^{C})}_{A} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) = {(ϕ_{1}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t),

\forall ε, \tilde{ε} \in {(C_{γ})}_{2} .

For the

C C C

α - F_{e}

(α - i r o n)

corresponding to the Austenite phase, such density functions are subject to the following constraints:

Defining

C_{α} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{j} \in {+ 1, - 1}, \forall j \in {1, 2, 3}},

{(C_{α})}_{1} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{1}, ε_{2} \in {+ 1, - 1} and ε_{3} = 0},

{(C_{α})}_{2} = {(ε_{1}, ε_{2}, ε_{3}), : ε_{1} = ε_{2} = 0 and ε_{3} \in {+ 1, - 1}},

we must have

ϕ_{A}^{α - F_{e}} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) = ϕ_{A}^{α - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t),

\forall ε, \tilde{ε} \in C_{α},

where

{\hat{δ}}_{z} \in R^{+}

is a small real parameter related to

α - F_{e}

crystalline structure dimensions.

We must have also,

{(ϕ_{2}^{C})}_{A} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) = {(ϕ_{2}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t),

\forall ε, \tilde{ε} \in {(C_{α})}_{1} \cup {(C_{α})}_{2} .

The other constraints for the densities are given by:

1.

For the Austenite phase:

(a): $\int_{Ω} {| ϕ_{p_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 26},$
(b): $\int_{Ω} {| ϕ_{e_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 26},$
(c): $\int_{Ω} {| ϕ_{N_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 30},$
(d): $\int_{Ω} {| ϕ_{A}^{γ - F_{e}} (x, y, z, t) |}^{2} d x = m_{A}^{γ},$
(e): $\int_{Ω} {| {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 6},$
(f): $\int_{Ω} {| {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 6},$
(g): $\int_{Ω} {| {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 6},$
(h): $\int_{Ω} {| {(ϕ_{1}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C},$

2.

For the Martensite phase:

(a): $\int_{Ω} {| ϕ_{p_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 26},$
(b): $\int_{Ω} {| ϕ_{e_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 26},$
(c): $\int_{Ω} {| ϕ_{N_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 30},$
(d): $\int_{Ω} {| ϕ_{A}^{α - F_{e}} (x, y, z, t) |}^{2} d x = m_{A}^{α},$
(e): $\int_{Ω} {| {(ϕ_{2}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x = m_{p}, \forall j \in {1, 6},$
(f): $\int_{Ω} {| {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x = m_{e}, \forall j \in {1, 6},$
(g): $\int_{Ω} {| {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x = m_{N}, \forall j \in {1, 6},$
(h): $\int_{Ω} {| {(ϕ_{2}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C} .$

3.

For the total

F_{e}

(iron) mass,

\int_{Ω} | ϕ_{1}^{γ} {(z, t) |}^{2} d z + \int_{Ω} {| ϕ_{2}^{γ} (z, t) |}^{2} d z = {(m_{F_{e}})}_{T},

4.

For the total Carbon mass

\int_{Ω} | ϕ_{1}^{C} {(z, t) |}^{2} d z + \int_{Ω} {| ϕ_{2}^{C} (z, t) |}^{2} d z = {(m_{C})}_{T} .

At this point we define the functional J which models such a pahse transition in question, where

J (ϕ, E) = G (\nabla ϕ) + F (ϕ) + F_{1} (ϕ) + J_{A u x} (ϕ, E)

where

\begin{matrix} G (\nabla ϕ) & = & \sum_{j = 1}^{26} \frac{{\hat{γ}}_{p}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{p_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{p_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{e}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{e_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{e_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{30} \frac{{\hat{γ}}_{N}^{γ - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{N_{j}}^{γ - F_{e}} \cdot \nabla ϕ_{N_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{p}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{p_{j}}^{α - F_{e}} \cdot \nabla ϕ_{p_{j}}^{α - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{26} \frac{{\hat{γ}}_{e}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{e_{j}}^{α - F_{e}} \cdot \nabla ϕ_{e_{j}}^{γ - F_{e}} d x d y d z d t \\ + \sum_{j = 1}^{30} \frac{{\hat{γ}}_{N}^{α - F_{e}}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla ϕ_{N_{j}}^{α - F_{e}} \cdot \nabla ϕ_{N_{j}}^{α - F_{e}} d x d y d z d t \\ + \frac{{\hat{γ}}_{A}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{A}^{γ - F_{e}} (y, z, t) \cdot \nabla ϕ_{A}^{γ - F_{e}} (y, z, t)) d y d z d t \\ + \frac{{\hat{γ}}_{A}^{α}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla ϕ_{A}^{α - F_{e}} (y, z, t) \cdot \nabla ϕ_{A}^{α - F_{e}} (y, z, t)) d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{p}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{p_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{p_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{e}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{e_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{e_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{N}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{1}^{C})}_{N_{j}} \cdot \nabla {(ϕ_{1}^{C})}_{N_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{p}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{p_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{p_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{e}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{e_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{e_{j}} d x d y d z d t \\ + \sum_{j = 1}^{6} \frac{{\hat{γ}}_{N}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} \nabla {(ϕ_{2}^{C})}_{N_{j}} \cdot \nabla {(ϕ_{2}^{C})}_{N_{j}} d x d y d z d t \\ + \frac{{\hat{γ}}_{A}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla {(ϕ_{1}^{C})}_{A} \cdot \nabla {(ϕ_{1}^{C})}_{A}) d y d z d t + \frac{{\hat{γ}}_{A}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla {(ϕ_{2}^{C})}_{A} \cdot \nabla {(ϕ_{2}^{C})}_{A}) d y d z d t \\ + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{γ}) \cdot \nabla (ϕ_{1}^{γ})) d z d t + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{α}) \cdot \nabla (ϕ_{1}^{α})) d z d t \\ + \frac{{\hat{γ}}_{T}^{C}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{1}^{C}) \cdot \nabla (ϕ_{1}^{C})) d z d t + \frac{{\hat{γ}}_{T}^{γ}}{2} \int_{0}^{t_{f}} \int_{Ω} (\nabla (ϕ_{2}^{C}) \cdot \nabla (ϕ_{2}^{C})) d z d t \end{matrix}

(131)

Also,

\begin{matrix} F (ϕ) \\ = & \frac{{\hat{α}}_{1}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{1} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} | | ϕ_{1} (ξ_{1}, ξ_{2}, ξ_{3}, t) |^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t \\ + \frac{{\hat{α}}_{2}}{2} \int_{0}^{t_{f}} \int_{Ω} \frac{| ϕ_{2} (x - ξ_{1}, y - ξ_{2}, z - ξ_{3}, t) |^{2} | | ϕ_{2} (ξ_{1}, ξ_{2}, ξ_{3}, t) |^{2}}{| (x, y, z) - (ξ_{1}, ξ_{2}, ξ_{3}) |} d x d y d z d ξ_{1} d ξ_{2} d ξ_{3} d t, \end{matrix}

F_{1} (ϕ) = - \int_{0}^{t_{f}} \int_{Ω} w^{2} (z, t) (| ϕ_{1} {(z, t) |}^{2} + ϕ_{2} (z, t) |^{2}) d z d t,

Finally,

J_{A u x} = J_{A u x_{1}} + J_{A u x_{2}} + J_{A u x_{3}} + J_{A u x_{4}} + J_{A u x_{5}}

, where

\begin{matrix} J_{A u x_{1}} & = & \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{p_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{e_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{e_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 1}^{30} \int_{0}^{t_{f}} \int_{Ω} E_{N_{j}}^{γ - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{N_{j}}^{γ - F_{e}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{p_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{p_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ + \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} E_{e_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{e_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ + \sum_{j = 1}^{30} \int_{0}^{t_{f}} \int_{Ω} E_{N_{j}}^{α - F_{e}} (y, z, t) (\int_{Ω} {| ϕ_{N_{j}}^{α - F_{e}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{A}^{γ - F_{e}} (y, t) (\int_{Ω} {| ϕ_{A}^{γ - F_{e}} (y, z, t) |}^{2} d y - m_{A}^{γ}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} E_{A}^{α - F_{e}} (y, t) (\int_{Ω} {| ϕ_{A}^{α - F_{e}} (y, z, t) |}^{2} d y - m_{A}^{α}) d z d t \end{matrix}

(132)

\begin{matrix} J_{A u x_{2}} & = & \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{p_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{e_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{N_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{p_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{p_{j}} (x, y, z, t) |}^{2} d x - m_{p}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{e_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{e_{j}} (x, y, z, t) |}^{2} d x - m_{e}) d y d z d t \\ \sum_{j = 1}^{26} \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{N_{j}} (y, z, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{N_{j}} (x, y, z, t) |}^{2} d x - m_{N}) d y d z d t \\ \int_{0}^{t_{f}} \int_{Ω} {(E_{1}^{C})}_{A} (y, t) (\int_{Ω} {| {(ϕ_{1}^{C})}_{A} (y, z, t) |}^{2} d y - m_{A}^{C}) d z d t \\ + \int_{0}^{t_{f}} \int_{Ω} {(E_{2}^{C})}_{A} (y, t) (\int_{Ω} {| {(ϕ_{2}^{C})}_{A} (y, z, t) |}^{2} d y - m_{A}^{C}) d z d t \end{matrix}

(133)

and,

\begin{matrix} J_{A u x_{3}} & = & \int_{0}^{t_{f}} E_{3}^{γ, α} (t) (\int_{Ω} (| ϕ_{1}^{γ} {(z, t) |}^{2} + | ϕ_{2}^{α} (z, t) |^{2}) d z - {(m_{F_{e}})}_{T}) d t \\ + \int_{0}^{t_{f}} E_{3}^{C} (t) (\int_{Ω} (| ϕ_{1}^{C} {(z, t) |}^{2} + | ϕ_{2}^{C} (z, t) |^{2}) d z - {(m_{C})}_{T}) d t . \end{matrix}

(134)

\begin{matrix} J_{A u x_{4}} \\ = & + \sum_{ε, \tilde{ε} \in C_{γ}} \int_{0}^{t_{f}} \int_{Ω} E_{4}^{ε, \tilde{ε}} (y, z, t) (ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ \sum_{ε, \tilde{ε} \in {(C_{γ})}_{1}} \int_{0}^{t_{f}} \int_{Ω} E_{5}^{ε, \tilde{ε}} (y, z, t) ϕ_{A}^{γ - F_{e}} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - ϕ_{A}^{γ - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in {(C_{γ})}_{2}} \int_{0}^{t_{f}} \int_{Ω} E_{6}^{ε, \tilde{ε}} (y, z, t) {(ϕ_{1}^{C})}_{A} (y, z_{1} + ε_{1} δ_{z}, z_{2} + ε_{2} δ_{z}, z_{3} + ε_{3} δ_{z}, t) \\ - {(ϕ_{1}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} δ_{z}, z_{2} + {\tilde{ε}}_{2} δ_{z}, z_{3} + {\tilde{ε}}_{3} δ_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in (C_{α})} \int_{0}^{t_{f}} \int_{Ω} E_{7}^{ε, \tilde{ε}} (y, z, t) (ϕ_{A}^{α - F_{e}} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) \\ - ϕ_{A}^{α - F_{e}} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t)) d y d z d t \\ + \sum_{ε, \tilde{ε} \in {(C_{α})}_{1} \cup {(C_{α})}_{2}} \int_{0}^{t_{f}} \int_{Ω} E_{8}^{ε, \tilde{ε}} (y, z, t) ({(ϕ_{2}^{C})}_{A} (y, z_{1} + ε_{1} {\hat{δ}}_{z}, z_{2} + ε_{2} {\hat{δ}}_{z}, z_{3} + ε_{3} {\hat{δ}}_{z}, t) \\ - {(ϕ_{2}^{C})}_{A} (y, z_{1} + {\tilde{ε}}_{1} {\hat{δ}}_{z}, z_{2} + {\tilde{ε}}_{2} {\hat{δ}}_{z}, z_{3} + {\tilde{ε}}_{3} {\hat{δ}}_{z}, t)) d y d z d t . \end{matrix}

(135)

Finally, for a field of displacements

u = (u_{1}, u_{2}, u_{3})

resulting from the action of a external load field

f = (f_{1}, f_{2}, f_{3})

and temperature variations, we define

\begin{matrix} J_{A u x_{5}} \\ = & \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} (Λ_{1} (x, t) H_{i j k l}^{1} ((e_{i j} (u) - e_{i j}^{1} (w)) (e_{k l} (u) - e_{k l}^{1} (w))) \\ + Λ_{2} (z, t) H_{i j k l}^{2} ((e_{i j} (u) - e_{i j}^{2} (w)) (e_{k l} (u) - e_{k l}^{2} (w)))) d x d t \\ - \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} ρ (x, t) u_{t} (x, t) \cdot u_{t} (x, t) d x d t \\ - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

(136)

where

e_{i j} (u) = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}),

ρ_{1} (z, t) = \int_{Ω} {| ϕ_{1} (x, y, z, t) |}^{2} d x d y,

ρ_{2} (z, t) = \int_{Ω} {| ϕ_{2} (x, y, z, t) |}^{2} d x d y,

ρ (z, t) = ρ_{1} (z, t) + ρ_{2} (z, t),

and

Λ_{1} (z, t) = \frac{ρ_{1} (z, t)}{ρ_{1} (z, t) + ρ_{2} (z, t)},

Λ_{2} (z, t) = \frac{ρ_{2} (z, t)}{ρ_{1} (z, t) + ρ_{2} (z, t)} .

Remark 24.1.

The system temperature is supposed to be directly proportional to

w {(z, t)}^{2}

, which in this model is a known function obtained experimentally. Finally, the strain tensors

{e_{i j}^{1} (w)}

and

{e_{i j}^{2} (w)}

refer to austenite and martensite phases, respectively. Such tensors also depend on the temperature and must be also obtained experimentally.

25. A note on classical free fields through a variational perspective

This section is strongly based on the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov.

Let

Ω = \hat{Ω} \times [0, T] \subset R^{4}

where

\hat{Ω} \subset R^{3}

is a bounded, open and connected set with a regular boundary denoted by

\partial \hat{Ω} .

Consider the Lagrangian density

L : R^{N} \times R^{N \times n} \to R

and an action

A : V \to R

where

A (u) = \int_{Ω} L (u, \nabla u) d x,

V = W_{0}^{1, 2} (Ω; R^{N})

We denote

\nabla u = \{\frac{\partial u_{i}}{\partial x_{j}}\}

and

\frac{\partial u_{i}}{\partial x_{j}} = {(u_{i})}_{x_{j}} .

Assume

u \in V

is such that

δ L (u, \nabla u) = 0,

so that

\frac{\partial L (u, \nabla u)}{\partial u_{i}} - \sum_{k = 1}^{n} \frac{d}{d x_{k}} (\frac{\partial L (u, \nabla u)}{\partial {(u_{i})}_{x_{k}}}) = 0, in Ω, \forall i \in {1, \dots, N} .

We define a change of variables

{(x^{'})}_{k} = x_{k} + δ x_{k},

where

x_{k} = (x_{0}, x_{1}, x_{2}, x_{3})

and

x_{0} = t

(here t denotes time).

Also

g_{j k} = 0, if j \neq k, g_{00} = - 1 and g_{11} = g_{22} = g_{33} = 1, {g^{j k}} = {g_{j k}}^{- 1},

δ x_{k} = \sum_{j = 1}^{N} X_{j}^{k} ε w^{j},

where

| ε | ≪ 1

denotes a small real parameter.

We define also

u_{i}^{'} (x^{'}) = u_{i} (x) + δ u_{i} (x),

where

δ u_{i} (x) = \sum_{j = 1}^{N} ψ_{i j} ε w^{j},

and

\bar{δ u_{i}} = u_{i}^{'} (x) - u_{i} (x) .

Observe that

\begin{matrix} δ u_{i} (x) & = & u_{i}^{'} (x) - u_{i} (x) \\ = & u_{i}^{'} (x^{'}) - u_{i}^{'} (x) + u_{i}^{'} (x) - u_{i} (x), \end{matrix}

(137)

so that

\begin{matrix} \bar{δ u_{i} (x)} & = & u_{i}^{'} (x) - u_{i} (x) \\ = & δ u_{i} (x) - (u_{i}^{'} (x^{'}) - u_{i}^{'} (x)) \\ \sum_{j = 1}^{N} ψ_{i j} ε w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i}^{'} ({\tilde{x}}^{i})}{d x_{k}} δ x_{k} \\ = & \sum_{j = 1}^{N} ψ_{i j} ε w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} δ x_{k} + O (ε^{2}) . \end{matrix}

(138)

Summarizing, we have got

\bar{δ u_{i} (x)} = ε (\sum_{j = 1}^{N} (ψ_{i j} w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} X_{j}^{K} w^{j})) + O (ε^{2}) .

Define now

\tilde{A} (u, φ_{1}, φ_{2}, ε) = \int_{Ω} L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] det J (x) d x .

where we have generically denoted

L [u] \equiv L (u, \nabla u),

L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] \equiv L (u (x + ε φ_{2} (x)) + ε φ_{1} (x), \nabla u (x + ε φ_{2} (x)) + ε \nabla φ_{1} (x)),

and

\begin{matrix} J (x) & = & \{\frac{\partial x_{j}^{'}}{\partial x_{k}}\} \\ = & \{\frac{\partial (x_{j} + ε {(φ_{2})}_{j} (x))}{\partial x_{k}}\} \\ = & \{δ_{j k} + ε \frac{\partial {(φ_{2})}_{j} (x)}{\partial x_{k}}\} . \end{matrix}

(139)

From such a last definition we have

det J (x) = 1 + ε \sum_{k = 1}^{n} \frac{\partial {(φ_{2})}_{k} (x)}{\partial x_{k}} + O (ε^{2}) .

so that

\frac{\partial det J (x)}{\partial ε} |_{ε = 0} = \sum_{k = 1}^{n} \frac{\partial {(φ_{2})}_{k} (x)}{\partial x_{k}},

At this point we define

δ A (u, φ_{1}, φ_{2}) = \frac{d}{d ε} (\tilde{A} (u, φ_{1}, φ_{2}, ε)) |_{ε = 0},

so that

\begin{matrix} δ A (u, φ_{1}, φ_{2}) & = & \int_{Ω} (\sum_{i = 1}^{N} (\frac{\partial L (u, \nabla u)}{\partial u_{i}} {(φ_{1})}_{i} \\ + \sum_{k = 1}^{n} (\frac{\partial L (u, \nabla u)}{\partial {(u_{i})}_{x_{k}}} {({(φ_{1})}_{i})}_{x_{k}}) \\ + \sum_{k = 1}^{n} \frac{δ L [u]}{δ u_{i}} \frac{\partial u_{i}}{\partial x_{k}} {(φ_{2})}_{k}) + \sum_{k = 1}^{n} L [u] \frac{\partial {(φ_{2})}_{k}}{\partial x_{k}}) d x . \end{matrix}

(140)

From this and

\frac{\partial L (u, \nabla u)}{\partial u_{i}} - \frac{d}{d x_{k}} (\frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}}) = 0, in Ω, \forall i \in {1, \dots, N},

we obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) & = & \sum_{i = 1}^{N} \sum_{k = 1}^{n} (\int_{Ω} \frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} {(φ_{1})}_{k})) d x \\ + \sum_{k = 1}^{n} \int_{Ω} \frac{d (L [u] {(φ_{2})}_{k})}{d x_{k}} d x . \end{matrix}

(141)

In particular, for

{(φ_{2})}_{k} = \sum_{j = 1}^{N} X_{j}^{k} w^{j}

and

{(φ_{1})}_{i} = \sum_{j = 1}^{N} (ψ_{i j} w^{j} - \sum_{k = 1}^{n} \frac{\partial u_{i}}{\partial x_{k}} X_{j}^{k} w^{j}),

we obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) \\ = & \sum_{i = 1}^{N} \sum_{k = 1}^{n} \int_{Ω} (\frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{l}} X_{j}^{l} w^{j})))) d x \\ + \sum_{k = 1}^{n} \int_{Ω} \frac{\partial L [u] X_{j}^{k} w^{j}}{d x_{k}} d x \\ = & \sum_{j = 1}^{N} (\sum_{k = 1}^{n} (\int_{Ω} \frac{d}{d x_{k}} (\sum_{i = 1}^{N} \frac{\partial L [u]}{\partial {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{l}} X_{j}^{l} w^{j})) \\ + L [u] X_{j}^{k} w^{j}) d x)) . \end{matrix}

(142)

Moreover, we define

θ_{k}^{j} = \sum_{i = 1}^{N} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (- ψ_{i j} + \sum_{l = 1}^{n} \frac{\partial u_{i}}{\partial x_{l}} X_{j}^{l})) - L (u) X_{j}^{k}

so that

δ A (u, φ_{1}, φ_{2}) = - \int_{Ω} \sum_{j = 1}^{N} \sum_{k = 1}^{n} \frac{d (θ_{j}^{k} w^{j})}{d x_{k}} d x,

\forall {w^{j}} \in C_{c}^{\infty} (Ω; R^{N}) .

In particular, for

ψ_{i j} = 0

and

X_{j}^{k} = δ_{j}^{k}

we obtain the Energy-Momentum tensor

T_{k}^{j}

, where

T_{k}^{j} \equiv θ_{k}^{j} = \sum_{i = 1}^{N} \sum_{l = 1}^{n} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} \frac{\partial u_{i}}{\partial x_{l}} δ_{j}^{l}) - L [u] δ_{j}^{k} .

25.1. The Angular-Momentum tensor

In this subsection we define the following change of variables

x_{k}^{'} = x_{k} + \sum_{m \neq k} g^{m m} x_{m} ε w^{k m},

where

w^{k m} = - w^{m k} .

With such relations in mind, we set

\begin{matrix} δ x_{k} & = & x_{k}^{'} - x_{k} \\ = & ε \sum_{l = 1}^{n} \sum_{m < l} w^{m l} (g^{l l} x_{l} g_{m}^{k} - g^{m m} x_{m} g_{l}^{k}) . \end{matrix}

(143)

We define also,

u_{i}^{'} (x^{'}) = u_{i} (x) + δ u_{i} (x)

where

δ u_{i} (x) = \sum_{l = 1}^{n} \sum_{j, p < l} A_{i (p l)}^{j} u_{j} (x) ε w^{p l} .

Moreover, we define

ψ_{i (m n)} = \sum_{j = 1}^{n} A_{i (m n)}^{j},

where

A_{i (p l)}^{j} = g_{i p} δ_{l}^{j} - g_{i}^{l} δ_{p}^{j} .

Hence,

ψ_{i (m n)} = \sum_{j = 1}^{n} A_{i (m n)}^{j} u_{j} (x) = g_{i n} u_{m} (x) - g_{j m} u_{n} (x) .

For the general variation, we define again

\tilde{A} (u, φ_{1}, φ_{2}, ε) = \int_{Ω} L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] det J (x) d x .

where we have generically denoted

L [u] \equiv L (u, \nabla u),

L [u (x + ε φ_{2} (x)) + ε φ_{1} (x)] \equiv L (u (x + ε φ_{2} (x)) + ε φ_{1} (x), \nabla u (x + ε φ_{2} (x)) + ε \nabla φ_{1} (x)),

\begin{matrix} J (x) & = & \{\frac{\partial x_{j}^{'}}{\partial x_{k}}\} \\ = & \{\frac{\partial (x_{j} + ε {(φ_{2})}_{j} (x))}{\partial x_{k}}\} \\ = & \{δ_{j k} + ε \frac{\partial {(φ_{2})}_{j} (x)}{\partial x_{k}}\} . \end{matrix}

(144)

and

δ A (u, φ_{1}, φ_{2}) = \frac{d}{d ε} (\tilde{A} (u, φ_{1}, φ_{2}, ε)) |_{ε = 0},

Moreover, we set

{(φ_{2})}_{k}^{m l} = w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}),

and

\bar{δ u_{i}} = u_{i}^{'} (x) - u_{i} (x) .

Thus,

\begin{matrix} δ u_{i} (x) & = & u_{i}^{'} (x) - u_{i} (x) \\ = & u_{i}^{'} (x^{'}) - u_{i}^{'} (x) + u_{i}^{'} (x) - u_{i} (x), \end{matrix}

(145)

so that

\begin{matrix} \bar{δ u_{i} (x)} = u_{i}^{'} (x) - u_{i} (x) \\ = & δ u_{i} (x) - (u_{i}^{'} (x^{'}) - u_{i}^{'} (x)) \\ δ u_{i} (x) - \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{d x_{k}} δ x_{k} + O (ε^{2}) \\ = & δ u_{i} (x) - \sum_{l = 1}^{n} \sum_{m < l} \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{k}} ε w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}) + O (ε^{2}) \\ = & ε (\sum_{l = 1}^{n} \sum_{j, k < l} A_{i (k l)}^{j} u_{j} (x) w^{k l} - \sum_{l = 1}^{n} \sum_{m < l} \sum_{k = 1}^{n} \frac{\partial u_{i} (x)}{\partial x_{k}} w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k})) + O (ε^{2}), \end{matrix}

With such results in mind, we define

\begin{matrix} {(φ_{1})}_{i}^{m l} & = & \sum_{j, k < l} A_{i (k l)}^{j} u_{j} (x) w^{m l} \\ - \sum_{k = 1}^{n} (\frac{\partial u_{i} (x)}{\partial x_{k}} w^{m l} (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k})) . \end{matrix}

(146)

Similarly as in the previous section, we may obtain

\begin{matrix} δ A (u, φ_{1}, φ_{2}) \\ = & \frac{d \tilde{A} (u, φ_{1}, φ_{2}, ε)}{d ε} |_{ε = 0} \\ = & \sum_{l = 1}^{n} \sum_{j, m < l} \sum_{k = 1}^{n} \sum_{i = 1}^{N} \int_{Ω} \frac{d}{d x_{k}} (\frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (A_{i {(l, m)}^{j}} u_{j} (x) + \frac{\partial u_{i}}{\partial x_{p}} g^{m m} x_{m} δ_{l}^{p} - \frac{\partial u_{i}}{\partial x_{p}} g^{l l} x_{l} δ_{m}^{p}) w^{m l}) d x \\ + \sum_{k = 1}^{n} \sum_{l = 1}^{n} \sum_{j, m < l} \sum_{i = 1}^{N} \int_{Ω} \frac{d}{d x_{k}} (L [u] (g^{l l} x_{l} δ_{m}^{k} - g^{m m} x_{m} δ_{l}^{k}) w^{m l}) d x \end{matrix}

(147)

Thus,

δ A (u, φ_{1}, φ_{2}) = - \sum_{k = 1}^{n} \sum_{m < l} \int_{Ω} \frac{d}{d x_{k}} (M_{m l}^{k} w^{m l}) d x,

where

\begin{matrix} M_{m l}^{k} & = & \sum_{i = 1}^{N} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} (A_{i l m}^{j} u_{j} - \frac{\partial u_{i}}{\partial x_{l}} g^{m m} x_{m} + \frac{\partial u_{i}}{\partial x_{m}} g^{l l} x_{l}) \\ + L [u] (g^{l l} x_{l} δ_{m}^{k} + g^{m m} x_{m} δ_{l}^{k}), \end{matrix}

(148)

so that

\begin{matrix} M_{l m}^{k} & = & (g^{m m} x_{m} T_{l}^{k} - g^{l l} x_{l} T_{m}^{k}) \\ - \sum_{i = 1}^{n} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} A_{i (l m)}^{j} u_{j} (x) \\ = & L_{m l}^{k} + S_{m l}^{k}, \end{matrix}

(149)

where

L_{m l}^{k} = (g^{m m} x_{m} T_{l}^{k} - g^{l l} x_{l} T_{m}^{k})

and

S_{m l}^{k} = - \sum_{i = 1}^{N} \sum_{j < l} \frac{\partial L [u]}{\partial {(u_{i})}_{x_{k}}} A_{i (l m)}^{j} u_{j} (x) .

The tensor

{L_{m l}^{k}}

is said to be the Orbital angular momentum tensor and

{S_{m l}^{k}}

is said to be Spin one.

25.2. A note on the solution of the Klein-Gordon equation

For

Ω = R^{4}

Ω_{1} = R^{3}

and denoting as usual by

i \in C

the imaginary unit, consider the Klein-Gordon equation in distributional sense

- \frac{\partial^{2} u}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} u}{\partial x_{j}^{2}} - m^{2} u = 0, in Ω,

where

u \in V = W^{1, 2} (Ω) .

Defining the Fourier transform of u, by

ϕ (p) = \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} e^{- i p \cdot x} u (x) d x,

in the momenta space, the last equation is equivalent to

(p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) = 0, in Ω,

where we have denoted

p = (p_{0}, p_{1}, p_{2}, p_{3}) \in R^{4},

and

x = (x_{0}, x_{1}, x_{2}, x_{3}) \in R 4

Observe that a general solution for this last equation is given by the wave function

\hat{ϕ} (p) = δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p),

where

ϕ \in W^{1, 2} (Ω) .

Indeed,

\begin{matrix} (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) \hat{ϕ} (p) & = & (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) \\ = & 0, in Ω . \end{matrix}

(150)

Here, we recall that generically for the Dirac delta function

δ (t),

we have

δ (t) = \{\begin{matrix} 0, & if t \neq 0, \\ + \infty, & if t = 0 . \end{matrix}

(151)

Observe that, for the scalar case in the previous section, we have

2 T^{00} = \sum_{j = 0}^{3} {(\frac{\partial u}{\partial x_{j}})}^{2} + m^{2} u .

Also, from

- \frac{\partial^{2} u}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} u}{\partial x_{j}^{2}} - m^{2} u = 0, in Ω,

we get

\int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x - \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x - m^{2} \int_{Ω} u^{2} d x = 0,

so that

\int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x = \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x + m^{2} \int_{Ω} u^{2} d x .

From such results, we may infer that

\begin{matrix} \int_{Ω} T^{00} d x & = & \int_{Ω} {(\frac{\partial u}{\partial t})}^{2} d x \\ = & {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2} \\ = & \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x + m^{2} \int_{Ω} u^{2} d x . \end{matrix}

(152)

On the other hand,

\begin{matrix} \sum_{j = 1}^{3} \int_{Ω} {(\frac{\partial u}{\partial x_{j}})}^{2} d x \\ = & \frac{1}{{(2 π)}^{3}} \sum_{j = 1}^{3} \int_{Ω} (\int_{Ω} i p_{j} \hat{ϕ} (p) e^{i p \cdot x} d p) (\int_{Ω} i p_{j}^{'} \hat{ϕ} (p^{'}) e^{i p^{'} \cdot x} d p^{'}) d x \\ = & \frac{1}{{(2 π)}^{3}} \sum_{j = 1}^{3} \int_{Ω} \int_{Ω} (- p_{j} p_{j}^{'} \hat{ϕ} (p) \hat{ϕ} (p^{'}) \int_{Ω} e^{i (p + p^{'}) \cdot x} d x) d p d p^{'} \\ = & \frac{1}{{(2 π)}^{3 / 2}} \sum_{j = 1}^{3} \int_{Ω} \int_{Ω} (- p_{j} p_{j}^{'} \hat{ϕ} (p) \hat{ϕ} (p^{'}) δ (p + p^{'})) d p d p^{'} \\ = & \frac{1}{{(2 π)}^{3 / 2}} \sum_{j = 1}^{3} \int_{Ω} (p_{j}^{2} \hat{ϕ} (p) \hat{ϕ} (- p)) d p . \end{matrix}

(153)

Thus, denoting

\hat{p} = (p_{1}, p_{2}, p_{3}),

d \hat{p} = d p_{1} d p_{2} d p_{3},

and

p_{0} (\hat{p}) = \sqrt{\sum_{j = 1}^{3} p_{j}^{2} + m^{2}},

we may infer that

\begin{matrix} \int_{Ω} T^{00} d x & = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\sum_{j = 1}^{3} p_{j}^{2} + m^{2}) \hat{ϕ} (p) \hat{ϕ} (- p)) d p \\ = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\sum_{j = 1}^{3} p_{j}^{2} + m^{2}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) ϕ (p) ϕ (- p)) d p \\ = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (p_{0} {(\hat{p})}^{2} ϕ (p_{0} (\hat{p}), \hat{p}) ϕ (- p_{0} (\hat{p}), - \hat{p})) d \hat{p} . \end{matrix}

(154)

Summarizing we have got

\begin{matrix} \int_{Ω} T^{00} d x & = & \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (p_{0} {(\hat{p})}^{2} ϕ (p_{0} (\hat{p}), \hat{p}) ϕ (- p_{0} (\hat{p}), - \hat{p})) d \hat{p} \\ = & {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2}, \end{matrix}

(155)

so that

\int_{Ω} T^{00} d x = {∥\frac{\partial u}{\partial t}∥}_{L^{2}}^{2}

may be expressed as a kind of average expectance of

p_{0}^{2}

related to the function

ϕ (p) .

25.3. A note on the Dirac equation

In this subsection we denote

Δ^{2} = \sum_{j = 0}^{3} g^{j j} L_{j} L_{j},

where

L_{j} = i g^{j j} \frac{\partial}{\partial x_{j}}, \forall j \in {0, 1, 2, 3} .

We recall that the relativistic Klein-Gordon equation may be written as

(Δ^{2} - m^{2}) u = 0, in Ω = R^{4} .

Moreover, for

4 \times 4

matrices

γ^{k}

indicated in the subsequent lines, we may obtain

{D_{i j}} u = [- i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) - m] [- i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) + m] u,

where

D_{i i} = Δ^{2} - m^{2}

and

D_{i j} = 0, if i \neq j, \forall i, j \in {0, 1, 2, 3} .

Here

u = {(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in V = W^{1, 2} (Ω; C^{4}) .

In such a case the fundamental Dirac equation stands for

[i (\sum_{j = 0}^{3} γ^{j} \frac{\partial}{\partial x_{j}}) - m] u = 0 \in R^{4}, in Ω .

Summarizing, if

{(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in V

is a solution of this last Dirac equation, then

u_{0}, u_{1}, u_{2}, u_{3}

are four solutions of the Klein-Gordon equation.

In the momentum configuration space, through the Fourier transform proprieties, the Dirac equation stands for

(\hat{p} + m) \hat{u} (p) = 0, in R^{4},

where

\hat{p} = \sum_{j = 0}^{3} g^{j j} p_{j} γ^{j} .

Observe that

\tilde{u} (p) = δ (\hat{p} + m) u (p)

corresponds to a general solution of the Dirac equation.

Indeed,

(\hat{p} + m) \tilde{u} (p) = (\hat{p} + m) δ (\hat{p} + m) u (p) = 0 \in R^{4}, in Ω .

On the other hand

\hat{u} (p) = δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) u (p)

correspond to four solutions of the Klein-Gordon equation.

At this point, we assume such a

\hat{u} (p)

corresponds to a solution of the Dirac equation as well.

Furthermore, here we recall that (please see the first chapter of the book [20], by N.N. Bogoliubov and D.V. Shirkov for details):

γ^{0} = \{\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}\},

(156)

γ^{1} = \{\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{matrix}\},

(157)

γ^{2} = \{\begin{matrix} 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ - i & 0 & 0 & 0 \end{matrix}\},

(158)

γ^{3} = \{\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \\ - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}\}

(159)

and

γ^{5} = \{\begin{matrix} 0 & 0 & - i & 0 \\ 0 & 0 & 0 & - i \\ - i & 0 & 0 & 0 \\ 0 & - i & 0 & 0 \end{matrix}\}

(160)

where we also denote

α_{j} = γ^{0} γ^{j}, \forall j \in {1, 2, 3},

σ_{j} = i γ^{5} γ^{0} γ^{j}, \forall j \in {1, 2, 3},

and

β = γ^{0} .

On the other hand, a variational formulation for the Dirac equation corresponds to the functional

A : V \to R

where

A (u) = \frac{1}{2} \int_{Ω} L (u, \nabla u) d x,

where

L (u, \nabla u) = i \sum_{j = 0}^{3} (u^{*} γ^{j} \frac{\partial u}{\partial x_{j}} - \frac{\partial u^{*}}{\partial x_{j}} γ^{j} u) - m^{2} u^{*} u,

where here

u = {(u_{0}, u_{1}, u_{2}, u_{3})}^{T} \in W^{1, 2} (Ω; C^{4}) .

From such statements and definitions, similarly as in the previous sections (please see [20] for details), we may obtain

T^{k l} = \frac{i}{2} g^{l l} (u^{*} γ^{k} \frac{\partial u}{\partial x_{l}} - \frac{\partial u^{*}}{\partial x_{l}} γ^{k} u),

and

S^{k, l m} = - (\frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}} A^{u, l m} u - u^{*} A^{u^{*}, l m} \frac{\partial L (u, \nabla u)}{\partial u_{x_{k}}}),

where

A^{u, l m} = \frac{i}{2} σ^{m l},

A^{u^{*}, l m} = \frac{i}{2} σ^{l m},

and where

σ^{l m} = \frac{γ^{l} γ^{k} - γ^{k} γ^{l}}{2},

so that

S^{k, l m} = \frac{1}{4} u^{*} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u .

Thus,

\begin{matrix} \int_{Ω} S^{k, l m} d x \\ = & \frac{1}{4} \int_{Ω} (u^{*} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u) d x \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3}} \int_{Ω} (\int_{Ω} \int_{Ω} (\hat{u} (p) e^{i p \cdot x} (γ^{k} σ^{l m} - σ^{l m} γ^{k}) \hat{u} (p^{'}) e^{i p^{'} \cdot x}) d p d p^{'}) d x \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} \int_{Ω} (\hat{u} (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) δ (p + p^{'}) \hat{u} (p^{'})) d p d p^{'} \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (\hat{u} (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) \hat{u} (- p)) d p \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω} (u (p) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) δ (p_{0}^{2} - \sum_{j = 1}^{3} p_{j}^{2} - m^{2}) u (- p)) d p \\ = & \frac{1}{4} \frac{1}{{(2 π)}^{3 / 2}} \int_{Ω_{1}} (u (p_{0} (\hat{p}), \hat{p}) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u (- p_{0} (\hat{p}), - \hat{p})) d \hat{p}, \end{matrix}

(161)

where

p_{0} (\hat{p}) = \sqrt{\sum_{j = 1}^{3} p_{j}^{2} + m^{2}} .

Summarizing, we have got

\int_{Ω} S^{k, l m} d x = \frac{1}{4 {(2 π)}^{3 / 2}} \int_{Ω_{1}} (u (p_{0} (\hat{p}), \hat{p}) (γ^{k} σ^{l m} - σ^{l m} γ^{k}) u (- p_{0} (\hat{p}), - \hat{p})) d \hat{p},

where

Ω_{1} = R^{3}

\hat{p} = (p_{1}, p_{2}, p_{3})

and

d \hat{p} = d p_{1} d p_{2} d p_{3} .

26. A note on quantum field operators

This section is strongly based on the chapter 3, page 53 of the book [21], by G.B. Folland.

Let

Ω = \hat{Ω} \times [0, T] \subset R^{4}

where

\hat{Ω} \subset R^{3}

is a open, bounded and connected set with a regular boundary denoted

\partial \hat{Ω} .

Define

V = W^{1, 2} (Ω)

and

V_{0} = W_{0}^{1, 2} (Ω) .

Consider an operator

H : V_{1} = V_{0} \cap W^{2, 2} (Ω) \to Y

where in a distributional sense,

H (u) = - \frac{\partial^{2} u}{\partial t^{2}} + \nabla^{2} u - m^{2} u,

and where

Y = Y^{*} = L^{2} (Ω) .

Suppose there exists operators

B_{1} : Y \to Y

and

B_{2} : Y \to Y

such that

B_{1} B_{2} (u) = H (u) + \frac{1}{2} u

and

B_{2} B_{1} (u) = H (u) - \frac{1}{2} u, \forall u \in V_{1} .

Assume also

ϕ_{0} \in V_{1}

is such that

∥ ϕ_{0} ∥_{L^{2}} = 1,

and

B_{1} ϕ_{0} = 0 .

Now define

ϕ_{k} = \frac{B_{2}^{k} (ϕ_{0})}{\sqrt{k!}}, \forall k \in N .

Observe that

[B_{1} B_{2}] = B_{1} B_{2} - B_{2} B_{1} = I_{d} .

We shall prove by induction that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

(162)

Indeed, for

k = 1

[B_{1}, B_{2}] = I_{d} = 1 B_{2}^{0},

so that (162) holds for

k = 1

Suppose now (162) holds for

k \in N,

so that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1} .

In order to complete the induction, it suffices to prove that (162) holds for

k + 1 .

Observe that

\begin{matrix} [B_{1}, B_{2}^{k + 1}] & = & (B_{1} B_{2}^{k + 1} - B_{2}^{k + 1} B_{1}) \\ = & (B_{1} B_{2}^{k}) B_{2} - B_{2}^{k + 1} B_{1} \\ = & (B_{2}^{k} B_{1} + k B_{2}^{k - 1}) B_{2} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k} (B_{1} B_{2}) + k B_{2}^{k} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k} (B_{2} B_{1} + I_{d}) + k B_{2} - B_{2}^{k + 1} B_{1} \\ = & B_{2}^{k + 1} B_{1} + B_{2}^{k} + k B_{2}^{k} - B_{2}^{k + 1} B_{1} \\ = & (k + 1) B_{2}^{k} . \end{matrix}

(163)

Thus, the induction is complete, so that

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

Moreover, we recall that

B_{1} ϕ_{0} = 0,

so that

\begin{matrix} B_{1} ϕ_{k} & = & B_{1} (\frac{B_{2}^{k} ϕ_{0}}{\sqrt{k!}}) \\ = & \frac{(B_{2}^{k} B_{1} + k B_{2}^{k - 1}) ϕ_{0}}{\sqrt{k!}} \\ = & \frac{k ϕ_{k - 1} \sqrt{(k - 1)!}}{\sqrt{k!}} \\ = & \frac{k ϕ_{k - 1}}{\sqrt{k}} \\ = & \sqrt{k} ϕ_{k - 1}, \forall k \in N . \end{matrix}

(164)

Summarizing, we have got

B_{1} ϕ_{k} = \sqrt{k} ϕ_{k - 1}, \forall k \in N .

Now, we shall prove that

B_{2} ϕ_{k} = \sqrt{k + 1} ϕ_{k + 1}, \forall k \in N .

Observe that

\begin{matrix} B_{2}^{k + 1} ϕ_{0} & = & ϕ_{k + 1} (\sqrt{(k + 1)!} \\ = & B_{2} (B_{2}^{k} ϕ_{0}) \\ = & (B_{2} ϕ_{k}) \sqrt{k!} . \end{matrix}

(165)

Summarizing, we have got

(B_{2} ϕ_{k}) \sqrt{k!} = ϕ_{k + 1} (\sqrt{(k + 1)!},

so that

(B_{2} ϕ_{k}) = \sqrt{k + 1} ϕ_{k + 1}, \forall k \in N .

Finally, from such results, we may infer that

\begin{matrix} B_{1} B_{2} ϕ_{k} & = & B_{1} (\sqrt{k + 1} ϕ_{k + 1}) \\ = & \sqrt{k + 1} B_{1} ϕ_{k + 1} \\ = & \sqrt{k + 1} \sqrt{k + 1} ϕ_{k} \\ = & (k + 1) ϕ_{k}, \forall k \in N . \end{matrix}

(166)

Similarly,

\begin{matrix} B_{2} B_{1} ϕ_{k} & = & B_{2} (\sqrt{k} ϕ_{k - 1}) \\ = & \sqrt{k} B_{2} ϕ_{k - 1} \\ = & \sqrt{k} \sqrt{k} ϕ_{k} \\ = & k ϕ_{k} . \end{matrix}

(167)

Therefore we have got

H ϕ_{k} = B_{1} B_{2} ϕ_{k} - \frac{1}{2} ϕ_{k} = (k + 1) ϕ_{k} - \frac{1}{2} ϕ_{k} = (k + \frac{1}{2}) ϕ_{k},

that is

H ϕ_{k} = (k + \frac{1}{2}) ϕ_{k}, \forall k \in N .

Thus, for each

k \in N

k + \frac{1}{2}

is an eigenvalue of H with corresponding eigenvector

ϕ_{k}

26.1. An application concerning the harmonic oscillator operator in quantum mechanics

In this section we have the aim of representing the relativistic Klein-Gordon equation through the creation and annihilation operations related to the harmonic oscillator in quantum mechanics.

Consider first the one-dimensional Hamiltonian, corresponding to the harmonic oscillator, namely

H = - \frac{ℏ}{2 m} \frac{d^{2}}{d x^{2}} + K \frac{x^{2}}{2},

which through an appropriate re-scale results into the following related Hamiltonian

H_{0}

, where

H_{0} = \frac{1}{2} (- \frac{d^{2}}{d x^{2}} + x^{2}) .

Define now the operators

B_{1} = A = \frac{1}{\sqrt{2}} (x + \frac{d}{d x}),

and

B_{2} = A^{*} = \frac{1}{\sqrt{2}} (x - \frac{d}{d x}) .

Clearly,

H_{0} = B_{1} B_{2} - \frac{I_{d}}{2} = B_{2} B_{1} + \frac{I_{d}}{2},

so that

[A, A^{*}] = [B_{1}, B_{2}] = B_{1} B_{2} - B_{2} B_{1} = I_{d} .

Similarly, as in the previous sections, by induction, we may obtain

[B_{1}, B_{2}^{k}] = k B_{2}^{k - 1}, \forall k \in N .

For

ϕ_{0} = π^{- 1 / 4} e^{- \frac{x^{2}}{2}},

we define

ϕ_{k} = \frac{1}{\sqrt{k}} B_{2}^{k} ϕ_{0}, \forall k \in N .

Also from the previous section, we may obtain

B_{2} ϕ_{k} = A^{*} ϕ_{k} = \sqrt{k + 1} ϕ_{k + 1},

B_{1} ϕ_{k} = A ϕ_{k} = \sqrt{k} ϕ_{k - 1}, \forall k \in N .

B_{2} B_{1} = A^{*} A ϕ_{k} = k ϕ_{k},

and

B_{1} B_{2} ϕ_{k} = A A^{*} ϕ_{k} = (k + 1) ϕ_{k}, \forall k \in N \cup {0} .

so that

H_{0} ϕ_{k} = (k + 1 / 2) ϕ_{k}, \forall k \in N .

Here we recall that

B_{1} ϕ_{0} = A ϕ_{0} = 0,

and

∥ ϕ_{0} ∥_{L^{2}} = 1 .

In reference [21], page 54 it is proven that such a sequence

{ϕ_{k}}

is an ortho-normal basis for

L^{2} (R) .

Finally, observe that for

R^{4}

we may define

{(B_{1})}_{j} = A_{j} = \frac{1}{\sqrt{2}} (\frac{\partial}{\partial x_{j}} + x_{j}),

and

{(B_{2})}_{j} = A_{j}^{*} = \frac{1}{\sqrt{2}} (- \frac{\partial}{\partial x_{j}} + x_{j}), \forall j \in {0, 1, 2, 3} .

Here generically,

x = (x_{0}, x_{1}, x_{2}, x_{3}) \in R^{4} .

Observe that clearly

\frac{\partial}{\partial x_{j}} = \frac{\sqrt{2}}{2} (A_{j} - A_{j}^{*}),

and

x_{j} I_{d} = \frac{\sqrt{2}}{2} (A_{j} + A_{j}^{*}), \forall j \in {0, 1, 2, 3} .

Denoting

x_{0} = t

where t stands for time, consider the relativistic Klein-Gordon equation,

- \frac{\partial^{2} ϕ}{\partial t^{2}} + \sum_{j = 1}^{3} \frac{\partial^{2} ϕ}{\partial x_{j}^{2}} - m^{2} ϕ = 0 .

From the previous results, we may represent such an equation by

(- \frac{1}{2} {(A_{0} - A_{0}^{*})}^{2} + \sum_{j = 1}^{3} \frac{1}{2} {(A_{j} - A_{j}^{*})}^{2} - m^{2} I_{d}) ϕ = 0 .

We highlight from the previous results we know the action of

A_{j}

and

A_{j}^{*}

on an appropriate basis of

L^{2} (R^{4})

obtained though an appropriate tensorial product of the bases

{{ϕ_{k} (x_{j})}, for j \in {0, 1, 2, 3}} .

We shall call the operators

A_{j}^{*}

and

A_{j}

as the creation and annihilation operators concerning the original harmonic operator in quantum mechanics.

To justify such a nomenclature, we recall that

A_{j}^{*} ϕ_{0} (x_{j}) = ϕ_{1} (x_{j})

and

A_{j} ϕ_{0} (x_{j}) = 0, \forall j \in {0, 1, 2, 3} .

27. A dual variational formulation for a related model

In this section we develop a concave dual variational formulation for a Ginzburg-Landau type equation.

Let

Ω \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a functional

J : V \to R

defined by

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(168)

where

γ > 0,

α > 0

β > 0

f \in L^{2} (Ω),

and

V = W_{0}^{1, 2} (Ω) .

We also denote

Y = Y^{*} = L^{2} (Ω) .

Define now

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

for some appropriate

K_{3} > 0

and,

J_{1} : V \times Y \to R

J_{1} (u, v_{0}^{*}) = J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x,

where

K_{1} = \frac{1}{4 α K_{3}^{2} + ε}

for some small parameter

0 < ε ≪ 1 .

Observe that

\begin{matrix} J (u, v_{0}^{*}) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}} \\ - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ \geq & inf_{u \in V_{1}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}}\} \\ + inf_{v \in Y} \{- {〈 v, v_{0}^{*} 〉}_{L^{2}} + \frac{α}{2} \int_{Ω} {(v - β)}^{2} d x\} \\ = & - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}) \\ \equiv & J^{*} (v_{0}^{*}), \forall u \in V_{1}, v_{0}^{*} \in Y^{*}, \end{matrix}

(169)

where we have denoted

F^{*} (v_{0}^{*}) = sup_{u \in V_{1}} {- {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - F (u, v_{0}^{*})},

F (u, v_{0}^{*}) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x - {〈 u, f 〉}_{L^{2}},

and

G (v) = \frac{α}{2} \int_{Ω} {(v - β)}^{2} d x,

\begin{matrix} G^{*} (v_{0}^{*}) & = & sup_{v \in Y} {{〈 v, v_{0}^{*} 〉}_{L^{2}} - G (v)} \\ = & \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(170)

Observe that

\frac{\partial F (u, v_{0}^{*})}{\partial u^{2}} = - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- \nabla^{2} + 2 v_{0}^{*})}^{2},

so that we define

B^{*} = {v_{0}^{*} \in Y^{*} : - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- \nabla^{2} + 2 v_{0}^{*})}^{2} > 0} .

With such assumptions and definitions in mind, we may prove the following theorem:

Theorem 27.1.

For

J^{*} (v_{0}^{*}) = - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}),

suppose

{\hat{v}}_{0}^{*} \in B^{*}

is such that

δ J^{*} ({\hat{v}}_{0}^{*}) = 0 .

Let

u_{0} \in Y

be such that

\frac{\partial H (u_{0}, {\hat{v}}_{0}^{*})}{\partial u} = 0,

where

H (u, v_{0}^{*}) = F (u, v_{0}^{*}) + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} .

Suppose

u_{0} \in V_{1} .

Under such hypotheses,

F^{*} ({\hat{v}}_{0}^{*}) = H (u_{0}, {\hat{v}}_{0}^{*}),

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) \\ = & sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) \\ = & J^{*} ({\hat{v}}_{0}^{*}) . \end{matrix}

(171)

Proof.

The proof that

F^{*} ({\hat{v}}_{0}^{*}) = H (u_{0}, {\hat{v}}_{0}^{*}),

is immediate from

{\hat{v}}_{0} \in B^{*} .

Moreover, the proof that

δ J (u_{0}) = 0,

and

J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = J^{*} ({\hat{v}}_{0}^{*})

may be done similarly as in the previous sections.

Observe that

J^{*} (v_{0}^{*}) = - F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}) = inf_{u \in V_{1}} {H (u, v_{0}^{*}) - G^{*} (v_{0}^{*})},

so that

J^{*}

is concave in

v_{0}^{*}

as the infimum of a family of concave functionals in

v_{0}^{*}

From this and

δ J^{*} ({\hat{v}}_{0}^{*}) = 0

we get

J^{*} ({\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) .

Furthermore observe that

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ \leq & J_{1} (u, v_{0}^{*}) \\ = & F (u, {\hat{v}}_{0}^{*}) + {〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - G^{*} ({\hat{v}}_{0}^{*}) \\ \leq & F (u, {\hat{v}}_{0}^{*}) + sup_{v_{0}^{*} \in Y^{*}} \{{〈 u^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - G^{*} ({\hat{v}}_{0}^{*})\} \\ = & F (u, {\hat{v}}_{0}^{*}) + G (u^{2}) \\ = & J_{1} (u, {\hat{v}}_{0}^{*}), \forall u \in V_{1} . \end{matrix}

(172)

Hence

J (u_{0}) = J_{1} (u_{0}, {\hat{v}}_{0}^{*}) = inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) .

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & J_{1} (u_{0}, {\hat{v}}_{0}^{*}) \\ = & inf_{u \in V_{1}} J_{1} (u, {\hat{v}}_{0}^{*}) \\ = & sup_{v_{0}^{*} \in Y^{*}} J^{*} (v_{0}^{*}) \\ = & J^{*} ({\hat{v}}_{0}^{*}) . \end{matrix}

(173)

The proof is complete. □

28. The generalized method of lines applied to fourth order differential equations

In this sections we develop an application of the generalized method of lines to a fourth order equation.

We start by addressing the following ordinary differential equation (ode):

ε \frac{d^{4} u (x)}{d x^{4}} - f = 0, in [0, 1],

with the boundary conditions

u (0) = u^{'} (0) = 0

and

u (1) = u^{'} (1) = 0 .

In terms of linear elasticity, such a boundary conditions corresponds to a bi-clamped beam.

In a finite difference context, this last equation corresponds to

ε (\frac{u_{n + 2} - 4 u_{n + 1} + 6 u_{n} - 4 u_{n - 1} + u_{n - 2}}{d^{4}}) - f_{n} = 0, \forall n \in {1, \dots N - 2},

where N is the number of nodes and

d = 1 / N

Considering that, from the boundary conditions,

u_{- 1} = u_{0} = 0,

for

n = 1

we get

6 u_{1} - 4 u_{2} + u_{3} = \frac{f_{1} d^{4}}{ε},

so that

u_{1} = a_{1} u_{2} + b_{1} u_{3} + c_{1},

where

a_{1} = 2 / 3, b_{1} - 1 / 6 and c_{1} = \frac{f_{1} d^{4}}{6 ε} .

Similarly, for

n = 2

, we obtain

- 4 u_{1} + 6 u_{2} - 4 u_{3} + u_{4} = \frac{f_{2} d^{4}}{ε} .

Hence, replacing the value of

u_{1}

previously obtained in this last equation, we have

- 4 (a_{1} u_{2} + b_{1} u_{3} + c_{1}) + 6 u_{2} - 4 u_{3} + u_{4} = \frac{f_{2} d^{4}}{ε},

so that

u_{2} = a_{2} u_{3} + b_{2} u_{4} + c_{2},

where defining

m_{12} = (6 - 4 a_{1})

, we have also

a_{2} = \frac{4 b_{1} + 4}{m_{12}},

b_{2} = - \frac{1}{m_{12}},

c_{2} = \frac{1}{m_{12}} (\frac{f_{2} d^{4}}{ε} + 4 c_{1}) .

Now reasoning inductively, for n, having

u_{n - 1} = a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1},

and

u_{n - 2} = a_{n - 2} u_{n - 1} + b_{n - 2} u_{n} + c_{n - 2}

we obtain

u_{n - 2} = a_{n - 2} (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + b_{n - 2} u_{n} + c_{n - 2},

so that from this and

u_{n + 2} - 4 u_{n + 1} + 6 u_{n} - 4 u_{n - 1} + u_{n - 2} = \frac{f_{n} d^{4}}{ε},

we obtain

\begin{matrix} a_{n - 2} (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + b_{n - 2} u_{n} + c_{n - 2} \\ - 4 (a_{n - 1} u_{n} + b_{n - 1} u_{n + 1} + c_{n - 1}) + 6 u_{n} - 4 u_{n + 1} + u_{n + 2} = \frac{f_{n} d^{4}}{ε}, \end{matrix}

(174)

so that

u_{n} = a_{n} u_{n + 1} + b_{n} u_{n + 1} + c_{n}

where defining

m_{12} = (a_{n - 2} (a_{n - 1}) + b_{n - 2} - 4 a_{n - 1} + 6)

we obtain

a_{n} = - \frac{1}{m_{12}} (a_{n - 2} b_{n - 1} - 4 b_{n - 1} - 4)

b_{n} = - \frac{1}{m_{12}},

and

c_{n} = \frac{1}{m_{12}} (a_{n - 2} c_{n - 1} + c_{n - 2} - 4 c_{n - 1} - \frac{f_{n} d^{4}}{ε}) .

Summarizing, we have got

u_{n} = a_{n} u_{n + 1} + b_{n} u_{n + 2} + c_{n}, \forall n \in {1, \cdot N - 2} .

Observe now that from the boundary conditions,

u_{N - 1} = u_{N} = 0 .

From these last two equations, we may obtain

u_{N - 2} = c_{N_{2}},

and

u_{N - 3} = a_{N - 3} u_{N - 2} + b_{N - 3} u_{N - 1} + c_{N - 3},

and so on up to obtaining

u_{1} = a_{1} u_{2} + b_{1} u_{3} + c_{1} .

The problem is then solved.

28.1. A numerical example

We develop a numerical example considering

ε = 1,

and

f \equiv 1, in [0, 1] .

Thus, we have solved the equation

ε \frac{d^{4} u (x)}{d x^{4}} - f = 0, in [0, 1],

with the boundary conditions

u (0) = u^{'} (0) = 0

and

u (1) = u^{'} (1) = 0 .

In a finite differences context, we have used

N = 100

nodes and

d = 1 / N .

For a solution

u (x)

, please see Figure 19.

In the next lines, we present the concerning software in MAT-LAB

**************

1.: clear all

m8=100;

d=1/m8;

e1=1.0;

for i=1:m8

f(i,1)=1.0;

end;

a(1)=2/3;

b(1)=-1/6;

c(1)=f(1,1)* $d^{4}$ /(6e1);

m12=(6-4*a(1));

a(2)=(4*b(1)+4)/m12;

b(2)=-1/m12;

c(2)=1/m12*(4*c(1)+f(2,1)* $d^{4}$ /e1);

for i=3:m8-2

m12=(a(i-2)*a(i-1)+b(i-2)-4*a(i-1)+6);

a(i)=-1/m12*(a(i-2)*b(i-1)-4*b(i-1)-4);

b(i)=-1/m12;

c(i)=1/m12*(f(i,1)* $d^{4}$ /e1-c(i-2)-a(i-2)*c(i-1)+4*c(i-1));

end;

u(m8,1)=0;

u(m8-1,1)=0;

for i=2:m8-1;

u(m8-i,1)=a(m8-i)*u(m8-i+1,1)+b(m8-i)*u(m8-i+2,1)+c(m8-i);

end;

for i=1:m8

x(i)=i*d;

end;

plot(x,u)

******************

29. A note on hyper-finite differences for the generalized method of lines

In this section we develop an application of the hyper finite differences method through an approximation of the generalized method of lines.

Consider the equation

\{\begin{matrix} - ε u^{''} (x) + α u^{3} - β u - f = 0, & in Ω = [0, 1], \\ u (0) = 0, & u (1) = 0 \end{matrix}

(175)

ε > 0

is small, in order to decrease the error concerning the approximations used we propose to divide the domain

Ω = [0, 1]

into

N_{1}

sub-intervals of same measure. Thus we define

x_{k} = \frac{k}{N_{1}}, \forall k \in {0, 1, \dots, N_{1}} .

For each sub-interval

I_{k} = [x_{k - 1}, x_{k}]

we are going to obtain an approximate solution of the equation in question with the general boundary conditions

u ((k - 1) / N_{1}) = U [k - 1],

and

u (k / N_{1}) = U [k] .

Denoting such a solution by

{u [i, k]}

where

x_{i} = \frac{k - 1}{N_{1}} + i d,

and

d = \frac{1}{m_{8} N_{1}},

where

m_{8}

is the fixed number of nodes in each interval

I_{k}

Observe that in a finite differences context, linearizing it about a initial solution

{u_{0} [i, k]},

the equation in question stands for:

\begin{matrix} - ε \frac{(u [i + 1, k] - 2 u [i, k] + u [i - 1, k])}{d^{2}} + 3 α u_{0} {[i, k]}^{2} u [i, k] - 2 α u_{0} {[i, k]}^{3} \\ - β u [i, k] - f [i, k] = 0, \forall i \in {1, \dots, m_{8} - 1} . \end{matrix}

(176)

In particular, for

i = 1

, we obtain

\begin{matrix} - ε \frac{(u [2, k] - 2 u [1, k] + u [0, k])}{d^{2}} + 3 α u_{0} {[1, k]}^{2} u [1, k] - 2 α u_{0} {[1, k]}^{3} \\ - β u [1, k] - f [1, k] = 0, \end{matrix}

(177)

so that

\begin{matrix} u [1, k] & = & a [1, k] u [2, k] + b [1, k] u [0, k] + c [1, k] T [1, k] \\ + e [1, k] + E_{r} [1, k], \end{matrix}

(178)

where

a [1, k] = 1 / 2,

b [1, k] = 1 / 2,

c [1, k] = 1 / 2,

e [1, k] = f [1, k] \frac{d^{2}}{2 ε},

T [1, k] = (- 3 α u_{0} {[1, k]}^{2} u [i, k] + 2 α u_{0} {[1, k]}^{3} - β u [1, k]) \frac{d^{2}}{ε},

and

E_{r} [1, k] = 0 .

Now reasoning inductively, having

\begin{matrix} u [i - 1, k] & = & a [i - 1, k] u [i, k] + b [i - 1, k] u [0, k] + c [i - 1, k] T [i - 1, k] \\ + e [i - 1, k] + E_{r} [i - 1, k], \end{matrix}

(179)

and

\begin{matrix} - ε \frac{(u [i + 1, k] - 2 u [i, k] + u [i - 1, k])}{d^{2}} + 3 α u_{0} {[i, k]}^{2} u [i, k] - 2 α u_{0} {[i, k]}^{3} \\ - β u [i, k] - f [i, k] = 0, \end{matrix}

(180)

so that

(u [i + 1, k] - 2 u [i, k] + u [i - 1, k]) + T [i, k] + f [i, k] \frac{d^{2}}{ε} = 0,

where,

T [i, k] = (- 3 α u_{0} {[i, k]}^{2} u [i, k] + 2 α u_{0} {[i, k]}^{3} + β u [i, k]) \frac{d^{2}}{ε},

we obtain

\begin{matrix} u [i, k] & = & a [i, k] u [i, k] + b [i, k] u [0, k] + c [i, k] T [i, k] \\ + e [i, k] + E_{r} [i, k], \end{matrix}

(181)

where,

a [i, k] = {(2 - a [i - 1, k])}^{- 1},

b [i, k] = a [i, k] b [i - 1, k],

c [i, k] = a [i, k] (c [i - 1, k] + 1),

e [i, k] = a [i, k] (e [i - 1, k] + \frac{f [i, k] d^{2}}{ε}),

and

E_{r} [i, k] = a [i, k] (E_{r} [i - 1, k]) + c [i, k] (T [i - 1, k] - T [i, k]) .

Observe that in particular for

i = m_{8} - 1

, we have

u [m 8, k] = U [k]

and

u [0, k] = U [k - 1],

so that from above, neglecting

E_{r} [1, k]

, we also obtain

\begin{matrix} u [m_{8} - 1, k] \approx a [m_{8} - 1] u [m_{8}, k] + b [m_{8} - 1, k] u [0, k] \\ + c [m_{8} - 1, k] T [m 8 - 1, k] (u [m_{8}, k], u [0, k]) + e [m_{8} - 1, k] \\ = & H_{m_{8} - 1} (U [k], U [k - 1]) . \end{matrix}

(182)

Similarly, for

i = m 8 - 2

we may obtain

\begin{matrix} u [m_{8} - 2, k] \approx a [m_{8} - 2] u [m_{8} - 1, k] + b [m_{8} - 2, k] u [0, k] \\ + c [m_{8} - 2, k] T [m 8 - 2, k] (u [m_{8} - 1, k], u [0, k]) + e [m_{8} - 2, k] \\ = & H_{m_{8} - 2} (U [k], U [k - 1]), \end{matrix}

(183)

and so on, up to finding

u [1, k] = H_{1} (U [k], U [k - 1]), \forall k \in {1, \dots, N_{1}} .

At this point we connect the sub-intervals by setting

U [0] = U [N_{1}] = 0

and obtaining

{U [1], \dots, U [N_{1} - 1]},

by solving the equations

\begin{matrix} - ε \frac{(u [m_{8} - 1, k] - 2 U [k] + u [1, k + 1])}{d^{2}} + 3 α u_{0} {[m 8, k]}^{2} U [k] - 2 α u_{0} {[m 8, k]}^{3} \\ - β U [k] - f [m_{8}, k] = 0, \forall k \in {1, \dots, N_{1} - 1} . \end{matrix}

(184)

Having obtained

{U [k], \forall k \in {1, \dots, N_{1} - 1}}

we may obtain the solution

{u [i, k]}

where

i \in {0, \dots, m_{8}}

and

k \in {1, \dots, N_{1}} .

The next step is to replace

{u_{0} [i, k]}

{u [i, k]}

and then to repeat the process until an appropriate convergence criterion is satisfied.

The problem is then approximately solved.

We have obtained numerical results for

ε = 0.001

f \equiv 1,

Ω

N_{1} = 10

m_{8} = 100

and

α = β = 1 .

For the related software in MATHEMATICA we have obtained

U [1], \dots, U [9],

Here the software and results:

**************************

1.: Clear[u, U, z, N1];

m8 = 100;

N1 = 10;

d = 1/m8/N1;

e1 = 0.001;

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = 1.01]];

A = 1.0;

B = 1.0;

a[1] = 1.0/2;

b[1] = 1.0/2;

c[1] = 1/2.0;

e[1] = $d^{2} / e 1 / 2.0$ ;

For[i = 2, i < m8, i++,

a[i] = 1/(2.0 - a[i - 1]);

b[i] = b[i - 1]*a[i];

c[i] = a[i]*(c[i - 1] + 1.0);

e[i] = $a [i] * (e [i - 1] + d^{2} / e 1)$ ;

];

For[k1 = 1, k1 < 10, k1++,

Print[k1];

Clear[U, z];

For[k = 1, k < N1 + 1, k++,

u[0, k] = U[k - 1];

u[m8, k] = U[k];

For[i = 1, i < m8, i++,

z = a[m8 - i]*u[m8 - i + 1, k] + b[m8 - i]*u[0, k] +

c[m8 - i]*(-3*A*uo[m8 - i + 1, k] $^{2}$ *u[m8 - i + 1, k] +

2*A*uo[m8 - i + 1, k] $^{3}$ + B*u[m8 - i + 1, k])* $d^{2} / e 1$ +

e[m8 - i];

u[m8 - i, k] = Expand[z]]];

U[0] = 0.0;

U[N1] = 0.0;

S = 0;

For[k = 1, k < N1, k++,

S = S + (e1*(-u[m8 - 1, k] + 2*U[k] - u[1, k + 1])/ $d^{2}$ +

3*A*U[k]*uo[m8, k] $^{2}$ - 2*A*uo[m8, k] $^{3}$ - B*U[k] - 1) $^{2}$ ];

Sol = NMinimize[S, U[1], U[2], U[3], U[4], U[5], U[6], U[7], U[8], U[9]];

For[k = 1, k < N1, k++,

w4[k] = U[k] Ṡol[[2, k]]];

For[k = 1, k < N1, k++,

U[k] = w4[k]];

For[k = 1, k < N1 + 1, k++,

For[i = 0, i < m8 + 1, i++,

uo[i, k] = u[i, k]]];

Print[U[5]]];

For[k = 0, k < N1 + 1, k++,

Print["U[", k, "]=", U[k]]]

U[0]=0.

U[1]=1.27567

U[2]=1.32297

U[3]=1.32466

U[4]=1.32472

U[5]=1.32472

U[6]=1.32472

U[7]=1.32472

U[8]=1.32472

U[9]=1.32471

U[10]=0.

**********************

Remark 29.1.

Observe that along the domain we have obtained approximately the constant value

u = 1.32472

. This is expected since

ε = 0.001

is small and such a value u is approximately the solution of equation

α u^{3} - β u - 1 = 0 .

30. Applications to the optimal shape design for a beam model

In this section, we present a numerical procedure for the shape optimization concerning the Bernoulli beam model.

Let

Ω = [0, 1] \subset R

corresponds to the horizontal axis of a straight beam with rectangular cross section

b \times h (x)

, that is, the beam has a variable thickness

h (x)

distributed along such a horizontal axis x, where

x \in [0, 1] .

Define now

V = {w \in W^{2, 2} (Ω) : w (0) = w (1) = 0},

which corresponds to a simply supported beam.

Consider the problem of minimizing in

V \times B

the functional

J (w, h) = \frac{1}{2} \int_{Ω} H (x) w_{x x} {(x)}^{2} d x

subject to

{(H (x) w_{x x} (x))}_{x x} - P (x) = 0, in Ω,

where

H (x) = \frac{h {(x)}^{3} b}{12} E,

h (x)

is variable beam thickness,

A (x) = b h (x)

corresponds to a rectangular cross section perpendicular to the x axis, and E is the young elasticity model.

Also, we define

B = \{h : [0, 1] \to R measurable : h_{m i n} \leq h (x) \leq h_{m a x} and \int_{0}^{1} h (x) \leq c_{0} h_{m a x}\},

where

0 < c_{0} < 1

and

C^{*} = {w \in V : {(H (x) w_{x x} (x))}_{x x} - P (x) = 0, in Ω} .

Observe that

\begin{matrix} inf_{(w, h) \in C^{*} \times B} J (w, h) \\ = & inf_{h \in B} \{inf_{w \in C^{*}} J (w, h)\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{\frac{1}{2} \int_{Ω} H (x) w_{x x} {(x)}^{2} d x - {〈 \hat{w}, {(H (x) w_{x x} (x))}_{x x} - P (x) 〉}_{L^{2}}\}\}\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{- \frac{1}{2} \int_{Ω} H (x) {\hat{w}}_{x x}^{2} d x + {〈 \hat{w}, P 〉}_{L^{2}}\}\} \\ = & inf_{h \in B} \{inf_{M \in D^{*}} \{\frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x\}\} . \end{matrix}

(185)

where

D^{*} = {M \in Y^{*} : M_{x x} - P = 0, in Ω, and M (0) = M (1) = 0} .

Summarizing, we have got

inf_{(w, h) \in C^{*} \times B} J (w, h) = inf_{(M, h) \in D^{*} \times B} \{\frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x\} .

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n = 1$ and

$h_{n} (x) = c_{0} h_{m a x} .$
Calculate $w_{n} \in V$ solution of equation

${(H_{n} (x) {(w_{n})}_{x x})}_{x x} = P (x),$

where

$H_{n} (x) = \frac{E b h_{n} {(x)}^{3}}{12} .$
Calculate $h_{n + 1} (x) \in B$ such that

$J^{*} (M_{n}, h_{n + 1}) = inf_{h \in B} J^{*} (M_{n}, h),$

where

$M_{n} = H_{n} {(w_{n})}_{x x},$

$J^{*} (M, h) = \frac{1}{2} \int_{Ω} \frac{M^{2}}{H (x)} d x .$
Set $n : = n + 1$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for

c_{0} = 0.65

E = 210 10^{7}

b = 0.1 m

P (x) = 36 10^{2} N

h_{m i n} = 0.072 m

and

h_{m a x} = 0.18 m .

We have also defined

h (x) = t (x) h_{m a x},

where

0.4 \leq t (x) \leq 1, a . e . in Ω .

For the optimal solution

w = w (x)

, please see Figure 20.

For a corresponding optimal solution

t = t (x)

, please see Figure 21.

Remark 30.1.

For such a simply-supported beam model, for the numerical solution of equation

{(H (x) w_{x x})}_{x x} = P,

with the boundary conditions

w (0) = w (1) = w^{''} (0) = w^{''} (1) = 0

firstly we have solved the equation

v_{x x} - P = 0

with the boundary conditions

v (0) = v (1) = 0 .

Subsequently, we have solved the equation

H (x) w_{x x} = v

with the boundary conditions

w (0) = w (1) = 0 .

Here we present the software developed in MAT-LAB.

******************

1.: clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210* $10^{7}$ ;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0* $10^{2}$ ;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)* $d^{2}$ +z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ and $(k < 10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5* $L {(i, 1)}^{3} * h o^{3}$ /12*e1;

f1(i,1)=v(i,1)/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ +z1(i-1));

end;

w(m8,1)=0;

for i=1:m8-1

w(m8-i,1)=m70(m8-i)*w(m8-i+1,1)+z1(m8-i);

end;

d2wo(1,1)=(-2*w(1,1)+w(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2wo(i,1)=(w(i+1,1)-2*w(i,1)+w(i-1,1))/ $d^{2}$ ;

end;

k9=1;

b14=1.0;

while $(b 14 > 10^{- 4}) and (k 9 < 120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

***************

With the auxiliary function "beamNov2023":

********************

1.: function S=beamNov2023(x)

global m8 d d2wo H e1 ho h1 xo b5

S=0;

for i=1:m8-1

S=S+1/ $(x {(i, 1)}^{3}) / h o^{3}$ /b5/e1*(H(i,1)* ${d 2 w o (i, 1))}^{2}$ *12;

end;

*****************************

We develop numerical results also for

V = W_{0}^{2, 2} (Ω) = {w \in W^{2, 2} (Ω) such that w (0) = w (1) = w^{'} (0) = w^{'} (1) = 0} .

Such boundary conditions corresponds to bi-clamped beam. The remaining data is equal to the previous example

For the optimal solution

w = w (x)

, please see Figure 22.

For a corresponding optimal solution

t = t (x)

, please see Figure 23.

Remark 30.2.

For such a bi-clamped beam model, for the numerical solution of equation

{(H (x) w_{x x})}_{x x} = P,

with the boundary conditions

w (0) = w (1) = w^{'} (0) = w^{'} (1) = 0,

firstly we have solved the equation

v_{x x} - P = 0

with the boundary conditions

v (0) = v (1) = 0 .

Subsequently, we solved the equation

H (x) w_{x x} = v + a x + b

with the boundary conditions

w (0) = w (1) = 0,

obtaining

a, b \in R

such that the boundary conditions

w^{'} (0) = w^{'} (1) = 0

are also satisfied.

Here we present the software developed in MAT-LAB.

*************************

1.: clear all

global m8 d d2wo H e1 ho h1 xo b5

m8=100;

d=1.0/m8;

b5=0.1;

e1=210* $10^{7}$ ;

ho=0.18;

A=zeros(m8-1,m8-1);

for i=1:m8-1

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.4*ones(m8-1,1);

ub=ones(m8-1,1);

b=zeros(m8-1,1);

b(1,1)=0.65*(m8-1);

for i=1:m8

f(i,1)=1.0;

L(i,1)=1/2;

P(i,1)=36.0* $10^{2}$ ;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)=1/m50(i)*(-P(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)*(-P(i,1)* $d^{2}$ +z(i-1));

end;

v(m8,1)=0;

for i=1:m8-1

v(m8-i,1)=m50(m8-i)*v(m8-i+1,1)+z(m8-i);

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4}) and (k < 10)$

k

k=k+1;

for i=1:m8-1

H(i,1)=b5* $L {(i, 1)}^{3} * h o^{3}$ /12*e1;

f1(i,1)=v(i,1)/H(i,1);

f2(i,1)=i*d/H(i,1);

f3(i,1)=1/H(i,1);

end;

i=1;

m12=2;

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ );

z2(i)=m70(i)*(-f2(i,1)* $d^{2}$ );

z3(i)=m70(i)*(-f3(i,1)* $d^{2}$ );

for i=2:m8-1

m12=2-m70(i-1);

m70(i)=1/m12;

z1(i)=m70(i)*(-f1(i,1)* $d^{2}$ +z1(i-1));

z2(i)=m70(i)*(-f2(i,1)* $d^{2}$ +z2(i-1));

z3(i)=m70(i)*(-f3(i,1)* $d^{2}$ +z3(i-1));

end;

w1(m8,1)=0;

w2(m8,1)=0;

w3(m8,1)=0;

for i=1:m8-1

w1(m8-i,1)=m70(m8-i)*w1(m8-i+1,1)+z1(m8-i);

w2(m8-i,1)=m70(m8-i)*w2(m8-i+1,1)+z2(m8-i);

w3(m8-i,1)=m70(m8-i)*w3(m8-i+1,1)+z3(m8-i);

end;

m3(1,1)=w2(1,1);

m3(1,2)=w3(1,1);

m3(2,1)=w2(m8-1,1);

m3(2,2)=w3(m8-1,1);

h3(1,1)=-w1(1,1);

h3(2,1)=-w1(m8-1,1);

h5(:,1)=inv(m3)*h3;

for i=1:m8

wo(i,1)=w1(i,1)+h5(1,1)*w2(i,1)+h5(2,1)*w3(i,1);

end;

d2wo(1,1)=(-2*wo(1,1)+wo(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2wo(i,1)=(wo(i+1,1)-2*wo(i,1)+wo(i-1,1))/ $d^{2}$ ;

end;

k9=1;

b14=1.0;

while $(b 14 > 10^{- 4}) and (k 9 < 120)$

k9

k9=k9+1;

X=fmincon(’beamNov2023’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(abs(xo-x3))

x3=xo;

for i=1:m8-1

L(i,1)=xo(i,1);

end;

end;

*****************************

Remark 30.3.

About the numerical results obtained for these two beam models, a final word of caution is necessary.

31. Applications to the optimal shape design for a plate model

In this section, we present a numerical procedure for the shape optimization concerning a thin plate model.

Let

Ω = [0, 1] \times [0, 1] \subset R^{2}

corresponds to the middle surface of a thin plate with a variable thickness

h (x, y)

Define now

V = {w \in W^{2, 2} (Ω) : w = 0 on \partial Ω},

which corresponds to a simply supported plate.

Consider the problem of minimizing in

V \times B

the functional

J (w, h) = \frac{1}{2} \int_{Ω} H (x, y) {(\nabla^{2} w (x, y))}^{2} d x

subject to

\nabla^{2} [(H (x, y) \nabla^{2} w (x, y))] - P (x, y) = 0, in Ω,

where

H (x, y) = \frac{h {(x, y)}^{3}}{12} E / (1 - w_{5}^{2}),

h = h (x, y)

is variable plate thickness, E is the young elasticity model and

w_{5} = 0.3

Also, we define

B = \{h : Ω \to R measurable : h_{m i n} \leq h (x, y) \leq h_{m a x} and \int_{Ω} h (x, y) \leq c_{0} h_{m a x}\},

where

0 < c_{0} < 1

and

C^{*} = {w \in V : \nabla^{2} [H (x, y) \nabla^{2} w (x, y))] - P (x, y) = 0, in Ω} .

Observe that

\begin{matrix} inf_{(w, h) \in C^{*} \times B} J (w, h) \\ = & inf_{h \in B} \{inf_{w \in C^{*}} J (w, h)\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{\frac{1}{2} \int_{Ω} H (x, y) {[\nabla^{2} w (x, y)]}^{2} d x - {〈 \hat{w}, \nabla^{2} [H (x, y) \nabla^{2} w (x, y)] - P (x, y) 〉}_{L^{2}}\}\}\} \\ = & inf_{h \in B} \{sup_{\hat{w} \in V} \{- \frac{1}{2} \int_{Ω} H (x, y) {[\nabla^{2} \hat{w} (x, y)]}^{2} d x + {〈 \hat{w}, P 〉}_{L^{2}}\}\} \\ = & inf_{h \in B} \{inf_{\tilde{M} \in D^{*}} \{\frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x\}\} . \end{matrix}

(186)

where

D^{*} = {\tilde{M} \in Y^{*} \nabla^{2} \tilde{M} - P = 0, in Ω, and \tilde{M} = 0, on Ω} .

Summarizing, we have got

inf_{(w, h) \in C^{*} \times B} J (w, h) = inf_{(\tilde{M}, h) \in D^{*} \times B} \{\frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x\} .

In order to obtain numerical results, we suggest the following primal dual procedure:

Set $n = 1$ and

$h_{n} (x) = c_{0} h_{m a x} .$
Calculate $w_{n} \in V$ solution of equation

$\nabla^{2} (H_{n} (x, y) \nabla^{2} w_{n} (x, y)) = P (x, y),$

where

$H_{n} (x, y) = \frac{E h_{n} {(x)}^{3}}{12 (1 - w_{5}^{2})} .$
Calculate $h_{n + 1} \in B$ such that

$J^{*} ({\tilde{M}}_{n}, h_{n + 1}) = inf_{h \in B} J^{*} ({\tilde{M}}_{n}, h),$

where

${\tilde{M}}_{n} = H_{n} (x, y) \nabla^{2} w_{n},$

$J^{*} (\tilde{M}, h) = \frac{1}{2} \int_{Ω} \frac{{\tilde{M}}^{2}}{H (x, y)} d x .$
Set $n : = n + 1$ and go to step 2 until an appropriate convergence criterion is satisfied.

We have developed numerical results for

c_{0} = 0.75

E = 200 10^{5}

P (x, y) = 2 10^{2} N

h_{m i n} = 0.45 * (0.12) m

and

h_{m a x} = 0.12 m .

We have also defined

h (x, y) = t (x, y) h_{m a x},

where

0.45 \leq t (x, y) \leq 1, a . e . in Ω .

For the optimal solution

w = w (x, y)

, please see Figure 24.

For a corresponding optimal solution

t = t (x, y)

, please see Figure 22.

Remark 31.1.

For such a simply-supported plate model, for the numerical solution of equation

\nabla^{2} [H (x, y) \nabla^{2} w (x, y)] = P,

with the boundary conditions

w = 0 on \partial Ω,

firstly we have solved the equation

\nabla^{2} v - P = 0

with the boundary conditions

v = 0 on \partial Ω .

Subsequently, we have solved the equation

H (x, y) \nabla^{2} w (x, y) = v (x, y)

with the boundary conditions

w = 0 on \partial Ω .

Here we present the software developed in MAT-LAB.

*********************

1.: clear all

global m8 d d2xwo d2ywo H e1 ho xo b5

m8=40;

d=1.0/m8;

w5=0.3;

e1=200* $10^{5}$ / $(1 - w 5^{2})$ ;

ho=0.12;

A=zeros( ${(m 8 - 1)}^{2}, {(m 8 - 1)}^{2}$ );

for i=1: ${(m 8 - 1)}^{2}$

A(1,i)=1.0;

xo(i,1)=0.55;

x3(i,1)=0.55;

end;

lb=0.45*ones( ${(m 8 - 1)}^{2}$ ,1);

ub=ones( ${(m 8 - 1)}^{2}$ ,1);

b=zeros( ${(m 8 - 1)}^{2}$ ,1);

b(1,1)=0.75* ${(m 8 - 1)}^{2};$

for i=1:(m8-1)

for j=1:m8-1

f(i,j,1)=1.0;

L(i,j,1)=1/2;

P(i,j,1)=2* $10^{2}$ ; end;

end;

for i=1:m8

wo(:,i)=0.001*ones(m8-1,1);

end;

m2=zeros(m8-1,m8-1);

for i=2:m8-2

m2(i,i)=-2.0;

m2(i,i-1)=1.0;

m2(i,i+1)=1.0;

end;

m2(1,1)=-2.0;

m2(1,2)=1.0;

m2(m8-1,m8-1)=-2.0;

m2(m8-1,m8-2)=1.0;

Id=eye(m8-1);

i=1;

m12=2*Id-m2* $d^{2} / d^{2}$ ; m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)* $d^{2}$ );

for i=2:m8-1

m12=2*Id-m2* $d^{2} / d^{2}$ -m50(:,:,i-1);

m50(:,:,i)=inv(m12);

z(:,i)=m50(:,:,i)*(-P(:,i,1)* $d^{2}$ +z(:,i-1));

end; v(:,m8)=zeros(m8-1,1);

for i=1:m8-1

v(:,m8-i)=m50(:,:,m8-i)*v(:,m8-i+1)+z(:,m8-i);

end;

k=1;

b12=1.0;

while ( $b 12 > 10^{- 4}$ ) and ( $k < 12$ )

k

k=k+1;

for i=1:m8-1

for j=1:m8-1

H(j,i,1)= $L {(j, i, 1)}^{3} * h o^{3}$ /12*e1;

f1(j,i,1)=v(j,i)/H(j,i,1);

end;

end;

i=1;

m12=2*Id-m2* $d^{2} / d^{2}$ ;

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)* $d^{2}$ );

for i=2:m8-1

m12=2*Id-m2* $d^{2} / d^{2}$ -m70(:,:,i-1);

m70(:,:,i)=inv(m12);

z1(:,i)=m70(:,:,i)*(-f1(:,i,1)* $d^{2}$ +z1(:,i-1));

end;

w(:,m8)=zeros(m8-1,1);

for i=1:m8-1

w(:,m8-i)=m70(:,:,m8-i)*w(:,m8-i+1)+z1(:,m8-i);

end;

d2xwo(:,1)=(-2*w(:,1)+w(:,2))/ $d^{2}$ ;

for i=2:m8-1

d2xwo(:,i)=(w(:,i+1)-2*w(:,i)+w(:,i-1))/ $d^{2}$ ;

end;

for i=1:m8-1

d2ywo(:,i)=m2*w(:,i)/ $d^{2}$ ;

end;

k9=1; b14=1.0;

while ( $b 14 > 10^{- 4}$ ) and ( $k 9 < 30$ )

k9

k9=k9+1;

X=fmincon(’beamNov2023A3’,xo,A,b, $[], []$ ,lb,ub);

b14=max(abs(xo-X))

xo=X;

end;

b12=max(max(abs(w-wo)))

wo=w;

x3=xo;

for i=1:m8-1

for j=1:m8-1

L(j,i,1)=xo((i-1)*(m8-1)+j,1);

end;

end;

end;

for i=1:m8-1

x8(i,1)=i*d;

end;

mesh(x8,x8,L);

*********************

With the auxiliary function "beamNov2023A3’, where

****************************

1.: function S=beamNov2023A3(x)

global m8 d d2xwo d2ywo H e1 ho xo b5

S=0;

for i=1:m8-1

for j=1:m8-1

x1(j,i)=x((m8-1)*(i-1)+j,1);

end;

end;

for i=1:m8-1

for j=1:m8-1

S=S+ $1 / ({(x 1 (j, i))}^{3}) / h o^{3} / e 1 * {(H (j, i, 1))}^{2} * {(d 2 x w o (j, i) + d 2 y w o (j, i))}^{2} * 12$ ;

end;

end;

********************************

Remark 31.2.

About the numerical results obtained for this plate model, a final word of caution is necessary.

32. A note on the first Maxwell equation of electromagnetism

Let

Ω_{1} \subset R^{3}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω_{1}

Suppose

E : Ω_{1} \to R^{3}

is an electric field of

C^{1}

class in

Ω .

Let

Ω \subset Ω_{1}

be also an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

S = \partial Ω

Observe that there exists a scalar field

V : Ω \to R

such that

\nabla^{2} V = div E, in Ω,

and

\nabla V \cdot n = 0, on S = \partial Ω .

Here

n

denotes the normal outward field to S.

Observe also that

\nabla^{2} V = div \nabla V = div E,

so that defining

h = \nabla V - E,

we have that

div h = 0, in Ω .

Hence, from such results and the divergence Theorem, we get

\begin{matrix} \int_{S} E \cdot n d S & = & \int_{S} (\nabla V) \cdot n d S - \int_{S} h \cdot n d S \\ = & - \int_{Ω} div h d V = 0 . \end{matrix}

(187)

Summarizing, we have got

\int_{S} E \cdot n d S = 0 .

Consider now a charge

q_{0}

localized at the center of a sphere

Ω_{2}

of radius

R > 0

and boundary

S_{2} = \partial Ω_{2} .

The electric field on the sphere surface generated by

q_{0}

is given by

E_{2} = \frac{1}{4 π ε_{0}} \frac{q_{0}}{R^{2}} n_{2},

where

n_{2}

is the normal outward field to

S_{2}

Clearly

\int_{S_{2}} E_{2} \cdot n_{2} d S_{2} = \frac{1}{4 π ε_{0}} \frac{q_{0}}{R^{2}} (4 π R^{2}) = \frac{q_{0}}{ε_{0}} .

Consider again the set

Ω

but now with a charge

q_{0}

localized at a point

x

inside the interior of

Ω

, which is denoted by

Ω^{0}

At first the electric field

E

generated by

q_{0}

is not of

C^{1}

class on

Ω .

However, there exists

R > 0

such that

B_{R} (x) \subset Ω = Ω^{0} .

Define

Ω_{3} = Ω ∖ B_{R} (x)

Therefore,

E

is of

C^{1}

class on

Ω_{3} .

Denoting the boundary of

Ω_{3}

S_{3}

, from the previous results, we may infer that

\int_{S_{3}} E \cdot n d S_{3} = 0,

so that

\begin{matrix} \int_{S_{3}} E \cdot n d S_{3} & = & \int_{S} E \cdot n d S - \int_{\partial B_{R} (x)} E \cdot n d S_{2} \\ = & \int_{S} E \cdot n d S - \frac{q_{0}}{ε_{0}} \\ = & 0 . \end{matrix}

(188)

Therefore, we have got

\int_{S} E \cdot n d S = \frac{q_{0}}{ε_{0}} .

Assume now on

Ω

we have a density of charges

ρ (x)

For a small volume

Δ V

consider a punctual charge

q_{0}

localized in

x \in Ω

such that

q_{0} \approx ρ (x) Δ V .

Denoting by

Δ E

the electric field generated by

q_{0}

, from the previous results we may infer that

\int_{S} Δ E \cdot n d S = \frac{q_{0}}{ε_{0}} \approx \frac{ρ (x) Δ V}{ε_{0}} .

Such an equation in its differential form, stands for:

\int_{S} d E \cdot n d S = \frac{ρ (x) d V}{ε_{0}} .

Integrating in

Ω

we may obtain

\begin{matrix} \int_{S} E \cdot n d S & = & \int_{S} \int_{Ω} d E \cdot n d V d S \\ = & \int_{Ω} \frac{ρ (x)}{ε_{0}} d V, \end{matrix}

(189)

so that

\int_{S} E \cdot n d S = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

From this and the Divergence Theorem, we have

\int_{S} E \cdot n d S = \int_{Ω} div E d V = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

Summarizing, we have got

\int_{Ω} div E d V = \int_{Ω} \frac{ρ (x)}{ε_{0}} d V .

This is the integral form of the first Maxwell equation of electromagnetism.

For this last equation, the set

Ω \subset Ω_{1}

is rather arbitrary so that for

Ω

as a ball of small radius

r > 0

with center at a point

x \in Ω_{1}

, from the Mean Value Theorem fot integrals and letting

r \to 0^{+}

, we obtain

div E = \frac{ρ}{ε_{0}}, in Ω_{1} .

This last equation stands for the differential form of the first Maxwell equation of electromagnetism.

Remark 32.1.

Summarizing, in this section we have formally obtained a mathematical deduction of the first Maxwell equation of electromagnetism.

33. A note on relaxation for a general model in the vectorial calculus of variations

Let

Ω \subset R^{n}

be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider a function

g : R^{N \times n} \to R

twice differentiable and such that

g (y) \to + \infty, as | y | \to + \infty .

Define a functional

G : V \to R

G (\nabla u) = \frac{1}{2} \int_{Ω} g (\nabla u) d x,

where

V = {W^{1, 2} (Ω; R^{N}) : u = u_{0} on \partial Ω} .

Moreover, for

f \in L^{2} (Ω; R^{N})

, define also

J (u) = G (\nabla u) - {〈 u, f 〉}_{L^{2}} .

We assume there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Observe that from the convex analysis basic theory, we have that

\begin{matrix} α & = & inf_{u \in^{V}} J (u) \\ = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {{(G \circ \nabla)}^{* *} (u) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

(190)

On the other hand

\begin{matrix} {(G \circ \nabla)}^{* *} (u) & \leq & H (u) \\ \equiv & inf_{(λ, (v, w)) \in [0, 1] \times B (u, λ)} {λ G (\nabla w) + (1 - λ) G (\nabla v)} \\ \leq & G (\nabla u), \end{matrix}

(191)

where

B (u, λ) = {(v, w) \in V : λ w + (1 - λ) v = u} .

From such results, we may infer that

inf_{u \in V} J^{* *} (u) = inf_{u \in V} {H (u) - {〈 u, f 〉}_{L^{2}}} = inf_{u \in V} J (u) .

Furthermore, observe that

λ \nabla w + (1 - λ) \nabla v = \nabla u,

so that

\begin{matrix} \nabla v & = & \nabla u + λ (\nabla v - \nabla w) \\ = & \nabla u + λ \nabla ϕ, \end{matrix}

(192)

where

ϕ = v - w \in W_{0}^{1, 2} (Ω; R^{N})

so that

\nabla ϕ = \nabla v - \nabla w,

and

\nabla w = \nabla v - \nabla ϕ .

Therefore,

\nabla w = \nabla v - \nabla ϕ = \nabla u + λ \nabla ϕ - \nabla ϕ = \nabla u - (1 - λ) \nabla ϕ .

Replacing such results into the expression of H, we have

H (u) = inf_{(λ, ϕ) \in [0, 1] \times V_{0}} {λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ)},

where

V_{0} = W_{0}^{1, 2} (Ω; R^{N}) .

Joining the pieces, we have got

\begin{matrix} inf_{u \in V} J (u) & = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} {H (u) - {〈 u, f 〉}_{L^{2}}} \\ = & inf_{(λ, ϕ, u) \in [0, 1] \times V_{0} \times V} {λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ) - {〈 u, f 〉}_{L^{2}}} . \end{matrix}

This last functional corresponds to a relaxation for the original non-convex functional.

The note is complete.

33.1. Some related numerical results

In this subsection we present numerical results for an one-dimensional model and related relaxed formulation.

For

Ω = [0, 1] \subset R

, consider the functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x,

V = {u \in W^{1, 2} (Ω) : u (0) = 0 and u (1) = 1 / 2},

f \in Y = Y^{*} = L^{2} (Ω) .

Based on the results of the previous section, denoting

V_{0} = W_{0}^{1, 2} (Ω),

we define the following relaxed functional

J_{1} : [0, 1] \times V \times V_{0} \to R,

where

J_{1} (λ, u, ϕ) = \frac{λ}{2} \int_{Ω} {({(u^{'} - (1 - λ) ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1 - λ}{2} \int_{Ω} {({(u^{'} + λ ϕ^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x .

Indeed, we have developed an algorithm for minimizing the following regularized functional

J_{2} : [0, 1] \times V \times V_{0} \to R,

where

J_{2} (λ, u, ϕ) = J_{1} (λ, u, ϕ) + \frac{ε_{3}}{2} \int_{Ω} {(u^{''})}^{2} d x,

for a small parameter

ε_{3} > 0

For the case in which

f (x) = sin (π x) / 2

, for the optimal solution u, please see Figure 26.

For the case in which

f (x) = cos (π x) / 2

, for the optimal solution u, please see Figure 27.

For the case in which

f (x) = 0

, for the optimal solution u, please see Figure 28.

We highlight to obtain the solution for this last case which

f = 0

is harder. A good solution was possible only using

x_{0} = 0

as the initial solution concerning the iterative process.

Here we present the software in MAT-LAB developed.

*****************

1.: clear all

global m8 d u e3

m8=100;

d=1/m8;

e3=0.0005;

for i=1:2*m8+1

xo(i,1)=0.36;

end;

b12=1.0;

k=1;

while $(b 12 > 10^{- 7}) and (k < 60)$

k

k=k+1;

X=fminunc(’funDecember2023’,xo);

b12=max(abs(xo-X))

xo=X;

u(m8/2)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

***********************

With the main function "funDecember2023"

*********************

1.: function S=funDecember2023(x)

global m8 d u e3

for i=1:m8

u(i,1)=x(i,1);

v(i,1)=x(i+m8,1);

yo(i,1)=sin(pi*i*d)/2;

end;

L=(1+sin(x(2*m8+1,1)))/2;

u(m8,1)=1/2;

v(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

for i=2:m8

du(i,1)=(u(i,1)-u(i-1,1))/d;

dv(i,1)=(v(i,1)-v(i-1,1))/d;

end;

d2u(1,1)=(-2*u(1,1)+u(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2u(i,1)=(u(i-1,1)-2*u(i,1)+u(i+1,1))/ $d^{2}$ ;

end;

S=0;

for i=1:m8

S=S+ $1 / 2 * L * {({(d u (i, 1) - (1 - L) * d v (i, 1))}^{2} - 1)}^{2}$ ;

S=S+ $1 / 2 * (1 - L) * {({(d u (i, 1) + L * d v (i, 1))}^{2} - 1)}^{2}$ ;

S=S+ ${(u (i, 1) - y o (i, 1))}^{2}$ ;

end;

for i=1:m8-1

S=S+e3* $d 2 u {(i, 1)}^{2}$ ;

end;

*******************

33.2. A related duality principle and concerning convex dual formulation

With the notation and statements of the previous sections in mind, consider the functionals

J : V \to R

and

J_{3} : [0, 1] \times V \times V_{0} \to R

where

J (u) = G (\nabla u) + \frac{1}{2} \int_{Ω} u \cdot u d x - {〈 u, f 〉}_{L^{2}},

and

\begin{matrix} J_{3} (λ, u, ϕ) & = & λ G (\nabla u - (1 - λ) \nabla ϕ) + (1 - λ) G (\nabla u + λ \nabla ϕ) \\ + \frac{λ}{2} \int_{Ω} (u - (1 - λ) ϕ) \cdot (u - (1 - λ) ϕ) d x \\ + \frac{(1 - λ)}{2} \int_{Ω} (u + λ ϕ) \cdot (u + λ ϕ) d x \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}} . \end{matrix}

(193)

Here we have denoted

V = {u \in W^{1, 2} (Ω; R^{N}) : u = u_{0} on \partial Ω = S},

V_{0} = W_{0}^{1, 2} (Ω; R^{N}),

Y = Y^{*} = L^{2} (Ω; R^{N \times n})

and

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{N}) .

Observe that

J^{* *} (u) \leq min_{(λ, ϕ) \in [0, 1] \times V_{0}} J_{3} (λ, u, ϕ) .

Moreover,

\begin{matrix} J_{3} (λ, u, ϕ) & = & - {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + λ G (\nabla u - (1 - λ) \nabla ϕ) \\ - {〈 \nabla u - (1 - λ) \nabla ϕ, v_{2}^{*} 〉}_{L^{2}} + (1 - λ) G (\nabla u + λ \nabla ϕ) \\ - {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} (u - (1 - λ) ϕ) \cdot (u - (1 - λ) ϕ) d x \\ - {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} (u + λ ϕ) \cdot (u + λ ϕ) d x \\ + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} \\ + {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}} . \end{matrix}

(194)

Therefore,

\begin{matrix} J_{3} (λ, u, ϕ) & \geq & inf_{v_{1} \in Y} {- {〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + λ G (v_{1})} \\ + inf_{v_{2} \in Y} {- {〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} + (1 - λ) G (v_{2})} \\ + inf_{v_{3} \in Y_{1}} \{- {〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} (v_{3}) \cdot (v_{3}) d x\} \\ + inf_{v_{4} \in Y_{1}} \{- {〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} (v_{4}) \cdot (v_{4}) d x\} \\ + inf_{(u, ϕ) \in V \times V_{0}} {{〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} + {〈 \nabla u - (1 - λ) \nabla ϕ, v_{1}^{*} 〉}_{L^{2}} \\ + {〈 u - (1 - λ) ϕ, v_{3}^{*} 〉}_{L^{2}} + {〈 u + λ ϕ, v_{4}^{*} 〉}_{L^{2}} \\ - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}} - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}}} \\ = & - λ G^{*} (\frac{v_{1}^{*}}{λ}) - (1 - λ) G^{*} (\frac{v_{2}^{*}}{(1 - λ)}) \\ - F_{3}^{*} (v_{3}^{*}, λ) - F_{4}^{*} (v_{4}^{*}, λ) \\ + \int_{S} {(v_{1}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S + \int_{S} {(v_{2}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S, \\ \forall λ \in (0, 1), u \in V, ϕ \in V_{0}, v^{*} \in A^{*}, \end{matrix}

(195)

where

G^{*} (v^{*}) = sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G (v)},

\begin{matrix} F_{3}^{*} (v_{3}^{*}, λ) & = & sup_{v_{3} \in Y_{1}} \{{〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} - \frac{λ}{2} \int_{Ω} v_{3} \cdot v_{3} d x\} \\ = & \frac{1}{2 λ} \int_{Ω} v_{3}^{*} \cdot v_{3}^{*} d x, \end{matrix}

(196)

\begin{matrix} F_{4}^{*} (v_{4}^{*}, λ) & = & sup_{v_{4} \in Y_{1}} \{{〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} - \frac{(1 - λ)}{2} \int_{Ω} v_{4} \cdot v_{4} d x\} \\ = & \frac{1}{2 (1 - λ)} \int_{Ω} v_{4}^{*} \cdot v_{4}^{*} d x . \end{matrix}

(197)

Furthermore,

A^{*} = A_{1}^{*} \cap A_{2}^{*}

where

A_{1}^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y^{*}]}^{2} \times {[Y_{1}^{*}]}^{2} : - div {(v_{1}^{*})}_{i} - div {(v_{2}^{*})}_{i} + {(v_{3}^{*})}_{i} + {(v_{4}^{*})}_{i} - f_{i} = 0, in Ω},

and

\begin{matrix} A_{2}^{*} & = & {v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y^{*}]}^{2} \times {[Y_{1}^{*}]}^{2} : \\ - (- 1 - λ) div {(v_{1}^{*})}_{i} - λ div {(v_{2}^{*})}_{i} + (- 1 - λ) {(v_{3}^{*})}_{i} + λ {(v_{4}^{*})}_{i} = 0, in Ω} . \end{matrix}

(198)

Summarizing, we have got

\begin{matrix} inf_{(λ, u ϕ) \in (0, 1) \times V \times V_{0}} J_{3} (λ, u, ϕ) \\ \geq & sup_{v^{*} \in A^{*}} \{inf_{λ \in (0, 1)} \{- λ G^{*} (\frac{v_{1}^{*}}{λ}) - (1 - λ) G^{*} (\frac{v_{2}^{*}}{(1 - λ)}) \\ - F_{3}^{*} (v_{3}^{*}, λ) - F_{4}^{*} (v_{4}^{*}, λ) + \int_{\partial Ω} {(v_{1}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S + \int_{\partial Ω} {(v_{2}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S\}\} . \end{matrix}

(199)

Remark 33.1.

We highlight this last dual function in

v^{*}

is convex (in fact concave) on the convex set

A^{*}

34. A general convex primal dual formulation for an originally non-convex primal one

Let

Ω \subset R^{3}

be an open bounded and connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

Consider the functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(200)

where

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

Y = Y^{*} = L^{2} (Ω) .

Define

F_{1} : V \to R

and

F_{2} : V \times Y^{*} \to R

F_{1} (u) = \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and

\begin{matrix} F_{2} (u, v_{0}^{*}) & = & - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{K}{2} \int_{Ω} u^{2} d x \\ + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(201)

Define also

F_{1}^{*} : Y^{*} \to R

and

F_{2}^{*} : Y^{*} \times Y^{*} \to R

\begin{matrix} F^{*} (v_{1}^{*}) & = & sup_{u \in V} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{1} (u)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} + f)}^{2}}{- γ \nabla^{2} + K} d x, \end{matrix}

(202)

and

\begin{matrix} F_{2}^{*} (v_{1}^{*}, v_{0}^{*}) & = & sup_{u \in V} {- {〈 u, v_{1}^{*} 〉}_{L^{2}} - F_{2} (u, v_{0}^{*})} \\ = & - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{2 v_{0}^{*} - K} d x \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(203)

v_{0}^{*} \in B^{*}

, where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2},

for some appropriate

K > 0

to be specified.

At this point we define

V_{1} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}},

D^{*} = {v_{1}^{*} \in Y^{*} : ∥ v_{1}^{*} ∥_{\infty} \leq 5 / 4 K},

for appropriate

K_{3} > 0

to be specified, and

J_{1}^{*} : D^{*} \times B^{*} \to R

J_{1}^{*} (v_{1}^{*}, v_{0}^{*}) = - F_{1}^{*} (v_{1}^{*}) + F_{2}^{*} (v_{1}^{*}, v_{0}^{*}) .

Moreover, we define

J_{2}^{*} : D^{*} \times B^{*} \times V \to R

J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*}) = J_{1}^{*} (v_{1}^{*}, v_{0}^{*}) + \frac{K_{1}}{2} ∥ v_{1}^{*} - (- γ \nabla^{2} + K) {u ∥}_{2}^{2} + \frac{1}{5 K_{3}^{2} α} {∥ v_{1}^{*} - (- 2 v_{0}^{*} + K) u ∥}_{2}^{2} .

Observe that

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} = O (1 / K^{2}) + K_{1} + \frac{1}{5 K_{3}^{2} α},

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} = {(- γ \nabla^{2} + K)}^{2} + \frac{1}{5 K_{3}^{2} α} {(- 2 v_{0}^{*} + K)}^{2},

and

\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}} = - (- γ \nabla^{2} + K) - \frac{1}{5 K_{3}^{2} α} (- 2 v_{0}^{*} + K) .

From such results, we may obtain

\begin{matrix} det {δ_{u, v_{1}^{*}}^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})} & = & \frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial {(v_{1}^{*})}^{2}} \frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u^{2}} - {(\frac{\partial^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{\partial u \partial v_{1}^{*}})}^{2} \\ = & \frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{5 α K_{3}^{2}} + O (\frac{K_{1}}{K^{2}}) . \end{matrix}

(204)

From such results, in particular for

K_{1} ≫ max {K, K_{3}, 1, α, β, γ, 1 / α, 1 / γ, 1 / β},

K ≫ max {K_{3}, 1, α, β, γ, 1 / α, 1 / γ, 1 / β},

and

K_{3} \approx 3,

we have that

J_{2}^{*}

is convex in

(u, v_{1}^{*})

and concave in

v_{0}^{*}

(V_{1} \times D^{*}) \times B^{*} .

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 34.1.

Let

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \in V_{1} \times D^{*} \times B^{*}

be such that

δ J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses,

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(205)

Proof.

The proof that

δ J (u_{0}) = 0

and

J (u_{0}) = J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} = J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}),

may be done similarly as in the previous sections and will not be repeated.

Furthermore, since

δ J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = 0,

and

J_{2}^{*}

is convex in

(u, v_{1}^{*})

and concave in

v_{0}^{*}

V_{1} \times D^{*} \times B^{*}

, we have

J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{1} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}),

and

J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{1} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*}) .

From such results and the Saddle Point Theorem we may infer that

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{1}{5 K_{3}^{2} α} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(u, v_{1}^{*}) \in V_{1} \times D^{*}} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(206)

The proof is complete.

□

35. Conclusion

In the first part of this article we have developed a relaxation proposal and duality principles suitable for a large class of models in physics and engineering.

In a second part we develop duality principles for the quasi-convex envelop of some vectorial models in the calculus of variations.

We highlight such dual variational formulations established are in general convex (in fact concave).

Data Availability Statement

Details on the software for numerical results avaialable upon request. e-mail: fabio.botelho@ufsc.br.

Conflicts of Interest

The author declares no conflict of interest concerning this article.

References

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Figure 1. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 1. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 2. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 2. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 3. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 3. Solution

u_{0} (x)

for the case

f (x) = 0

Figure 4. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 4. Solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 5. Density

t (x, y)

for the Case A.

Figure 5. Density

t (x, y)

for the Case A.

Figure 6. Density

t (x, y)

for the Case B.

Figure 6. Density

t (x, y)

for the Case B.

Figure 7. Solution

u (x) = v_{3}^{*} (x) / β

for the example 1.

Figure 7. Solution

u (x) = v_{3}^{*} (x) / β

for the example 1.

Figure 8. Solution

u (x) = v_{3}^{*} (x) / β

for the example 2.

Figure 8. Solution

u (x) = v_{3}^{*} (x) / β

for the example 2.

Figure 9. Solution

u_{0} (x)

for the example A.

Figure 9. Solution

u_{0} (x)

for the example A.

Figure 10. Solution

u_{0} (x)

for the example B.

Figure 10. Solution

u_{0} (x)

for the example B.

Figure 11. Solution

ϕ_{N} (x)

for the

ω = 1.8

Figure 11. Solution

ϕ_{N} (x)

for the

ω = 1.8

Figure 12. Solution

ϕ_{S} (x)

for the

ω = 1.8

Figure 12. Solution

ϕ_{S} (x)

for the

ω = 1.8

Figure 13. Solution

ϕ_{N} (x)

for the

ω = 1.0

Figure 13. Solution

ϕ_{N} (x)

for the

ω = 1.0

Figure 14. Solution

ϕ_{S} (x)

for the

ω = 1.0

Figure 14. Solution

ϕ_{S} (x)

for the

ω = 1.0

Figure 15. Solution

ϕ_{u} (x)

for

ω = 1

Figure 15. Solution

ϕ_{u} (x)

for

ω = 1

Figure 16. Solution

ϕ_{v} (x)

for

ω = 1

Figure 16. Solution

ϕ_{v} (x)

for

ω = 1

Figure 17. Solution

ϕ_{u} (x)

for

ω = 15

Figure 17. Solution

ϕ_{u} (x)

for

ω = 15

Figure 18. Solution

ϕ_{v} (x)

for

ω = 15

Figure 18. Solution

ϕ_{v} (x)

for

ω = 15

Figure 19. Solution

u (x)

for the example B.

Figure 19. Solution

u (x)

for the example B.

Figure 20. Optimal solution

w (x)

for a simply supported beam.

Figure 20. Optimal solution

w (x)

for a simply supported beam.

Figure 21. Optimal shape solution

t (x)

for a simply supported beam.

Figure 21. Optimal shape solution

t (x)

for a simply supported beam.

Figure 22. Optimal solution

w (x)

for a bi-clamped beam.

Figure 22. Optimal solution

w (x)

for a bi-clamped beam.

Figure 23. Optimal shape solution

t (x)

for a bi-clamped beam.

Figure 23. Optimal shape solution

t (x)

for a bi-clamped beam.

Figure 24. Optimal solution

w (x, y)

for a simply supported plate.

Figure 24. Optimal solution

w (x, y)

for a simply supported plate.

Figure 25. Optimal shape solution

t (x, y)

for a simply supported plate.

Figure 25. Optimal shape solution

t (x, y)

for a simply supported plate.

Figure 26. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 26. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 27. Optimal solution

u (x)

for the case

f (x) = cos (π x) / 2

Figure 27. Optimal solution

u (x)

for the case

f (x) = cos (π x) / 2

Figure 28. Optimal solution

u (x)

for the case

f (x) = 0

Figure 28. Optimal solution

u (x)

for the case

f (x) = 0

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