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

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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 Silva Botelho^*

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Submitted:

25 June 2024

Posted:

26 June 2024

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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

MSC: 49N15; 35A15; 49J40

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

(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

𝜕 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 𝜕 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

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

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

and

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

that is

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

and

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

so that

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

and

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

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

and

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

that is

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

and

$- {(v_{0}^{*})}_{n} + \frac{𝜕 F_{3} (u_{n + 1}, ϕ_{n + 1})}{𝜕 ϕ} = 0$
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.

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

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 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

𝜕 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 𝜕 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

𝜕 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{𝜕 G ({({\hat{v}}^{*})}^{'})}{𝜕 {(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 2.

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.

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

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

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

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

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

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

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

$i = 1;$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$x (i) = i * d;$

$e n d;$

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

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

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

𝜕 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{𝜕 w}{𝜕 n} = 0 on 𝜕 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{𝜕 J_{3} (u)}{𝜕 w} & = & \frac{𝜕 J (u)}{𝜕 w} + \frac{𝜕 J_{5} (u)}{𝜕 w} \\ \approx & \frac{𝜕 J (u)}{𝜕 w} + O (\pm 3.0), \end{matrix}

(22)

and

\begin{matrix} \frac{𝜕^{2} J_{3} (u)}{𝜕 w^{2}} & = & \frac{𝜕^{2} J (u)}{𝜕 w^{2}} + \frac{𝜕^{2} J_{5} (u)}{𝜕 w^{2}} \\ \approx & \frac{𝜕^{2} J (u)}{𝜕 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{𝜕^{2} J (u)}{𝜕 w^{2}},

which has increased of order

O (700 - 1000) .

Moreover the difference between the approximate and exact equation

\frac{𝜕 J (u)}{𝜕 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 𝜕 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 3.

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

𝜕 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 𝜕 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 𝜕 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_{𝜕 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_{𝜕 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_{𝜕 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

𝜕 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 𝜕 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 4.

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

𝜕 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{𝜕^{2} F (u_{x}, u_{y})}{𝜕 v_{1} 𝜕 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{𝜕 F_{1} (v_{1}, v_{2})}{𝜕 v_{1}},

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

v_{2}^{*} = \frac{𝜕 F_{2} (v_{1}, v_{2})}{𝜕 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{𝜕 F_{1}^{*} (v^{*})}{𝜕 v_{1}^{*}} + \frac{v_{1}^{*} + f}{ε} - \frac{𝜕 φ}{𝜕 y} = 0 .

(38)

From the variation of

J_{1}^{*}

v_{2}^{*}

we obtain

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

and

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

so that

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

and

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

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

\begin{matrix} - ε {(\frac{𝜕 F_{1}^{*} (v^{*})}{𝜕 v_{1}^{*}})}_{x} - ε {(\frac{𝜕 F_{1}^{*} (v^{*})}{𝜕 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{𝜕 F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{𝜕 v_{1}})}_{x} + {(\frac{𝜕 F_{1} (u_{x} - φ_{y}, u_{y} + φ_{x})}{𝜕 v_{2}})}_{y} + f = 0,

so that

{(\frac{𝜕 F (u_{x} - φ_{y}, u_{y} + φ_{x})}{𝜕 v_{1}})}_{x} + {(\frac{𝜕 F (u_{x} - φ_{y}, u_{y} + φ_{x})}{𝜕 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

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

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

and

\frac{𝜕^{2} J_{1}^{*} (u, v_{0}^{*})}{𝜕 u 𝜕 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{𝜕^{2} J_{1}^{*} (u, v_{0}^{*})}{𝜕 {(v_{0}^{*})}^{2}} \frac{𝜕^{2} J_{1}^{*} (u, v_{0}^{*})}{𝜕 u^{2}} - {(\frac{𝜕^{2} J_{1}^{*} (u, v_{0}^{*})}{𝜕 u 𝜕 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 5.

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 1.

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

𝜕 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 6.

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

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

and

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

so that

\begin{matrix} det \{\frac{𝜕^{2} J_{1}^{*} (u, v_{0}^{*}, v_{1}^{*})}{𝜕 u 𝜕 v_{1}^{*}}\} \\ = & \frac{𝜕^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 {(v_{1}^{*})}^{2}} \frac{𝜕^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u^{2}} - {(\frac{𝜕^{2} J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u 𝜕 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 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{𝜕^{2} J_{1}^{*} (u_{0}, {\hat{v}}_{1}^{*}, v_{0}^{*})}{𝜕 {(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

𝜕 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

𝜕 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 3.

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{𝜕 J_{1} (u_{0}, t_{0})}{𝜕 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

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

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

$J_{1} (u_{n}, t_{n + 1}) = inf_{t \in B} J_{1} (u_{n}, t) .$
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.

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

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:

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.
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

𝜕 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

𝜕 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{𝜕^{2} J_{1} (u)}{𝜕 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}

(67)

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

𝜕 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{𝜕 F_{1} (u)}{𝜕 u} = - γ \nabla^{2} u,

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

v_{3}^{*} = \frac{𝜕 F_{3} (u)}{𝜕 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 𝜕 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{𝜕^{2} J^{*} (v^{*})}{𝜕 {(v_{2}^{*})}^{2}} = 2;

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

\frac{𝜕^{2} J^{*} (v^{*})}{𝜕 v_{2}^{*} 𝜕 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{𝜕^{2} J^{*} (v^{*})}{𝜕 v_{2}^{*} 𝜕 v_{3}^{*}}) & = & \frac{𝜕^{2} J^{*} (v^{*})}{𝜕 {(v_{2}^{*})}^{2}} \frac{𝜕^{2} J^{*} (v^{*})}{𝜕 {(v_{3}^{*})}^{2}} - {(\frac{𝜕^{2} J^{*} (v^{*})}{𝜕 v_{2}^{*} 𝜕 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.

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.
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

𝜕 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{𝜕^{2} G (u)}{𝜕 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{𝜕 G (u)}{𝜕 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{𝜕 G (u)}{𝜕 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}

(78)

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}

(79)

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.

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.
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}

(81)

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 7.

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{𝜕 r_{N} (x, t)}{𝜕 t} \cdot \frac{𝜕 r_{N} (x, t)}{𝜕 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{𝜕 r_{D}}{𝜕 t} \cdot \frac{𝜕 r_{D}}{𝜕 t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{T} |}^{2} \frac{𝜕 r_{T}}{𝜕 t} \cdot \frac{𝜕 r_{T}}{𝜕 t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{𝜕 r_{H_{e}}}{𝜕 t} \cdot \frac{𝜕 r_{H_{e}}}{𝜕 t} d x d y \\ + \frac{1}{2} \int_{Ω} {| ϕ_{N} |}^{2} \frac{𝜕 r_{N}}{𝜕 t} \cdot \frac{𝜕 r_{N}}{𝜕 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.

$\int_{Ω} {| ϕ_{N}^{D} (x, y) |}^{2} d x = m_{N},$
$\int_{Ω} {| ϕ_{N_{1}}^{T} (x, y) |}^{2} d x = m_{N},$
$\int_{Ω} {| ϕ_{N_{2}}^{T} (x, y) |}^{2} d x = m_{N},$
$\int_{Ω} {| ϕ_{N_{1}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
$\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y) |}^{2} d x = m_{N},$
$\int_{Ω} {| ϕ_{p}^{D} (y) |}^{2} d y = λ_{1} c m_{p},$
$\int_{Ω} {| ϕ_{p}^{T} (y) |}^{2} d y = λ_{1} c m_{p},$
$\int_{Ω} {| ϕ_{2 P}^{H_{e}} (y) |}^{2} d y = λ_{2} (2 c m_{p}),$

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

We have also the following complementing constraints,

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

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 8.

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 9.

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

𝜕 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:

For the Deuterium field

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

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

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

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

$ϕ_{e} = ϕ_{e} (x, y, t) .$

Furthermore, we define also the related densities

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

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

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

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

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

𝜕 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

𝜕 Ω

, we assume there is a part

Ω_{1} \subset 𝜕 Ω

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

We assume also there is another part

𝜕 Ω_{2} \subset 𝜕 Ω

such that

𝜕 Ω_{1} \cap 𝜕 Ω_{2} = \emptyset

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

𝜕 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:

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
$m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
$m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
$\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

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

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{𝜕 r_{D}}{𝜕 t} \cdot \frac{𝜕 r_{D}}{𝜕 t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{T} |}^{2} \frac{𝜕 r_{T}}{𝜕 t} \cdot \frac{𝜕 r_{T}}{𝜕 t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{H_{e}} |}^{2} \frac{𝜕 r_{H_{e}}}{𝜕 t} \cdot \frac{𝜕 r_{H_{e}}}{𝜕 t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{N} |}^{2} \frac{𝜕 r_{N}}{𝜕 t} \cdot \frac{𝜕 r_{N}}{𝜕 t} d x d y d t \\ + \frac{1}{2} \int_{0}^{t_{f}} \int_{Ω} {| ϕ_{e} |}^{2} \frac{𝜕 r_{e}}{𝜕 t} \cdot \frac{𝜕 r_{e}}{𝜕 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{𝜕 r_{D}}{𝜕 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{𝜕 r_{T}}{𝜕 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{𝜕 r_{H_{e}}}{𝜕 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{𝜕 r_{e}}{𝜕 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:

The momentum conservation equation for the fluid motion

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

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

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,

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

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

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

and

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

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Mass conservation equation:

$\frac{𝜕 ρ}{𝜕 t} + div (ρ u) = 0 .$
Energy equation

$ρ \frac{D e}{D t} + \nabla_{x} ({\hat{E}}_{1}) \cdot u + {\hat{E}}_{2} + P (div u) = \frac{𝜕 Q}{𝜕 t} - div q,$

where we assume the Fourier law

$q = - K \nabla T,$

where $T = T (x, t)$ is the scalar field of temperature and Q is a standard heat function.

Also,

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

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where the densities ${\hat{E}}_{1}$ and ${\hat{E}}_{2}$ are defined through the expressions of $F (ϕ)$ and $F_{2}$ so that

$F (ϕ) = \int_{0}^{t_{f}} \int_{Ω} {\hat{E}}_{1} d x d t$

and

$F_{2} = \int_{0}^{t_{f}} \int_{Ω} {\hat{E}}_{2} d x d t .$

Here we recall that since $r_{D}$ is highly oscillating in t we approximately have

$u \cdot r_{D} \approx 0$

in a weak or measure sense. The same remark is valid for the other internal velocity fields.

Moreover,

$\frac{D e}{D t} = \frac{𝜕 e}{𝜕 t} + u_{j} \frac{𝜕 e}{𝜕 x_{j}} .$
$P = F_{7} (ρ, T),$

for an appropriate scalar function $F_{7}$ .
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)

${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$

(g)

${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$

(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)

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$

(b)

$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$

(c)

$(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$

(d)

$(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$

(e)

${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$

(f)

$m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$
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)

$\int_{Ω} {| ϕ_{N_{2}}^{H_{e}} (x, y, t) |}^{2} d x = m_{N} .$
For the induced electric field, we must have

$\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} {| ϕ_{p}^{D} |}^{2} (u + \frac{𝜕 r_{D}}{𝜕 t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{𝜕 r_{T}}{𝜕 t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{𝜕 r_{H_{e}}}{𝜕 t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{𝜕 r_{e} (x, y, t)}{𝜕 t} d x)) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{𝜕}{𝜕 t} (curl A - B_{0}) = 0, \end{matrix}$

(96)

where ${\hat{K}}_{p}$ and ${\hat{K}}_{e}$ are appropriate real constants related to the respective charges.
A Maxwell equation:

$div B = 0,$

where

$B = B_{0} - curl A .$
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

$F (ϕ) = \int_{0}^{t_{f}} \int_{Ω} f_{5} (ϕ, x, ξ, t) d x d ξ d t,$

we have also

$E_{ρ} = \{\int_{Ω} \frac{𝜕 f_{5} (ϕ, x, ξ, t)}{𝜕 x_{k}} d ξ\} .$

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{𝜕 u_{k}}{𝜕 t} + u_{j} \frac{𝜕 u_{k}}{𝜕 x_{j}}) - ρ f_{k} + \frac{𝜕 P}{𝜕 x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{𝜕 ρ}{𝜕 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} + \nabla_{x} ({\hat{E}}_{1}) \cdot u + P (div u) - \frac{𝜕 Q}{𝜕 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{𝜕 r_{D}}{𝜕 t}) \\ + {\hat{K}}_{p} {| ϕ_{p}^{T} |}^{2} (u + \frac{𝜕 r_{T}}{𝜕 t}) \\ + {\hat{K}}_{p} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{𝜕 r_{H_{e}}}{𝜕 t}) \\ + {\hat{K}}_{e} \int_{Ω} {| ϕ_{e} (x, y, t) |}^{2} (u (y, t) + \frac{𝜕 r_{e} (x, y, t)}{𝜕 t} d x)) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{𝜕}{𝜕 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:

For the Deuterium field

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

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

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

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

$ϕ_{e} = ϕ_{e} (x, y, t) .$

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

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

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

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

Also,

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

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
$m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
$m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
$(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
$(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
$\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
$m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
$m_{e} (t) = \int_{Ω} | ϕ_{p}^{D} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} | ϕ_{p}^{T} {(x, t) |}^{2} d x \frac{m_{e}}{m_{p}} + \int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t) |}^{2} d x \frac{m_{e}}{m_{p}} .$

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{𝜕 f_{5} (ϕ, x, ξ)}{𝜕 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

𝜕 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:

For a single Deuterium atom indexed by s:

$| ϕ_{D} {(x, y, t, s) |}^{2} = | ϕ_{p}^{D} {(y, t, s) |}^{2} + | ϕ_{N}^{D} {(x, y, t, s) |}^{2} {| ϕ_{p}^{D} (y, t, s) |}^{2} \frac{1}{m_{p}},$
For a single Tritium atom indexed by s:

$| ϕ_{T} {(x, y, t, s) |}^{2} = | ϕ_{p}^{T} {(y, t, s) |}^{2} + (| ϕ_{N_{1}}^{T} {(x, y, t, s) |}^{2} + | ϕ_{N_{2}}^{T} {(x, y, t, s) |}^{2}) | ϕ_{p}^{T} {(y, t, s) |}^{2} \frac{1}{m_{p}},$
For a single Helium atom indexed by s:

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

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

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

Furthermore, we define also the related densities

$ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
$ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

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

𝜕 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

𝜕 Ω

, we assume there is a part

Ω_{1} \subset 𝜕 Ω

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

We assume also there is another part

𝜕 Ω_{2} \subset 𝜕 Ω

such that

𝜕 Ω_{1} \cap 𝜕 Ω_{2} = \emptyset

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

𝜕 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:

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
$m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
$m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
$(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
$(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
$\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} m_{H_{e}})_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d S d τ,$
$m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
$\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

(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{𝜕 r_{D}}{𝜕 t} \cdot \frac{𝜕 r_{D}}{𝜕 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{𝜕 r_{T}}{𝜕 t} \cdot \frac{𝜕 r_{T}}{𝜕 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{𝜕 r_{H_{e}}}{𝜕 t} \cdot \frac{𝜕 r_{H_{e}}}{𝜕 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{𝜕 r_{N}}{𝜕 t} \cdot \frac{𝜕 r_{N}}{𝜕 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{𝜕 r_{e}}{𝜕 t} \cdot \frac{𝜕 r_{e}}{𝜕 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{𝜕 r_{D}}{𝜕 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{𝜕 r_{T}}{𝜕 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{𝜕 r_{H_{e}}}{𝜕 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{𝜕 r_{e}}{𝜕 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{𝜕 N_{D} (t)}{𝜕 t})}^{2} d t + \frac{ε_{T}}{2} \int_{0}^{t_{f}} {(\frac{𝜕 N_{D} (t)}{𝜕 t})}^{2} d t \\ + \frac{ε_{N}}{2} \int_{0}^{t_{f}} {(\frac{𝜕 N_{N} (t)}{𝜕 t})}^{2} d t + \frac{ε_{H_{e}}}{2} \int_{0}^{t_{f}} {(\frac{𝜕 N_{H_{e}} (t)}{𝜕 t})}^{2} d t \\ + \frac{ε_{e}}{2} \int_{0}^{t_{f}} {(\frac{𝜕 N_{e} (t)}{𝜕 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:

The momentum conservation equation for the fluid motion

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

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

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,

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

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

$\begin{matrix} F_{E} = {{(F_{E})}_{k}} = \\ (K_{p} (\int_{0}^{N_{D} (t)} | ϕ_{p}^{D} |^{2} d s + \int_{0}^{N_{T} (t)} | ϕ_{p}^{T} |^{2} d s + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} d s) + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} d s) E, \end{matrix}$

and

$\begin{matrix} F_{M} & = & {{(F_{M})}_{k}} \\ = & (K_{p} (\int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{𝜕 r_{D}}{𝜕 t}) d s \\ \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{𝜕 r_{T}}{𝜕 t}) d s \\ + \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{𝜕 r_{H_{e}}}{𝜕 t}) d s) \\ + K_{e} \int_{0}^{N_{e} (t)} {| ϕ_{e} |}^{2} (u + \frac{𝜕 r_{e}}{𝜕 t}) d s) \times B . \end{matrix}$

(109)
Mass conservation equation:

$\frac{𝜕 ρ}{𝜕 t} + div (ρ u) = 0 .$
Energy equation

$ρ \frac{D e}{D t} + \nabla_{x} ({\hat{E}}_{1}) \cdot u + {\hat{E}}_{2} + P (div u) = \frac{𝜕 Q}{𝜕 t} - div q,$

where we assume the Fourier law

$q = - K \nabla T,$

where $T = T (x, t)$ is the scalar field of temperature and Q is a standard heat function.

Also,

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

(110)

where the densities ${\hat{E}}_{1}$ and ${\hat{E}}_{2}$ are defined through the expressions of $F (ϕ)$ and $F_{2}$ so that

$F (ϕ) = \int_{0}^{t_{f}} \int_{Ω} {\hat{E}}_{1} d x d t$

and

$F_{2} = \int_{0}^{t_{f}} \int_{Ω} {\hat{E}}_{2} d x d t .$

Here we recall that since $r_{D}$ is highly oscillating in t we approximately have

$u \cdot r_{D} \approx 0$

in a weak or measure sense. The same remark is valid for the other internal velocity fields.

Moreover,

$\frac{D e}{D t} = \frac{𝜕 e}{𝜕 t} + u_{j} \frac{𝜕 e}{𝜕 x_{j}} .$
$P = F_{7} (ρ, T),$

for an appropriate scalar function $F_{7}$ .
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)

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ)) u \cdot n d S d τ,$

(b)

$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$

(c)

$(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$

(d)

$(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$

(e)

${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$

(f)

${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$

(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)

${(m_{e})}_{T} (t) = m_{e} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ .$

(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)
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)

$\int_{Ω} {| ϕ_{2 p}^{H_{e}} (x, t, s) |}^{2} d x = 2 m_{p},$
$m_{D} (t) = m_{p} N_{D} (t) + m_{N} N_{D} (t)$

$m_{T} (t) = m_{p} N_{T} (t) + m_{N} N_{T} (t),$

$m_{H_{e}} (t) = 2 m_{p} N_{H_{e}} (t) + 2 m_{N} N_{H_{e}} (t),$

$m_{e} (t) = m_{e} N_{D} (t) + m_{e} N_{T} (t) + 2 m_{e} N_{H_{e}} (t) .$
For the induced electric field, we must have

$\begin{matrix} curl E_{i n d} + \frac{1}{c} curl ({\hat{K}}_{p} \int_{0}^{N_{D} (t)} {| ϕ_{p}^{D} |}^{2} (u + \frac{𝜕 r_{D}}{𝜕 t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{𝜕 r_{T}}{𝜕 t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{𝜕 r_{H_{e}}}{𝜕 t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{𝜕 r_{e} (x, y, t)}{𝜕 t} d x) d s) \\ \times (curl A - B_{0}) - \frac{1}{c} \frac{𝜕}{𝜕 t} (curl A - B_{0}) = 0, \end{matrix}$

(112)

where ${\hat{K}}_{p}$ and ${\hat{K}}_{e}$ are appropriate real constants related to the respective charges.
A Maxwell equation:

$div B = 0,$

where

$B = B_{0} - curl A .$
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

$F (ϕ) = \int_{0}^{t_{f}} \int_{Ω} f_{5} (ϕ, x, t ξ, s) d x d ξ d s d t,$

we have also

$E_{ρ} = \{\int_{Ω} \frac{𝜕 f_{5} (ϕ, x, t, ξ, s)}{𝜕 x_{k}} d ξ d s\} .$

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{𝜕 u_{k}}{𝜕 t} + u_{j} \frac{𝜕 u_{k}}{𝜕 x_{j}}) - ρ f_{k} + \frac{𝜕 P}{𝜕 x_{k}} - τ_{k j, j} - {(F_{E})}_{k} - {(F_{M})}_{k}〉}_{L^{2}} \\ + {〈Λ_{4}, \frac{𝜕 ρ}{𝜕 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} + \nabla_{x} ({\hat{E}}_{1}) \cdot u + P (div u) - \frac{𝜕 Q}{𝜕 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}

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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{𝜕 r_{D}}{𝜕 t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{T} (t)} {| ϕ_{p}^{T} |}^{2} (u + \frac{𝜕 r_{T}}{𝜕 t}) d s \\ + {\hat{K}}_{p} \int_{0}^{N_{H_{e}} (t)} {| ϕ_{2 p}^{H_{e}} |}^{2} (u + \frac{𝜕 r_{H_{e}}}{𝜕 t}) d s \\ + {\hat{K}}_{e} \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} (u (y, t) + \frac{𝜕 r_{e} (x, y, t, s)}{𝜕 t} d x) d s) \\ {\times (curl A - B_{0}) - \frac{1}{c} \frac{𝜕}{𝜕 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:

For the Deuterium field

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

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

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

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

$ϕ_{e} = ϕ_{e} (x, y, t, s) .$

$ρ_{D} (y, t) = \int_{0}^{N_{D} (t)} \int_{Ω} {| ϕ_{D} (x, y, t, s) |}^{2} d x d s,$
$ρ_{T} (y, t) = \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{T} (x, y, t, s) |}^{2} d x d s,$

$ρ_{H_{e}} (y, t) = \int_{0}^{N_{H_{e}} (t)} \int_{Ω} {| ϕ_{H_{e}} (x, y, t, s) |}^{2} d x d s,$

$ρ_{N} (x, t) = \int_{0}^{N_{N} (t)} {| ϕ_{N} (x, t, s) |}^{2} d s,$

$ρ_{e} (y, t) = \int_{0}^{N_{e} (t)} \int_{Ω} {| ϕ_{e} (x, y, t, s) |}^{2} d x d s .$

Also,

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

${(m_{H_{e}, N})}_{T} (t) = m_{H_{e}, N} (t) + \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{H_{e}} (x, τ) + ρ_{N} (x, τ)) u \cdot n d S d τ,$
$m_{H_{e}, N} (t) = m_{H_{e}} (t) + m_{N} (t),$
$m_{H_{e}} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x,$
$m_{N} (t) = \int_{Ω} ρ_{N} (x, t) d x,$
$(m_{D}) (t) = {(m_{D})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{D} (x, τ)) u \cdot n d S d τ - α_{D} {(m_{H_{e}, N})}_{T} (t),$
$(m_{T}) (t) = {(m_{T})}_{0} - \int_{0}^{t} \int_{𝜕 Ω_{1} \cup 𝜕 Ω_{2}} (ρ_{T} (x, τ)) u \cdot n d S d τ - α_{T} {(m_{H_{e}, N})}_{T} (t),$
${(m_{H_{e}})}_{T} (t) = \int_{Ω} ρ_{H_{e}} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{H_{e}} (x, τ) u \cdot n d Γ d τ,$
${(m_{N})}_{T} (t) = \int_{Ω} ρ_{N} (x, t) d x + \int_{0}^{t} \int_{𝜕 Ω_{2}} ρ_{N} (x, τ) u \cdot n d Γ d τ,$
$\frac{{(m_{N})}_{T} (t)}{{(m_{H_{e}})}_{T} (t)} = \frac{α_{N}}{α_{H_{e}}},$

so that

$α_{N} {(m_{H_{e}})}_{T} (t) = α_{H_{e}} {(m_{N})}_{T} (t),$
${(m_{e})}_{T} (t) = m_{e} (t) - \int_{0}^{t} \int_{𝜕 Ω_{2}} (ρ_{e} (x, τ)) u \cdot n d Γ d τ,$
$m_{e} (t) = \int_{Ω} ρ_{e} (x, t) d x .$
$\begin{matrix} m_{e} (t) & = & \int_{0}^{N_{D} (t)} \int_{Ω} | ϕ_{p}^{D} {(y, t, s) |}^{2} d y d y d s \frac{m_{e}}{m_{p}} + \int_{0}^{N_{T} (t)} \int_{Ω} {| ϕ_{p}^{T} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} \\ + \int_{0}^{N_{p} (t)} \int_{Ω} {| ϕ_{2 p}^{H_{e}} (y, t, s) |}^{2} d y d s \frac{m_{e}}{m_{p}} . \end{matrix}$

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Finally,

E = E_{i n d} + E_{ρ},

and where generically denoting

F (ϕ) = \int_{0}^{t_{f}} \int_{Ω} f_{5} (ϕ, x, t, ξ, s) d x d ξ d s,

we have also

E_{ρ} = \{\int_{Ω} \frac{𝜕 f_{5} (ϕ, x, t, ξ, s)}{𝜕 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

𝜕 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{𝜕 r_{u}}{𝜕 t} \cdot \frac{𝜕 r_{u}}{𝜕 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} .

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.
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

𝜕 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:

$\int_{Ω} {| ϕ_{p_{j k l}} (x, y, z, t) |}^{2} d x = m_{p_{j k l}},$
$\int_{Ω} {| ϕ_{A_{k l}} (y, z, t) |}^{2} d y = m_{A_{k l}},$
$\int_{Ω} {| ϕ_{M_{l}} (z, t) |}^{2} d z = m_{M_{l}} .$

At this point we denote for the atoms

A_{1}

A_{2}

of a hydrogen molecule:

$m_{e_{j}} = m_{e}$ : mass of electron $e_{j}$ in the atom $A_{j}$ , where $j \in {1, 2} .$
$m_{p_{j}} = m_{p}$ : 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 10.

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

𝜕 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 11.

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}

(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

𝜕 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:

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)

$\int_{Ω} {| {(ϕ_{1}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C},$
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)

$\int_{Ω} {| {(ϕ_{2}^{C})}_{A} (x, y, z, t) |}^{2} d x = m_{A}^{C} .$
For the total $F_{e}$ (iron) mass,

$\int_{Ω} | ϕ_{1}^{γ} {(z, t) |}^{2} d z + \int_{Ω} {| ϕ_{2}^{γ} (z, t) |}^{2} d z = {(m_{F_{e}})}_{T},$
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{𝜕 u_{i}}{𝜕 x_{j}} + \frac{𝜕 u_{j}}{𝜕 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 12.

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

𝜕 \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{𝜕 u_{i}}{𝜕 x_{j}}\}

and

\frac{𝜕 u_{i}}{𝜕 x_{j}} = {(u_{i})}_{x_{j}} .

Assume

u \in V

is such that

δ L (u, \nabla u) = 0,

so that

\frac{𝜕 L (u, \nabla u)}{𝜕 u_{i}} - \sum_{k = 1}^{n} \frac{d}{d x_{k}} (\frac{𝜕 L (u, \nabla u)}{𝜕 {(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{𝜕 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{𝜕 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{𝜕 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{𝜕 x_{j}^{'}}{𝜕 x_{k}}\} \\ = & \{\frac{𝜕 (x_{j} + ε {(φ_{2})}_{j} (x))}{𝜕 x_{k}}\} \\ = & \{δ_{j k} + ε \frac{𝜕 {(φ_{2})}_{j} (x)}{𝜕 x_{k}}\} . \end{matrix}

(139)

From such a last definition we have

det J (x) = 1 + ε \sum_{k = 1}^{n} \frac{𝜕 {(φ_{2})}_{k} (x)}{𝜕 x_{k}} + O (ε^{2}) .

so that

\frac{𝜕 det J (x)}{𝜕 ε} |_{ε = 0} = \sum_{k = 1}^{n} \frac{𝜕 {(φ_{2})}_{k} (x)}{𝜕 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{𝜕 L (u, \nabla u)}{𝜕 u_{i}} {(φ_{1})}_{i} \\ + \sum_{k = 1}^{n} (\frac{𝜕 L (u, \nabla u)}{𝜕 {(u_{i})}_{x_{k}}} {({(φ_{1})}_{i})}_{x_{k}}) \\ + \sum_{k = 1}^{n} \frac{δ L [u]}{δ u_{i}} \frac{𝜕 u_{i}}{𝜕 x_{k}} {(φ_{2})}_{k}) + \sum_{k = 1}^{n} L [u] \frac{𝜕 {(φ_{2})}_{k}}{𝜕 x_{k}}) d x . \end{matrix}

(140)

From this and

\frac{𝜕 L (u, \nabla u)}{𝜕 u_{i}} - \frac{d}{d x_{k}} (\frac{𝜕 L (u, \nabla u)}{𝜕 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{𝜕 L [u]}{𝜕 {(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{𝜕 u_{i}}{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{𝜕 u_{i} (x)}{𝜕 x_{l}} X_{j}^{l} w^{j})))) d x \\ + \sum_{k = 1}^{n} \int_{Ω} \frac{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{k}} (\sum_{j = 1}^{N} (ψ_{i j} - \sum_{l = 1}^{n} \frac{𝜕 u_{i} (x)}{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{x_{k}}} (- ψ_{i j} + \sum_{l = 1}^{n} \frac{𝜕 u_{i}}{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{x_{k}}} \frac{𝜕 u_{i}}{𝜕 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{𝜕 x_{j}^{'}}{𝜕 x_{k}}\} \\ = & \{\frac{𝜕 (x_{j} + ε {(φ_{2})}_{j} (x))}{𝜕 x_{k}}\} \\ = & \{δ_{j k} + ε \frac{𝜕 {(φ_{2})}_{j} (x)}{𝜕 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{𝜕 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{𝜕 u_{i} (x)}{𝜕 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{𝜕 u_{i} (x)}{𝜕 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{𝜕 u_{i} (x)}{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{x_{k}}} (A_{i {(l, m)}^{j}} u_{j} (x) + \frac{𝜕 u_{i}}{𝜕 x_{p}} g^{m m} x_{m} δ_{l}^{p} - \frac{𝜕 u_{i}}{𝜕 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{𝜕 L [u]}{𝜕 {(u_{i})}_{x_{k}}} (A_{i l m}^{j} u_{j} - \frac{𝜕 u_{i}}{𝜕 x_{l}} g^{m m} x_{m} + \frac{𝜕 u_{i}}{𝜕 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{𝜕 L [u]}{𝜕 {(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{𝜕 L [u]}{𝜕 {(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{𝜕^{2} u}{𝜕 t^{2}} + \sum_{j = 1}^{3} \frac{𝜕^{2} u}{𝜕 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{𝜕 u}{𝜕 x_{j}})}^{2} + m^{2} u .

Also, from

- \frac{𝜕^{2} u}{𝜕 t^{2}} + \sum_{j = 1}^{3} \frac{𝜕^{2} u}{𝜕 x_{j}^{2}} - m^{2} u = 0, in Ω,

we get

\int_{Ω} {(\frac{𝜕 u}{𝜕 t})}^{2} d x - \sum_{j = 1}^{3} \int_{Ω} {(\frac{𝜕 u}{𝜕 x_{j}})}^{2} d x - m^{2} \int_{Ω} u^{2} d x = 0,

so that

\int_{Ω} {(\frac{𝜕 u}{𝜕 t})}^{2} d x = \sum_{j = 1}^{3} \int_{Ω} {(\frac{𝜕 u}{𝜕 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{𝜕 u}{𝜕 t})}^{2} d x \\ = & {∥\frac{𝜕 u}{𝜕 t}∥}_{L^{2}}^{2} \\ = & \sum_{j = 1}^{3} \int_{Ω} {(\frac{𝜕 u}{𝜕 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{𝜕 u}{𝜕 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{𝜕 u}{𝜕 t}∥}_{L^{2}}^{2}, \end{matrix}

(155)

so that

\int_{Ω} T^{00} d x = {∥\frac{𝜕 u}{𝜕 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{𝜕}{𝜕 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{𝜕}{𝜕 x_{j}}) - m] [- i (\sum_{j = 0}^{3} γ^{j} \frac{𝜕}{𝜕 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{𝜕}{𝜕 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{𝜕 u}{𝜕 x_{j}} - \frac{𝜕 u^{*}}{𝜕 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{𝜕 u}{𝜕 x_{l}} - \frac{𝜕 u^{*}}{𝜕 x_{l}} γ^{k} u),

and

S^{k, l m} = - (\frac{𝜕 L (u, \nabla u)}{𝜕 u_{x_{k}}} A^{u, l m} u - u^{*} A^{u^{*}, l m} \frac{𝜕 L (u, \nabla u)}{𝜕 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

𝜕 \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{𝜕^{2} u}{𝜕 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{𝜕}{𝜕 x_{j}} + x_{j}),

and

{(B_{2})}_{j} = A_{j}^{*} = \frac{1}{\sqrt{2}} (- \frac{𝜕}{𝜕 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{𝜕}{𝜕 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{𝜕^{2} ϕ}{𝜕 t^{2}} + \sum_{j = 1}^{3} \frac{𝜕^{2} ϕ}{𝜕 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

𝜕 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{𝜕 F (u, v_{0}^{*})}{𝜕 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 4.

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{𝜕 H (u_{0}, {\hat{v}}_{0}^{*})}{𝜕 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

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

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:

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

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 13.

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 14.

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.

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

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":

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

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 15.

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.

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

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 16.

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 𝜕 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 17.

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 𝜕 Ω,

firstly we have solved the equation

\nabla^{2} v - P = 0

with the boundary conditions

v = 0 on 𝜕 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 𝜕 Ω .

Here we present the software developed in MAT-LAB.

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

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

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

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 18.

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

𝜕 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 = 𝜕 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 = 𝜕 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} = 𝜕 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_{𝜕 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 19.

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

𝜕 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 𝜕 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.

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

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"

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

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 𝜕 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_{𝜕 Ω} {(v_{1}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S + \int_{𝜕 Ω} {(v_{2}^{*})}_{i j} n_{j} {(u_{0})}_{i} d S\}\} . \end{matrix}

(199)

Remark 20.

We highlight this last dual function in

v^{*}

is convex (in fact concave) on the convex set

A^{*}

33.3. A Numerical Example

For

Ω = [0, 1] \subset R

consider a functional

J : V \to R

where

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

(200)

where

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

Y = Y^{*} = L^{2} (Ω)

and

f \in Y .

Define

G : Y \to R

and

F : V \to R

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

and

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

respectively.

Denoting

V_{0} = W_{0}^{1, 2} (Ω),

define also

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

\begin{matrix} J_{1} (u, ϕ, λ) & = & λ G (u^{'} - (1 - λ) ϕ^{'}) + (1 - λ) G (u^{'} + λ ϕ^{'}) \\ + λ F (u - (1 - λ) ϕ) + (1 - λ) F (u + λ ϕ) \\ - {〈 u, f 〉}_{L^{2}} . \end{matrix}

(201)

Observe that

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

(202)

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

(203)

\begin{matrix} {(λ F)}^{*} (v_{3}^{*}) & = & sup_{v_{3} \in Y} {{〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} - λ F (v_{3})} \\ = & λ F^{*} (\frac{v_{3}^{*}}{λ}) \\ = & \frac{1}{2 λ} \int_{Ω} {(v_{3}^{*})}^{2} d x, \end{matrix}

(204)

and

\begin{matrix} {((1 - λ) F)}^{*} (v_{4}^{*}) & = & sup_{v_{4} \in Y} {{〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} - (1 - λ) F (v_{4})} \\ = & (1 - λ) F^{*} (\frac{v_{4}^{*}}{(1 - λ)}) \\ = & \frac{1}{2 (1 - λ)} \int_{Ω} {(v_{4}^{*})}^{2} d x . \end{matrix}

(205)

Denoting

v^{*} = (v_{1}^{*}, \dots, v_{4}^{*}) \in {[Y^{*}]}^{4}

, define

J^{*} : {[Y^{*}]}^{4} \times (0, 1) \to R

\begin{matrix} J_{1}^{*} (v^{*}, λ) & = & - λ G^{*} (\frac{v_{1}^{*}}{λ}) - (1 - λ) G^{*} (\frac{v_{2}^{*}}{(1 - λ)}) \\ - λ F^{*} (\frac{v_{3}^{*}}{λ}) - (1 - λ) F^{*} (\frac{v_{4}^{*}}{(1 - λ)}) \\ + v_{1}^{*} (1) u (1) + v_{2}^{*} (1) u (1) . \end{matrix}

(206)

Similarly as in the previous section, we may obtain

inf_{u \in V} J (u) \geq inf_{λ \in (0, 1)} \{sup_{v^{*} \in A^{*}} J^{*} (v^{*}, λ)\},

where

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

A_{1}^{*} = {v^{*} \in Y^{*} : {(v_{1}^{*})}^{'} + {(v_{2}^{*})}^{'} - v_{3}^{*} - v_{4}^{*} + f = 0, in Ω},

and

A_{2}^{*} = {(v^{*}, λ) \in {[Y^{*}]}^{4} \times (0, 1) : - (1 - λ) {(v_{1}^{*})}^{'} + λ {(v_{2}^{*})}^{'} + (1 - λ) v_{3}^{*} - λ v_{4}^{*} = 0, in Ω} .

From such expressions of

A_{1}^{*}

and

A_{2}^{*}

we may obtain

v_{3}^{*} = {(v_{1}^{*})}^{'} + λ f,

and

v_{4}^{*} = {(v_{2}^{*})}^{'} + (1 - λ) f .

Replacing such expressions for

v_{3}^{*}

and

v_{4}^{*}

into the expression of

J^{*}

, and from now and on denoting

v^{*} = (v_{1}^{*}, v_{2}^{*}) \in {[Y^{*}]}^{2}

, we may obtain

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

where

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

(207)

Consequently, we have got

inf_{u \in V} J (u) \geq sup_{v^{*} \in {[Y^{*}]}^{2}} \{inf_{λ \in (0, 1)} J_{1}^{*} (v^{*}, λ)\} .

In order to obtain numerical results we have designed the following algorithm:

Set $n = 1$ and $λ_{n} = 1 / 2$ .
Calculate ${(v^{*})}_{n} \in {[Y^{*}]}^{2}$ such that

$J_{1}^{*} ({(v^{*})}_{n}, λ_{n}) = sup_{v^{*} \in {[Y^{*}]}^{2}} J_{1}^{*} (v^{*}, λ_{n}) .$
Calculate $λ_{n + 1} \in (0, 1)$ such that

$J_{1}^{*} ({(v^{*})}_{n}, λ_{n + 1}) = inf_{λ \in (0, 1)} J_{1}^{*} ({(v^{*})}_{n}, λ) .$
Set $n : = n + 1$ and go to item (2) until the satisfaction of an appropriate convergence criterion.

We have developed numerical results for the following cases

$f (x) = sin (π x) / 2,$
$f (x) = cos (π x) / 2,$
$f (x) = 0 .$

Observe that for the optimal point we have

v_{3}^{*} = u - (1 - λ) ϕ,

and

v_{4}^{*} = u + λ ϕ,

so that

u = λ v_{3}^{*} + (1 - λ) v_{4}^{*} .

For the optimal solution

u_{0} (x)

found for the cases (1), (2) and (3), please see the Figure 29, Figure 30 and Figure 31, respectively.

Here we present the concerning software in MAT-LAB.

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

clear all

global m8 d L v1 v2 v3 v4 yo dv1 dv2 e1

m8=140;

d=1/m8;

e1=0.0001;

L=1/2;

for i=1:2*m8

xo(i,1)=0.01;

end;

for i=1:m8

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

end;

x1=1/2;

k=1;

b12=1;

while $(b 12 > 10^{- 4})$ and $(k < 100)$

k

k=k+1;

X1=fminunc(’funFeb24’,xo);

b12=max(abs(X1-xo))

xo=X1;

X2=fminunc(’funFeb24A’,x1);

x1=X2;

L=(sin(x1)+1)/2;

L

end;

u(m8,1)=1/2;

for i=1:m8-1

u(i,1)=L*v3(i,1)+(1-L)*v4(i,1);

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

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

Here the auxiliary function "funFeb24"

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

function S=funFeb24(x)

global m8 d L v1 v2 v3 v4 yo dv1 dv2 e1

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(m8+i,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8

S=S+ $1 / 2 / s q r t (L^{2} + e 1) * v 1 {(i, 1)}^{2} + s q r t (v 1 {(i, 1)}^{2} + e 1)$ ;

S=S+ $1 / 2 / s q r t ({(1 - L)}^{2} + e 1) * v 2 {(i, 1)}^{2} + s q r t (v 2 {(i, 1)}^{2} + e 1)$ ;

end;

for i=1:m8-1

v3(i,1)=dv1(i,1)+L*yo(i,1);

v4(i,1)=dv2(i,1)-(L-1)*yo(i,1);

S=S+ $1 / 2 / s q r t (L^{2} + e 1) * v 3 {(i, 1)}^{2} + 1 / 2 / s q r t ({(1 - L)}^{2} + e 1) * v 4 {(i, 1)}^{2}$ ;

end;

S=S-(v1(m8,1)+v2(m8,1))/d/2;

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

Finally, the auxiliary function "funFeb24A"

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

function S1=funFeb24A(y)

global m8 d L v1 v2 v3 v4 yo e1

L=(sin(y)+1)/2;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8

S=S+ $1 / 2 / s q r t (L^{2} + e 1) * v 1 {(i, 1)}^{2} + s q r t (v 1 {(i, 1)}^{2} + e 1)$ ;

S=S+ $1 / 2 / s q r t ({(1 - L)}^{2} + e 1) * v 2 {(i, 1)}^{2} + s q r t (v 2 {(i, 1)}^{2} + e 1)$ ;

end;

for i=1:m8-1

v3(i,1)=dv1(i,1)+L*yo(i,1);

v4(i,1)=dv2(i,1)-(L-1)*yo(i,1);

S=S+ $1 / 2 / s q r t (L^{2} + e 1) * v 3 {(i, 1)}^{2} + 1 / 2 / s q r t ({(1 - L)}^{2} + e 1) * v 4 {(i, 1)}^{2}$ ;

end;

S=S-(v1(m8,1)+v2(m8,1))/d/2;

S1=-S;

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

34. One More 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

𝜕 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 𝜕 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}

(208)

On the other hand

\begin{matrix} {(G \circ \nabla)}^{* *} (u) & \leq & H (u) \\ \equiv & inf_{(λ, (v_{1}, \dots, v_{m})) \in B \times B_{1} (u, λ)} \{\sum_{j = 1}^{m} λ_{j} G (\nabla v_{j})\} \\ \leq & G (\nabla u), \end{matrix}

(209)

where

B = \{λ = (λ_{1}, \dots, λ_{m}) \in R^{m} : λ_{j} \geq 0, \forall j \in {1, \dots, m}, and \sum_{j = 1}^{m} λ_{j} = 1\},

and

B_{1} (u, λ) = \{v = (v_{1}, \dots, v_{m}) \in {[V]}^{m} : \sum_{j = 1}^{m} λ_{j} v_{j} = 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

\sum_{j = 1}^{m} λ_{j} \nabla v_{j} = \nabla u,

and

λ_{m} = 1 - \sum_{j = 1}^{m - 1} λ_{j},

so that

\begin{matrix} \nabla v_{m} & = & \nabla u - \sum_{j = 1}^{m - 1} λ_{j} (\nabla v_{j} - \nabla v_{m}) \\ = & \nabla u + \sum_{j = 1}^{m - 1} λ_{j} \nabla ϕ_{j}, \end{matrix}

(210)

where

ϕ_{j} = - v_{j} + v_{m} \in W_{0}^{1, 2} (Ω; R^{N})

so that

\nabla ϕ_{j} = - \nabla v_{j} + \nabla v_{m},

and

\nabla v_{m} = \nabla v_{j} + \nabla ϕ_{j}, \forall j \in {1, \dots, m} .

Therefore,

\nabla v_{j} = \nabla v_{m} - \nabla ϕ_{j} = \nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - \nabla ϕ_{j} .

Replacing such results into the expression of H, we have

H (u) = inf_{(λ, ϕ) \in B \times {(V_{0})}^{m - 1}} \{\sum_{j = 1}^{m - 1} λ_{j} G (\nabla u + \sum_{k = 1}^{m - 1} \nabla ϕ_{k} - \nabla ϕ_{j}) + λ_{m} G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k})\},

where we recall that

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_{(u, λ, ϕ) \in V \times B \times {(V_{0})}^{m - 1}} \{\sum_{j = 1}^{m - 1} λ_{j} G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - \nabla ϕ_{j}) \\ + λ_{m} G (\nabla u + \sum_{k = 1}^{m} λ_{k} \nabla ϕ_{k}) \\ - {〈 u, f 〉}_{L^{2}}\} . \end{matrix}

This last functional corresponds to a relaxation for the original non-convex functional.

The note is complete.

34.1. 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} : B \times V \times {[V_{0}]}^{m} \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) & = & \sum_{j = 1}^{m} λ G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - \nabla ϕ_{j}) \\ + λ_{m} G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k}) \\ + \sum_{j = 1}^{m - 1} \frac{λ_{j}}{2} \int_{Ω} (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - \nabla ϕ_{j}) \cdot (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - \nabla ϕ_{j}) d x \\ + \frac{(λ_{m})}{2} \int_{Ω} (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}) \cdot (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}) d x \\ - \sum_{j = 1}^{m - 1} λ_{j} {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - ϕ_{j}, f〉}_{L^{2}} \\ - (λ_{m}) {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}, f〉}_{L^{2}} . \end{matrix}

(211)

Here we have denoted

V = {u \in W^{1, 2} (Ω; R^{N}) : u = u_{0} on 𝜕 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 B \times {(V_{0})}^{m - 1}} J_{3} (λ, u, ϕ) .

Moreover,

\begin{matrix} J_{3} (λ, u, ϕ) & = & - \sum_{j = 1}^{m - 1} {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - \nabla ϕ_{j}, {(v_{1}^{*})}_{j}〉}_{L^{2}} \\ + \sum_{j = 1}^{m - 1} λ_{j} G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - \nabla ϕ_{j}) \\ - {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k}, {(v_{1}^{*})}_{m}〉}_{L^{2}} + λ_{m} G (\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k}) \\ - \sum_{j = 1}^{m - 1} {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - ϕ_{j}, {(v_{3}^{*})}_{j}〉}_{L^{2}} \\ + \sum_{j = 1}^{m - 1} \frac{λ_{j}}{2} \int_{Ω} (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - ϕ_{j}) \cdot (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - ϕ_{j}) d x \\ - {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}, {(v_{3}^{*})}_{m}〉}_{L^{2}} \\ + \frac{λ_{m}}{2} \int_{Ω} (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}) \cdot (u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}) d x \\ + \sum_{j = 1}^{m - 1} {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - \nabla ϕ_{j}, {(v_{1}^{*})}_{j}〉}_{L^{2}} \\ + {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k}, {(v_{1}^{*})}_{m}〉}_{L^{2}} \\ + \sum_{j = 1}^{m - 1} {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k} - ϕ_{j}, {(v_{3}^{*})}_{j}〉}_{L^{2}} \\ + {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}, {(v_{3}^{*})}_{m}〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}} \end{matrix}

(212)

Therefore,

\begin{matrix} J_{3} (λ, u, ϕ) & \geq & inf_{v_{1} \in {[Y]}^{m - 1}} \{\sum_{j = 1}^{m - 1} (- {〈 {(v_{1})}_{j}, {(v_{1}^{*})}_{j} 〉}_{L^{2}} + λ_{j} G ({(v_{1})}_{j}))\} \\ + inf_{{(v_{1})}_{m} \in Y} {- {〈 {(v_{1})}_{m}, {(v_{1}^{*})}_{m} 〉}_{L^{2}} + λ_{m} G ({(v_{1})}_{m})} \\ + inf_{v_{3} \in {[Y_{1}]}^{m - 1}} \{\sum_{j = 1}^{m - 1} (- {〈 {(v_{3})}_{j}, {(v_{3}^{*})}_{j} 〉}_{L^{2}} + \frac{λ_{j}}{2} \int_{Ω} {(v_{3})}_{j} \cdot {(v_{3})}_{j} d x)\} \\ + inf_{{(v_{3})}_{m} \in Y_{1}} \{- {〈 {(v_{3})}_{m}, {(v_{3}^{*})}_{m} 〉}_{L^{2}} + \frac{λ_{m}}{2} \int_{Ω} {(v_{3})}_{j} \cdot {(v_{3})}_{j} d x\} \\ + inf_{(u, ϕ) \in V \times {(V_{0})}^{m - 1}} \{\sum_{j = 1}^{m - 1} {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k} - ϕ_{j}, {(v_{1}^{*})}_{j}〉}_{L^{2}} \\ + {〈\nabla u + \sum_{k = 1}^{m - 1} λ_{k} \nabla ϕ_{k}, {(v_{1}^{*})}_{m}〉}_{L^{2}} \\ + \sum_{j = 1}^{m - 1} {〈u + \sum_{k = 1}^{m - 1} \nabla ϕ_{k} - ϕ_{j}, v_{3}^{*}〉}_{L^{2}} \\ + {〈u + \sum_{k = 1}^{m - 1} λ_{k} ϕ_{k}, {(v_{3}^{*})}_{m}〉}_{L^{2}} - {〈 u, f 〉}_{L^{2}}\} \\ = & - \sum_{j = 1}^{m - 1} λ_{j} G^{*} (\frac{{(v_{1}^{*})}_{j}}{λ_{j}}) - λ_{m} G^{*} (\frac{{(v_{1}^{*})}_{m}}{λ_{m}}) \\ - \sum_{j = 1}^{m - 1} {(F_{3})}_{j}^{*} ({(v_{3}^{*})}_{j}, λ_{j}) - {(F_{3})}_{m}^{*} ({(v_{3}^{*})}_{m}, λ_{m}) \\ + \sum_{k = 1}^{m} \int_{S} {({(v_{1}^{*})}_{k})}_{i j} n_{j} {(u_{0})}_{i} d S, \\ \forall λ \in B, u \in V, ϕ \in {(V_{0})}^{m - 1}, v^{*} \in A^{*}, \end{matrix}

(213)

where

G^{*} (v^{*}) = sup_{v \in Y} {{〈 v, v^{*} 〉}_{L^{2}} - G (v)},

\begin{matrix} {(F_{3})}_{j}^{*} ({(v_{3}^{*})}_{j}, λ_{j}) & = & sup_{v_{3} \in Y_{1}} \{{〈 {(v_{3})}_{j}, {(v_{3}^{*})}_{j} 〉}_{L^{2}} - \frac{λ_{j}}{2} \int_{Ω} {(v_{3})}_{j} \cdot {(v_{3})}_{j} d x\} \\ = & \frac{1}{2 λ_{j}} \int_{Ω} {(v_{3})}_{j}^{*} \cdot {(v_{3})}_{j}^{*} d x, \forall j \in {1, \dots, m} . \end{matrix}

(214)

Furthermore,

A^{*} = A_{1}^{*} \cap A_{2}^{*} (λ)

where

A_{1}^{*} = \{v^{*} = (v_{1}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{m} \times {[Y_{1}^{*}]}^{m} : - \sum_{j = 1}^{m} (div {({(v_{1}^{*})}_{j})}_{i} + {({(v_{3}^{*})}_{j})}_{i}) - f_{i} = 0, in Ω\},

and

\begin{matrix} A_{2}^{*} (λ) & = & \{v^{*} = (v_{1}^{*}, v_{3}^{*}) \in {[Y^{*}]}^{m} \times {[Y_{1}^{*}]}^{m} : \\ λ_{k} \sum_{j = 1}^{m} div {({(v_{1}^{*})}_{j})}_{i} - div {({(v_{1}^{*})}_{k})}_{i} - λ_{k} \sum_{j = 1}^{m} {({(v_{3}^{*})}_{j})}_{i} + {({(v_{3}^{*})}_{k})}_{i} = 0, \\ in Ω, \forall k \in {1, \dots, m - 1}, \forall i \in {1, \dots, N}\} . \end{matrix}

(215)

Summarizing, we have got

\begin{matrix} inf_{(λ, u ϕ) \in B \times V \times {(V_{0})}^{m - 1}} J_{3} (λ, u, ϕ) \\ \geq & inf_{λ \in B} \{sup_{v^{*} \in A^{*}} \{- \sum_{j = 1}^{m} λ_{j} G^{*} (\frac{{(v_{1}^{*})}_{j}}{λ_{j}}) \\ - \sum_{j = 1}^{m} {(F_{3}^{*})}_{j} ({(v_{3}^{*})}_{j}, λ_{j}) + \sum_{k = 1}^{m} \int_{𝜕 Ω} {({(v_{1}^{*})}_{k})}_{i j} n_{j} {(u_{0})}_{i} d S\}\} . \end{matrix}

(216)

Remark 21.

We highlight this last dual function in

v^{*}

is convex (in fact concave) on the convex set

A^{*}

35. A General Convex Primal Dual Formulation with a Restriction 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

𝜕 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}

(217)

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}

(218)

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}

(219)

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}

(220)

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

A^{+} = {u \in V : u f \geq 0, in Ω},

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

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}^{*} : V_{1} \times D^{*} \times B^{*} \to R

\begin{matrix} 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}{10 α K_{3}^{2}} {∥ v_{1}^{*} - (- 2 v_{0}^{*} + K) u ∥}_{2}^{2} \end{matrix}

(221)

Observe that

\frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 {(v_{1}^{*})}^{2}} = - \frac{1}{- γ \nabla^{2} + K} - \frac{1}{2 v_{0}^{*} - K} + K_{1} + \frac{1}{5 α K_{3}^{2}},

\frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u^{2}} = K_{1} {(- γ \nabla^{2} + K)}^{2} + \frac{1}{5 α K_{3}^{2}} {(- 2 v_{0}^{*} + K)}^{2},

and

\frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u 𝜕 v_{1}^{*}} = - K_{1} (- γ \nabla^{2} + K) - \frac{1}{5 α K_{3}^{2}} (- 2 v_{0}^{*} + K) .

Now we set

K_{1}, K, K_{3}

such that

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 .

From such results and constant choices, we may obtain

\begin{matrix} det {δ_{u, v_{1}^{*}}^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})} & = & \frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 {(v_{1}^{*})}^{2}} \frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u^{2}} - {(\frac{𝜕^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})}{𝜕 u 𝜕 v_{1}^{*}})}^{2} \\ = & O (\frac{K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{5 α K_{3}^{2}} + 2 K_{1} (- γ \nabla^{2} + 2 v_{0}^{*})) + O (\frac{K_{1}}{K}), \\ in V_{1} \times D^{*} \times B^{*} . \end{matrix}

(222)

Define now

C^{*} = \{v_{0}^{*} \in Y^{*} : \frac{{(- γ \nabla^{2} + 2 v_{0}^{*})}^{2}}{5 α K_{3}^{2}} + 2 (- γ \nabla^{2} + 2 v_{0}^{*}) > \frac{c_{0}}{K} I_{d}\},

where we assume that

c_{0} > 0

is such that if

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

, then

det {δ_{u, v_{1}^{*}}^{2} J_{2}^{*} (u, v_{1}^{*}, v_{0}^{*})} > 0, in B^{*} \cap C^{*} .

Finally, we also suppose the concerning constants are such that

B^{*} \cap C^{*}

is convex.

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 5.

Let

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

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{K_{1}}{2} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} {∥ - γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - 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}

(223)

Proof.

The proof that

δ J (u_{0}) = - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f = 0

and

J (u_{0}) = 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,

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

and

J_{2}^{*}

is 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_{2}^{*} (u, v_{1}^{*}, {\hat{v}}_{0}^{*}),

and

J_{2}^{*} (u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{2}^{*} (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{K_{1}}{2} {∥ - 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}

(224)

Finally, from evident convexity,

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{K_{1}}{2} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} {∥ - γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f ∥}_{2}^{2}\} . \end{matrix}

(225)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & J (u_{0}) + \frac{K_{1}}{2} {∥ - γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f ∥}_{2}^{2} \\ = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} {∥ - γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - 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}

(226)

The proof is complete.

□

36. A General Convex Dual Formulation for an Originally Non-Convex Primal One

In this section we develop a convex dual formulation for an originally non-convex primal formulation.

Let

Ω \subset R^{3}

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

𝜕 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}

(227)

where

α > 0, β > 0, γ > 0

V = W_{0}^{1, 2} (Ω)

and

Y = Y^{*} = L^{2} (Ω) .

At the moment, fix a matrix

K_{1} > 0

and

K > 0

to be specified.

Define

F_{1} : V \to R

F_{2} : V \to R

and

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

\begin{matrix} F_{1} (u) & = & \frac{γ}{4} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x \\ - {〈 u, f 〉}_{L^{2}}, \end{matrix}

(228)

\begin{matrix} F_{2} (u) & = & \frac{γ}{4} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K}{2} \int_{Ω} u^{2} d x, \end{matrix}

(229)

F_{3} (u, v_{0}^{*}) = - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + K \int_{Ω} u^{2} d x + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x + {〈 u, f 〉}_{L^{2}} .

Define also

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

and

F_{2}^{*} : 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}}{- \frac{γ}{2} \nabla^{2} + K} d x, \end{matrix}

(230)

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

(231)

At this point we also define

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2},

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

A^{+} = {u \in V : u f \geq 0, in Ω},

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

D^{*} = {v^{*} \in Y^{*} : ∥ v^{*} ∥_{\infty} \leq 5 / 4 K},

for an appropriate

K_{3} > 0

to be specified.

Furthermore, we define

F_{3}^{*} : D^{*} \times D^{*} \times B^{*} \to R

\begin{matrix} F_{3}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*}) & = & sup_{u \in V} {- {〈 u, v_{1}^{*} + v_{2}^{*} 〉}_{L^{2}} - F_{3} (u, v_{0}^{*})} \\ = & - \frac{1}{2} \int_{Ω} \frac{{(v_{1}^{*} + v_{2}^{*} - f)}^{2}}{2 v_{0}^{*} - 2 K} d x \\ - \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x - β \int_{Ω} v_{0}^{*} d x . \end{matrix}

(232)

Moreover, we define

J_{1}^{*} : D^{*} \times D^{*} \times B^{*} \to R

J_{1}^{*} (u, v_{1}^{*}, v_{0}^{*}) = - F_{1}^{*} (v_{1}^{*}) - F_{2} (v_{2}^{*}) + F_{3}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})

and

J_{2}^{*} : D^{*} \times D^{*} \times B^{*} \to R

\begin{matrix} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*}) & = & J_{1}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*}) \\ + \frac{K_{1}}{2} \int_{Ω} {(v_{1}^{*} - v_{2}^{*})}^{2} d x \\ + \frac{K^{2}}{2} \int_{Ω} {(\frac{v_{1}^{*}}{- \frac{γ}{2} \nabla^{2} + K} - \frac{v_{1}^{*} + v_{2}^{*} - f}{- 2 v_{0}^{*} + 2 K})}^{2} d x . \end{matrix}

(233)

Now observe that

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

and

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

and

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

We set

K_{1} ≫ K

K ≫ K_{3},

and

K_{3} \approx \sqrt{3}

. Moreover, after a re-scale if necessary, we assume

α \approx 0.15 .

From such results and constant choices, with the help of the software MATHEMATICA, we may obtain

\begin{matrix} det {δ_{v_{1}^{*}, v_{2}^{*}}^{2} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})} & = & \frac{𝜕^{2} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})}{𝜕 {(v_{1}^{*})}^{2}} \frac{𝜕^{2} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})}{𝜕 u^{2}} - {(\frac{𝜕^{2} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})}{𝜕 u 𝜕 v_{1}^{*}})}^{2} \\ = & O (2 K_{1} ({(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} + 4 (- γ \nabla^{2} + 2 v_{0}^{*}))) . \end{matrix}

(234)

Define now

H (v_{0}^{*}) \equiv 2 ({(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} + 4 (- γ \nabla^{2} + 2 v_{0}^{*})),

Observe that we may obtain

c_{0} > 0

such that if

v_{0}^{*} \in (C^{*} \times B^{*})

, then

det {δ_{v_{1}^{*}, v_{2}^{*}}^{2} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})} > 0,

where

C^{*} = {v_{0}^{*} \in Y^{*} : H (v_{0}^{*}) \geq c_{0} I_{d}} .

Furthermore, we assume

K > 0

and

c_{0} > 0

are such that

C^{*} \cap B^{*}

is convex.

With such statements, definitions and results in mind, we may prove the following theorem.

Theorem 6.

Let

({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \in D^{*} \times D^{*} \times (B^{*} \cap C^{*})

be such that

δ J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0 .

Under such hypotheses,

δ J (u_{0}) = 0,

and

\begin{matrix} J (u_{0}) \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(v_{1}^{*}, v_{2}^{*}) \in D^{*} \times D^{*}} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(235)

Proof.

The proof that

J (u_{0}) = 0,

- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f = 0,

and

J (u_{0}) = J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}),

may be done similarly as in the previous sections and will not be repeated.

Furthermore, since

δ J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = 0,

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

and

J_{2}^{*}

is concave in

v_{0}^{*}

D \times D^{*} \times B^{*}

, we have

J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*} {\hat{v}}_{0}^{*}) = inf_{(v_{1}^{*}, v_{2}^{*}) \in D^{*} \times D^{*}} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, {\hat{v}}_{0}^{*}),

and

J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in B^{*}} J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, v_{0}^{*}) .

From such results and the Saddle Point Theorem we may infer that

\begin{matrix} J (u_{0}) & = & J_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \\ = & sup_{v_{0}^{*} \in B^{*}} \{inf_{(v_{1}^{*}, v_{2}^{*}) \in D^{*} \times D^{*}} J_{2}^{*} (v_{1}^{*}, v_{2}^{*}, v_{0}^{*})\} \\ = & J_{2}^{*} {\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(236)

The proof is complete.

□

37. A Note on the Special Relativistic Physics

Consider in

R^{3}

two observers O and

O^{'}

and related referential Cartesian frames

O (x, y, z)

and

O^{'} (x^{'}, y^{'}, z^{'})

respectively.

Suppose a particle moves from a point

(x_{0}, y_{0}, z_{0})

to a point

(x_{0} + Δ x, y_{0} + Δ y, z_{0} + Δ z)

related to

O (x, y, z)

on a time interval

Δ t

Denote

I_{1} = Δ x^{2} + Δ y^{2} + Δ z^{2},

and

I_{2} = Δ t

In a Newtonian physics context, we have

I_{1} = Δ x^{2} + Δ y^{2} + Δ z^{2} = Δ {x^{'}}^{2} + Δ {y^{'}}^{2} + Δ {z^{'}}^{2},

and

I_{2} = Δ t = Δ t^{'},

that is,

I_{1}

and

I_{2}

remain invariant.

However, through experiments in higher energy physics, it was discovered that in fact is

I_{3}

which remains invariant (this had been previously proposed in the Einstein special relativity theory in 1905), where

I_{3} = - c^{2} Δ t^{2} + Δ x^{2} + Δ y^{2} + Δ z^{2},

so that

- c^{2} Δ t^{2} + Δ x^{2} + Δ y^{2} + Δ z^{2} = - c^{2} Δ {t^{'}}^{2} + Δ {x^{'}}^{2} + Δ {y^{'}}^{2} + Δ {z^{'}}^{2} = I_{3},

for any pair of observers O and

O^{'} .

Here c denotes the speed of light, and in the case in which

v, v^{'} ≪ c

we have the Newtonian approximation

Δ t^{'} \approx Δ t .

From the expression of

I_{3}

we obtain

\begin{matrix} - c^{2} \frac{Δ {t^{'}}^{2}}{Δ t^{2}} + \frac{Δ {x^{'}}^{2}}{Δ t^{2}} + \frac{Δ {y^{'}}^{2}}{Δ t^{2}} + \frac{Δ {z^{'}}^{2}}{Δ t^{2}} \\ = & - c^{2} \frac{Δ t^{2}}{Δ t^{2}} + \frac{Δ x^{2}}{Δ t^{2}} + \frac{Δ y^{2}}{Δ t^{2}} + \frac{Δ z^{2}}{Δ t^{2}} . \end{matrix}

(237)

Thus,

\begin{matrix} - c^{2} \frac{Δ {t^{'}}^{2}}{Δ t^{2}} + (\frac{Δ {x^{'}}^{2}}{Δ {t^{'}}^{2}} + \frac{Δ {y^{'}}^{2}}{Δ {t^{'}}^{2}} + \frac{Δ {z^{'}}^{2}}{Δ {t^{'}}^{2}}) \frac{Δ {t^{'}}^{2}}{Δ t^{2}} \\ = & - c^{2} + \frac{Δ x^{2}}{Δ t^{2}} + \frac{Δ y^{2}}{Δ t^{2}} + \frac{Δ z^{2}}{Δ t^{2}} \end{matrix}

(238)

so that

{(\frac{Δ t^{'}}{Δ t})}^{2} = \frac{c^{2} - (\frac{Δ x^{2}}{Δ t^{2}} + \frac{Δ y^{2}}{Δ_{t}^{2}} + \frac{Δ z^{2}}{Δ t^{2}})}{c^{2} - (\frac{Δ {x^{'}}^{2}}{Δ {t^{'}}^{2}} + \frac{Δ {y^{'}}^{2}}{Δ {t^{'}}^{2}} + \frac{Δ {z^{'}}^{2}}{Δ {t^{'}}^{2}})} .

Letting

Δ t, Δ t^{'} \to 0

, we obtain

{(\frac{𝜕 t^{'}}{𝜕 t})}^{2} = \frac{1 - \frac{v^{2}}{c^{2}}}{1 - \frac{{(v^{'})}^{2}}{c^{2}}} .

In particular for constant v and

v^{'} = 0

we have

{(\frac{Δ t^{'}}{Δ t})}^{2} = 1 - \frac{v^{2}}{c^{2}},

so that

Δ t^{'} = \sqrt{1 - \frac{v^{2}}{c^{2}}} Δ t .

Consider now that O is at rest and

O^{'}

has a constant velocity

v e_{1}

where

{e_{1}, e_{2}, e_{3}}

is the canonical basis for

R^{3}

related to

O .

Consider

O (x, y, z)

and

O^{'} (x, y, z)

such that the axis

x^{'}

coincide with the axis x, axis

y^{'}

is parallel to axis y and axis

z^{'}

is parallel to

z .

Since v is constant, we have

v = \frac{Δ x}{Δ t},

and

v^{'} = 0 .

Assuming

x (0) = 0,

and the initial time

t = 0

, we have

Δ x = x,

and

Δ t = t

so that

t^{'} = \sqrt{1 - \frac{v^{2}}{c^{2}}} t,

so that

t^{'} = \frac{1 - \frac{v^{2}}{c^{2}}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} t = \frac{(t - \frac{v v t}{c^{2}})}{\sqrt{1 - \frac{v^{2}}{c^{2}}}},

and thus

t^{'} = \frac{(t - \frac{v x}{c^{2}})}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} .

On the other hand we have

v^{'} = 0

We may easily check that the solution

x^{'} = \frac{x - v t}{\sqrt{1 - \frac{v^{2}}{c^{2}}}},

lead us to

v^{'} = 0

Indeed,

\frac{Δ x^{'} \sqrt{1 - \frac{v^{2}}{c^{2}}}}{Δ t^{'}} = \frac{Δ x^{'}}{Δ t},

so that, considering that v is constant, we obtain

\frac{d x^{'}}{d t} = \frac{\frac{d (x - v t)}{d t}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{\frac{d x}{d t} - v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{v - v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = 0,

that is,

\frac{d x^{'}}{d t} = 0 .

Thus,

\frac{d (x^{'} \sqrt{1 - \frac{v^{2}}{c^{2}}})}{d t^{'}} = 0,

so that

x^{'} \sqrt{1 - \frac{v^{2}}{c^{2}}} = c_{1}

for some constant

c_{1} \in R

so that

x^{'} = c_{2},

for some

c_{2} \in R .

Therefore

v^{'} = \frac{d x^{'}}{d t^{'}} = 0 .

Summarizing, for the Newton mechanics we have

t^{'} = t

x^{'} = x - v t,

y^{'} = y,

and

z^{'} = z .

On the other hand, for the special relativity context, we have the following Lorentz relations

t^{'} = \frac{(t - \frac{v x}{c^{2}})}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} .

x^{'} = \frac{x - v t}{\sqrt{1 - \frac{v^{2}}{c^{2}}}},

y^{'} = y,

and

z^{'} = z .

37.1. The Kinetics Energy for the Special Relativity Context

Consider the motion of a particle system described by the position field

r : Ω \times [0, T] \to R^{4},

where

Ω \subset R^{3}

[0, T]

is a time interval and

r (x, y, z, t) = (c t, X_{1} (x, y, z, t), X_{2} (x, y, z, t), X_{3} (x, y, z, t)) .

In my understanding, this is the special relativity theory context.

The related density field is denoted by

ρ : Ω \times [0, T] \to R^{+},

where

ρ (x, y, z, t) = m_{0} {| ϕ (x, y, z, t) |}^{2},

m_{0}

is total system mass at rest, and

ϕ : Ω \times [0, T] \to C

is a wave function such that

\int_{Ω} {| ϕ (x, y, z, t) |}^{2} d x = 1, \forall t \in [0, T] .

The Kinetics energy differential is given by

d E_{c} = - d m \frac{𝜕 r}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 t},

where

\begin{matrix} \frac{𝜕 r}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 t} & = & (c, \frac{𝜕 X_{1}}{𝜕 t}, \frac{𝜕 X_{2}}{𝜕 t}, \frac{𝜕 X_{3}}{𝜕 t}) \cdot (c, \frac{𝜕 X_{1}}{𝜕 t}, \frac{𝜕 X_{2}}{𝜕 t}, \frac{𝜕 X_{3}}{𝜕 t}) \\ = & - c^{2} + {(\frac{𝜕 X_{1}}{𝜕 t})}^{2} + {(\frac{𝜕 X_{2}}{𝜕 t})}^{2} + {(\frac{𝜕 X_{3}}{𝜕 t})}^{2} \\ = & - c^{2} + v^{2}, \end{matrix}

(239)

where

v^{2} = {(\frac{𝜕 X_{1}}{𝜕 t})}^{2} + {(\frac{𝜕 X_{2}}{𝜕 t})}^{2} + {(\frac{𝜕 X_{3}}{𝜕 t})}^{2} .

Moreover,

d m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} {| ϕ (x, y, z, t) |}^{2} d x d y d z,

so that

\begin{matrix} d E_{c} & = & \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} (c^{2} - v^{2}) {| ϕ (x, y, z, t) |}^{2} d x d y d z \\ = & m_{0} c \sqrt{c^{2} - v^{2}} {| ϕ |}^{2} d x d y d z . \end{matrix}

(240)

Thus,

E_{c} (t) = \int_{Ω} d E_{c} = \int_{Ω} m_{0} c \sqrt{c^{2} - v^{2}} {| ϕ |}^{2} d x d y d z .

In particular for a constant v (not varying in

(x, y, z, t)

), we obtain

E_{c} (t) = m_{0} c \sqrt{c^{2} - v^{2}} .

Hence if

v ≪ c,

we have

E_{c} (t) \approx m_{0} c^{2} .

This is the most famous Einstein equation previously published in his article of 1905.

37.2. The Kinetics energy for the general relativity context

In a general relativity theory context, the motion of a particle system will be specified by a field

(r \circ \hat{u}) : Ω \times [0, T] \to R^{4}

where

(r \circ \hat{u}) (x, t) = (c t, X_{1} (\hat{u} (x, t)), X_{2} (\hat{u} (x, t)), X_{3} (\hat{u} (x, t))),

where

\hat{u} (x, t) = (u_{0} (t), u_{1} (x, t), u_{2} (x, t), u_{3} (x, t)),

u_{0} (t) = t,

x = (x_{1}, x_{2}, x_{3}) \in Ω \subset R^{3},

and

t \in [0, T]

, where

[0, T]

is a time interval.

The corresponding density is represented by

(ρ \circ \hat{u}) : Ω \times [0, T] \to R^{+},

where

(ρ \circ \hat{u}) (x, t) = m_{0} {| ϕ (\hat{u} (x, t)) |}^{2},

m_{0}

is total system mass at rest and

ϕ : Ω \times [0, T] \to C

is a complex wave function such that

\int_{Ω} | ϕ (\hat{u} (x, t)) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x = 1, \forall t \in [0, T]

where

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

g_{j} = \frac{𝜕 r}{𝜕 u_{j}}

g_{j k} = g_{j} \cdot g_{k}, \forall j, k \in {0, 1, 2, 3} .

and

g = det {g_{j k}}

Now observe that

\begin{matrix} \frac{𝜕 r}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 t} & = & \frac{𝜕 r}{𝜕 u_{j}} \frac{𝜕 u_{j}}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 u_{k}} \frac{𝜕 u_{k}}{𝜕 t} \\ = & \frac{𝜕 r}{𝜕 u_{j}} \cdot \frac{𝜕 r}{𝜕 u_{k}} \frac{𝜕 u_{j}}{𝜕 t} \frac{𝜕 u_{k}}{𝜕 t} \\ = & g_{j k} \frac{𝜕 u_{j}}{𝜕 t} \frac{𝜕 u_{k}}{𝜕 t} . \end{matrix}

(241)

Observe that

\frac{𝜕 r}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 t} = g_{j k} \frac{𝜕 u_{j}}{𝜕 t} \frac{𝜕 u_{k}}{𝜕 t} = - c^{2} + v^{2} .

Moreover, the Kinetics energy differential is given by

d E_{c} = - d m \frac{𝜕 r}{𝜕 t} \cdot \frac{𝜕 r}{𝜕 t},

where

d m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} | ϕ (\hat{u} (x, t)) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x,

so that the total Kinetics energy is expressed by

E_{c} = \int_{0}^{T} \int_{Ω} d E_{c} d t,

that is,

\begin{matrix} E_{c} & = & \int_{0}^{T} \int_{Ω} \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} (c^{2} - v^{2}) | ϕ (\hat{u} (x, t) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x d t \\ = & \int_{0}^{T} \int_{Ω} m_{0} c \sqrt{c^{2} - v^{2}} | ϕ (\hat{u} (x, t)) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x d t \\ = & \int_{0}^{T} \int_{Ω} m_{0} c \sqrt{- g_{j k} \frac{𝜕 u_{j}}{𝜕 t} \frac{𝜕 u_{k}}{𝜕 t}} | ϕ (\hat{u} (x, t)) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x d t . \end{matrix}

(242)

Summarizing, for the general relativity theory context

E_{c} = \int_{0}^{T} \int_{Ω} m_{0} c \sqrt{- g_{j k} \frac{𝜕 u_{j}}{𝜕 t} \frac{𝜕 u_{k}}{𝜕 t}} | ϕ (\hat{u} (x, t)) |^{2} \sqrt{- g} | det {{\hat{u}}^{'} (x, t)} | d x d t .

38. About an Energy Term Related to the Manifold Curvature Variation

In this section we consider a particle system motion represented by a field

r : Ω \to R^{4}

C^{2}

class where here

Ω = \hat{Ω} \times [0, T],

\hat{Ω} \subset R^{3}

is an open, bounded and connected set, and

[0, T]

is a time interval.

More specifically, point-wise we denote

r (u) = (c t, X_{1} (u), X_{2} (u), X_{3} (u)),

where

u_{0} = t,

and

u = (u_{0}, u_{1}, u_{2}, u_{3}) \in Ω .

Now, define

g_{j} = \frac{𝜕 r (u)}{𝜕 u_{j}},

and

g_{j k} = g_{j} \cdot g_{k}, \forall j, k \in {0, 1, 2, 3} .

Moreover

{g^{j k}} = {g_{j k}}^{- 1},

and

g = det {g_{j k}} .

We assume

\{\frac{𝜕 r (u)}{𝜕 u_{j}}, for j \in {0, 1, 2, 3}\}

is a basis for

R^{4}

\forall u \in Ω .

At this point we define the Christofel symbols, denoted by

Γ_{j k}^{l}

, by

Γ_{j k}^{l} = \frac{1}{2} g^{l p} \{\frac{𝜕 g_{k p}}{𝜕 u_{j}} + \frac{𝜕 g_{j p}}{𝜕 u_{k}} - \frac{𝜕 g_{j k}}{𝜕 u_{p}}\}, \forall j, k, l \in {0, 1, 2, 3} .

Theorem 7.

Considering these last previous statements and definitions, we have that

\frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} = Γ_{j k}^{l} \frac{𝜕 r (u)}{𝜕 u_{l}}, \forall j, k \in {0, 1, 2, 3}, \forall u \in Ω .

Proof.

Fix

u \in Ω

and

j, k, m \in {0, 1, 2, 3} .

Observe that

\begin{matrix} Γ_{j k}^{l} g_{l m} & = & \frac{1}{2} g_{m l} g^{l p} \{\frac{𝜕 g_{k p}}{𝜕 u_{j}} + \frac{𝜕 g_{j p}}{𝜕 u_{k}} - \frac{𝜕 g_{j k}}{𝜕 u_{p}}\} \\ = & \frac{1}{2} δ_{m}^{p} \{\frac{𝜕 g_{k p}}{𝜕 u_{j}} + \frac{𝜕 g_{j p}}{𝜕 u_{k}} - \frac{𝜕 g_{j k}}{𝜕 u_{p}}\} \\ = & \frac{1}{2} \{\frac{𝜕 g_{k m}}{𝜕 u_{j}} + \frac{𝜕 g_{j m}}{𝜕 u_{k}} - \frac{𝜕 g_{j k}}{𝜕 u_{m}}\} \\ = & \frac{1}{2} \{\frac{𝜕}{𝜕 u_{j}} (\frac{𝜕 r (u)}{𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}}) + \frac{𝜕}{𝜕 u_{k}} (\frac{𝜕 r (u)}{𝜕 u_{j}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}}) - \frac{𝜕}{𝜕 u_{m}} (\frac{𝜕 r (u)}{𝜕 u_{j}} \cdot \frac{𝜕 r (u)}{𝜕 u_{k}})\} \\ = & \frac{1}{2} \{\frac{𝜕^{2} r (u)}{𝜕 u_{k} 𝜕 u_{j}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} + \frac{𝜕^{2} r (u)}{𝜕 u_{m} 𝜕 u_{j}} \cdot \frac{𝜕 r (u)}{𝜕 u_{k}} \\ + \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} + \frac{𝜕^{2} r (u)}{𝜕 u_{m} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{j}} \\ - \frac{𝜕^{2} r (u)}{𝜕 u_{m} 𝜕 u_{j}} \cdot \frac{𝜕 r (u)}{𝜕 u_{k}} - \frac{𝜕^{2} r (u)}{𝜕 u_{m} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{j}}\} \\ = & \frac{1}{2} \{\frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} + \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}}\} \\ = & \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} . \end{matrix}

(243)

Summarizing, we have got

Γ_{j k}^{l} \frac{𝜕 r (u)}{𝜕 u_{l}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} = Γ_{j k}^{l} g_{l m} = \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{m}} .

Since

\{\frac{𝜕 r (u)}{𝜕 u_{j}}, for j \in {0, 1, 2, 3}\},

is a basis for

R^{4}

, we may infer that

\frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} = Γ_{j k}^{l} \frac{𝜕 r (u)}{𝜕 u_{l}}, \forall j, k \in {0, 1, 2, 3}, \forall u \in Ω .

The proof is complete.

□

38.1. The energy term related to curvature variation

We define such an energy term, denoted by

E_{q}

, as

E_{q} (ϕ, r) = \frac{1}{2} \int_{Ω} g^{j k} g^{l p} \frac{𝜕}{𝜕 u_{j}} (ϕ \frac{𝜕 r (u)}{𝜕 u_{k}}) \cdot \frac{𝜕}{𝜕 u_{l}} (ϕ^{*} \frac{𝜕 r (u)}{𝜕 u_{p}}) \sqrt{- g} d u,

where

d u = d u_{1} d u_{2} d u_{3} d u_{0} .

Here

ϕ : Ω \to C

is a complex wave function representing the scalar density field.

Now observe that

\begin{matrix} \frac{𝜕}{𝜕 u_{j}} (ϕ \frac{𝜕 r (u)}{𝜕 u_{k}}) \cdot \frac{𝜕}{𝜕 u_{l}} (ϕ^{*} \frac{𝜕 r (u)}{𝜕 u_{p}}) \\ = & (\frac{𝜕 ϕ}{𝜕 u_{j}} \frac{𝜕 r (u)}{𝜕 u_{k}} + ϕ \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}}) \cdot (\frac{𝜕 ϕ^{*}}{𝜕 u_{l}} \frac{𝜕 r (u)}{𝜕 u_{p}} + ϕ^{*} \frac{𝜕^{2} r (u)}{𝜕 u_{l} 𝜕 u_{p}}) \\ = & \frac{𝜕 ϕ}{𝜕 u_{j}} \frac{𝜕 ϕ^{*}}{𝜕 u_{j}} g_{k p} + {| ϕ |}^{2} \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕^{2} r (u)}{𝜕 u_{l} 𝜕 u_{p}} \\ + ϕ \frac{𝜕 ϕ^{*}}{𝜕 u_{l}} \frac{𝜕^{2} r (u)}{𝜕 u_{j} 𝜕 u_{k}} \cdot \frac{𝜕 r (u)}{𝜕 u_{p}} \\ + ϕ^{*} \frac{𝜕 ϕ}{𝜕 u_{j}} \frac{𝜕^{2} r (u)}{𝜕 u_{l} 𝜕 u_{p}} \cdot \frac{𝜕 r (u)}{𝜕 u_{k}} \\ = & \frac{𝜕 ϕ}{𝜕 u_{j}} \frac{𝜕 ϕ^{*}}{𝜕 u_{j}} g_{k p} + {| ϕ |}^{2} Γ_{j k}^{m} Γ_{l p}^{o} g_{m o} \\ + ϕ \frac{𝜕 ϕ^{*}}{𝜕 u_{l}} Γ_{j k}^{s} g_{s p} + ϕ^{*} \frac{𝜕 ϕ}{𝜕 u_{j}} Γ_{l p}^{r} g_{r k} . \end{matrix}

(244)

From such results, we may infer that

\begin{matrix} E_{q} (ϕ, r) & = & \frac{1}{2} \int_{Ω} g^{j k} \frac{𝜕 ϕ}{𝜕 u_{j}} \frac{𝜕 ϕ^{*}}{𝜕 u_{k}} \sqrt{- g} d u \\ + \frac{1}{2} \int_{Ω} g^{j k} g^{l p} Γ_{j k}^{r} Γ_{l p}^{s} g_{r s} {| ϕ |}^{2} \sqrt{- g} d u \\ + \frac{1}{2} \int_{Ω} g^{j k} Γ_{j k}^{l} (ϕ \frac{𝜕 ϕ^{*}}{𝜕 u_{l}} + ϕ^{*} \frac{𝜕 ϕ}{𝜕 u_{l}}) \sqrt{- g} d u . \end{matrix}

(245)

39. A Note on the Definition of Temperature

The main results in this section may be found in similar form in the book [16], page 261.

Consider a system with

N = \sum_{j = 1}^{N_{0}} N_{j}

and suppose each set of

N_{j}

particles has a set of

C_{j}

possible states.

Therefore, the number of states of such

N_{j}

particles is given by

Δ Γ_{j} = \frac{{(C_{j})}^{N_{j}}}{N_{j}!},

where we have considered simple permutations as equivalent states.

Define

S_{j} = ln (Δ Γ_{j}),

and define the system entropy, denoted by S, as

S = A (\sum_{j = 1}^{N_{0}} S_{j}),

where

A > 0

is a normalizing constant.

Thus,

S = A \sum_{j = 1}^{N_{0}} ln (\frac{{(C_{j})}^{N_{j}}}{N_{j}!}),

so that

S = A (\sum_{j = 1}^{N_{0}} (N_{j} ln (C_{j}) - ln (N_{j}!))) .

N_{j}

is large enough, we have the following approximation

ln (N_{j}!) \approx N_{j} ln (N_{j}) .

In particular for

C_{j} = 1, \forall j \in {1, \dots, N_{0}}

we obtain

S = A (\sum_{j = 1}^{N_{0}} S_{j}) \approx - A (\sum_{j = 1}^{N_{0}} N_{j} ln (N_{j})),

At this point we define the following local density

{\hat{N}}_{j}

where

{\hat{N}}_{j} (x, t) = \frac{| ϕ_{j} {(x, t) |}^{2}}{{| ϕ (x, t) |}^{2}} N,

where

{| ϕ (x, t) |}^{2} = \sum_{j = 1}^{N_{0}} {| ϕ_{j} (x, t) |}^{2} .

Here,

ϕ_{j} : Ω \to C

denotes the wave function of the particles corresponding to the system part

N_{j}

The final definition of Entropy is given by

S (x, t) = A (\sum_{j = 1}^{N_{0}} S_{j} (x, t))

where

\begin{matrix} S_{j} (x, t) & = & - {\hat{N}}_{j} (x, t) ln ({\hat{N}}_{j} (x, t)) \\ = & - \frac{| ϕ_{j} {(x, t) |}^{2}}{{ϕ (x, t) |}^{2}} N ln (\frac{| ϕ_{j} {(x, t) |}^{2}}{{ϕ (x, t) |}^{2}} N) . \end{matrix}

(246)

Here we highlight the position field for each particle system part

N_{j}

is given by

{\hat{r}}_{j} (x, t) = x + r_{j} (x, t),

where

r_{j}

is related to the internal energy, that is, related to the atomic/electronic vibrational motion linked with the concept of temperature, as specified in the next lines.

The total kinetics energy is given by

E (x, t) = - \frac{1}{2} \sum_{j = 1}^{N_{0}} m_{p_{j}} {| ϕ_{j} (x, t) |}^{2} \frac{𝜕 r_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 r_{j} (x, t)}{𝜕 t} .

At this point, we define the scalar field of temperature, denoted by

T (x, t)

, such as symbolically

\frac{𝜕 S}{𝜕 E} = \frac{1}{T (x, t)} .

More specifically, we define

T (x, t) = \sum_{j = 1}^{N_{0}} \frac{\frac{𝜕 E}{𝜕 ϕ_{j}}}{\frac{𝜕 S}{𝜕 ϕ_{j}}},

so that

T (x, t) = \frac{- \frac{1}{2} \sum_{j = 1}^{N_{0}} m_{p_{j}} ϕ_{j} (x, t) \frac{𝜕 r_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 r_{j} (x, t)}{𝜕 t}}{- A \frac{ϕ_{j} N}{{| ϕ |}^{2}} ln (\frac{| ϕ_{j} |^{2} N}{{| ϕ |}^{2}} + 1)} .

39.1. A Note on Basic Thermodynamics

Consider a solid

Ω \subset R^{3}

where such a

Ω

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

𝜕 Ω .

Denoting by

[0, T]

a time interval, consider a particle system where the field of displacements is given by

r_{j} (x, t) = r (x, t) + u (x, t) + {(r_{3})}_{j} (x, t),

where

r : Ω \times [0, T] \to R

is a macroscopic displacement field,

u : Ω \times [0, T] \to R

is the elastic displacement field and

{(r_{3})}_{j} : Ω \times [0, T] \to R

denotes the displacement field related to the atomic and electronic vibration motion concerning the concept of temperature, as specified in the previous section.

In particular for the case in which

r (x, t) = x,

we define the heat functional, denoted by W, as

\begin{matrix} W & = & \frac{1}{2} \int_{0}^{T} \int_{Ω} ρ (x, t) \frac{𝜕 u (x, t)}{𝜕 t} \cdot \frac{𝜕 u (x, t)}{𝜕 t} d x d t \\ - \int_{0}^{T} \int_{Ω} F \cdot u d x d t \\ + \frac{1}{2} \int_{0}^{T} \int_{Ω} H_{i j k l} e_{i j} (u) e_{k l} (u) d x d t \\ + \frac{1}{2} \sum_{j = 1}^{N_{0}} \int_{0}^{T} \int_{Ω} m_{p_{j}} {| ϕ_{j} (x, t) |}^{2} \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} d x d t, \end{matrix}

(247)

where

ρ (x, t) = \sum_{j = 1}^{N_{0}} m_{p_{j}} {| ϕ_{j} (x, t) |}^{2}

is the point wise total density,

\frac{1}{2} \int_{0}^{T} \int_{Ω} H_{i j k l} e_{i j} (u) e_{k l} (u) d x d t

is a standard elastic inner energy for small displacements

u,

F (x, t)

is the resulting field of external forces acting point wise on

Ω

, and for the term

\frac{1}{2} \sum_{j = 1}^{N_{0}} \int_{0}^{T} \int_{Ω} m_{p_{j}} {| ϕ_{j} (x, t) |}^{2} \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} d x d t

we are refereing to the definitions and notations of the previous section.

At this point we denote

E_{i n} = \frac{1}{2} \sum_{j = 1}^{N_{0}} \int_{0}^{T} \int_{Ω} m_{p_{j}} {| ϕ_{j} (x, t) |}^{2} \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} d x d t,

and

\begin{matrix} E_{T} & = & \frac{1}{2} \int_{0}^{T} \int_{Ω} ρ (x, t) \frac{𝜕 u (x, t)}{𝜕 t} \cdot \frac{𝜕 u (x, t)}{𝜕 t} d x d t \\ - \int_{0}^{T} \int_{Ω} F \cdot u d x d t \\ + \frac{1}{2} \int_{0}^{T} \int_{Ω} H_{i j k l} e_{i j} (u) e_{k l} (u) d x d t . \end{matrix}

(248)

Hence

W = E_{T} + E_{i n}

and from the previous section we may generically denote

δ E_{i n} = T δ S,

Therefore

δ W = δ E_{T} + δ E_{i n} = δ E_{T} + T δ S .

For a standard reversible process we must have

δ E_{T} = 0

so that

δ W = T δ S .

For a general case in which other types of internal energy (such as

E_{q}

indicated in the previous sections and even

E_{i n}

) are partially and irreversibly converted into a

E_{T}

type of energy, in which

δ E_{T} \neq 0,

we may have

δ W < T δ S .

Remark 22.

Indeed, in general the vibrational motion related to

E_{i n}

is of relativistic nature so that in fact we would need to consider

E_{i n} = \frac{1}{2} \sum_{j = 1}^{N_{0}} \int_{0}^{T} \int_{Ω} m_{p_{j}} c {| ϕ_{j} (x, t) |}^{2} \sqrt{c^{2} - \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t} \cdot \frac{𝜕 {(r_{3})}_{j} (x, t)}{𝜕 t}} \sqrt{- g_{j}} d x d t .

40. A Formal Proof of Castigliano Theorem

In this section we present the mathematical formalism of a result in elasticity theory known as the Castigliano’s Theorem.

Let

Ω \subset R^{3}

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

𝜕 Ω .

In a context of linear elasticity, consider the functional

J : V \to R

where

J (u) = E_{i n} - {〈 u_{i}, f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}) P_{i j},

u = (u_{1}, u_{2}, u_{3}) \in W_{0}^{1, 2} (Ω; R^{3}) \equiv V,

f = (f_{1}, f_{2}, f_{3}) \in L^{2} (Ω; R^{3})

Y = Y^{*} = L^{2} (Ω; R^{3}),

and

P_{i j} \in R, \forall i \in {1, 2, 3}, j \in {1, \dots, N}

for some

N \in N .

Here we have denoted

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

e_{i j} (u) = \frac{1}{2} (\frac{𝜕 u_{i}}{𝜕 x_{j}} + \frac{𝜕 u_{j}}{𝜕 x_{i}}) .

Moreover

H_{i j k l}

is a fourth order positive definite and constant tensor.

Observe that the variation of J in

u_{i}

give us the following Euler-Lagrange equation

- {(H_{i j k l} e_{k l} (u))}_{, j} - f_{i} - \sum_{j = 1}^{N} P_{i j} δ (x_{j}) = 0, in Ω .

(249)

Symbolically such a system stands for

\frac{𝜕 J (u)}{𝜕 u_{i}} = 0, \forall i \in {1, 2, 3},

so that

\frac{𝜕 (E_{i n} - {〈 u_{i}, f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}) P_{i j})}{𝜕 u_{i}} = 0, \forall i \in {1, 2, 3} .

(250)

We denote

u \in V

solution of (249) by

u = u (f, P),

so that multiplying the concerning extremal equation by

u_{i}

and integrating by parts, we get

\begin{matrix} H_{1} (u (f, P), f, P) & = & 2 E_{i n} (u (f, P)) - {〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}, f, P) P_{i j} \\ = & 0, \forall f \in Y^{*}, P \in R^{3 N} . \end{matrix}

(251)

Therefore

\frac{d}{d P_{i j}} (H_{1} (u (f, P), f, P)) = 0,

so that

2 \frac{d E_{i n}}{d P_{i j}} - \frac{d}{d P_{i j}} ({〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} + \sum_{j = 1}^{N} u_{i} (x_{j}, f, p) P_{i j} 〉_{L^{2}}) = 0,

that is

\begin{matrix} \frac{d E_{i n}}{d P_{i j}} + {〈\frac{𝜕 (E_{i n} - {〈 u_{i}, f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}) P_{i j})}{𝜕 u_{k}}, \frac{𝜕 u_{k}}{𝜕 P_{i j}}〉}_{L^{2}} \\ - \frac{𝜕}{𝜕 P_{i j}} ({〈 u_{i}, f_{i} 〉}_{L^{2}} + \sum_{j = 1}^{N} u_{i} (x_{j}) P_{i j}) \\ = & 0 . \end{matrix}

(252)

From this and (249) we obtain

\frac{d E_{i n}}{d P_{i j}} - u_{i} (x_{j}) = 0,

so that

u_{i} (x_{j}) = \frac{d E_{i n}}{d P_{i j}} = \frac{d}{d P_{i j}} (\frac{1}{2} \int_{Ω} H_{i j k l} e_{i j} (u (f, P)) e_{k l} (u (f, P)) d x),

\forall i \in {1, 2, 3}, \forall j \in {1, \dots, N} .

With such results in mind, we have proven the following theorem.

Theorem 8

(Castigliano). Considering the notations and definitions in this section, we have

u_{i} (x_{j}) = \frac{d E_{i n}}{d P_{i j}} = \frac{d}{d P_{i j}} (\frac{1}{2} \int_{Ω} H_{i j k l} e_{i j} (u (f, P)) e_{k l} (u (f, P)) d x),

\forall i \in {1, 2, 3}, \forall j \in {1, \dots, N} .

40.1. A Generalization of Castigliano Theorem

In this subsection we present a more general version of the Castigliano theorem.

Considering the context of last section, we recall that

\begin{matrix} H_{1} (u (f, P), f, P) & = & 2 E_{i n} (u (f, P)) - {〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}, f, P) P_{i j} \\ = & 0, \forall f \in Y^{*}, P \in R^{3 N} . \end{matrix}

(253)

Therefore, for

x_{k} \in Ω

such that

x_{k} \neq x_{j}, \forall j \in {1, \dots, N},

we have

{〈\frac{d}{d f_{i}} (H_{1} (u (f, P), f, P)), δ (x_{k})〉}_{L^{2}} = 0,

so that

\begin{matrix} 2 {〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}} \\ - {〈\frac{d}{d f_{i}} ({〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} + \sum_{j = 1}^{N} u_{i} (x_{j}, f, p) P_{i j} 〉_{L^{2}}), δ (x_{k})〉}_{L^{2}} \\ = & 0, \end{matrix}

(254)

that is

\begin{matrix} {〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}} \\ + {〈\frac{d}{d u_{k}} (E_{i n} (u (f, P)) - {〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}, f, p) P_{i j}) \frac{d u_{k}}{d f_{i}}, δ (x_{k})〉}_{L^{2}} \\ - {〈\frac{𝜕}{𝜕 f_{i}} ({〈 u_{i} (f, P), f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}, f, p) P_{i j}), δ x (x_{k})〉}_{L^{2}} \\ = & 0 . \end{matrix}

(255)

From such results, we may obtain

{〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}} - {〈 u_{i} (x), δ (x_{k}) 〉}_{L^{2}} = 0,

so that

{〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}} - u_{i} (x_{k}) = 0,

that is

u_{i} (x_{k}) = {〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}},

\forall i \in {1, 2, 3}, \forall x_{k} \in Ω such that x_{k} \neq x_{j}, \forall j \in {1, \dots, N} .

With such results in mind, we have proven the following theorem.

Theorem 9

(The Generalized Castigliano Theorem). Considering the notations and definitions in this section, we have

u_{i} (x_{k}) = {〈\frac{d}{d f_{i}} (E_{i n} (u (f, P))), δ (x_{k})〉}_{L^{2}},

\forall i \in {1, 2, 3}, \forall x_{k} \in Ω such that x_{k} \neq x_{j}, \forall j \in {1, \dots, N} .

40.2. The Virtual Work Principle

Considering the definitions, results and statements of the previous section and subsection, we may easily prove the following theorem.

Theorem 10

(The virtual work principle). Let

x_{l} \in Ω

such that

x_{l} \neq x_{j}, \forall j \in {1, \dots, N} .

For a virtual constant load

P_{l k} \in R

x_{l}

at the direction of

u_{k} (x_{l}),

define now

J : V \to R

where

J (u) = E_{i n} - {〈 u_{i}, f_{i} 〉}_{L^{2}} - \sum_{j = 1}^{N} u_{i} (x_{j}) P_{i j} - P_{l k} u_{k} (x_{l}) .

Under such hypotheses,

u_{k} (x_{l}) = {(\frac{d E_{i n} (u (f, P, P_{l k}))}{d P_{l k}})}_{P_{l k} = 0},

\forall k \in {1, 2, 3}, \forall x_{l} \in Ω such that x_{l} \neq x_{j}, \forall j \in {1, \dots, N} .

Proof.

The proof is exactly the same as in the Castigliano Theorem in the previous section except by setting the virtual load

P_{l k} = 0

in the end of this calculation and will not be repeated. □

41. Duality for a General Relaxed Primal Variational Formulation

Let

Ω \subset R^{3}

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

𝜕 Ω .

Consider a functional

J : V \to R

where

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}},

where

V = W_{0}^{1, 2} (Ω),

γ > 0,

α > 0

β > 0

Y = Y^{*} = L^{2} (Ω)

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{3})

, and

f \in L^{2} (Ω) .

We define the associated relaxed functional

J_{1} : V \times V \times (0, 1),

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

(256)

Moreover, we define,

F_{1} : V \times V \times (0, 1) \to R,

F_{2} : V \times V \times (0, 1) \to R,

F_{3} : V \times V \times (0, 1) \to R,

F_{4} : V \times V \times (0, 1) \to R,

F_{5} : V \times V \times (0, 1) \to R,

and

F_{6} : V \times V \times (0, 1) \to R,

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

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

F_{3} (u, ϕ, λ) = \frac{λ α}{2} \int_{Ω} {({(u - (1 - λ) ϕ)}^{2} - β)}^{2} d x,

F_{4} (u, ϕ, λ) = \frac{(1 - λ) α}{2} \int_{Ω} {({(u + λ ϕ)}^{2} - β)}^{2} d x,

F_{5} (u, ϕ, λ) = - λ {〈 u - (1 - λ) ϕ, f 〉}_{L^{2}},

F_{6} (u, ϕ, λ) = - (1 - λ) {〈 u + λ ϕ, f 〉}_{L^{2}},

respectively.

Observe that

\begin{matrix} J_{1} (u, ϕ, u) & = & F_{1} (u, ϕ, λ) + F_{2} (u, ϕ, λ) \\ F_{3} (u, ϕ, λ) + F_{4} (u, ϕ, λ) \\ F_{5} (u, ϕ, λ) + F_{6} (u, ϕ, λ), \end{matrix}

(257)

Thus,

\begin{matrix} J_{1} (u, ϕ, u) & \geq & F_{1} (u, ϕ, λ) + F_{2} (u, ϕ, λ) \\ + {〈 {(u - (1 - λ) ϕ)}^{2} - β, v_{3}^{*} 〉}_{L^{2}} \\ + {〈 {(u + λ ϕ)}^{2} - β, v_{4}^{*} 〉}_{L^{2}} \\ F_{5} (u, ϕ, λ) + F_{6} (u, ϕ, λ) \\ F_{3} (u, ϕ, λ) + F_{4} (u, ϕ, λ) \\ + inf_{v_{3} \in Y} {- {〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} + {\tilde{F}}_{3} (v_{3}, λ)} \\ + inf_{v_{4} \in Y} {- {〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} + {\tilde{F}}_{4} (v_{4}, λ)} \end{matrix}

(258)

where

{\tilde{F}}_{3} (v_{3}, λ) = \frac{λ α}{2} \int_{Ω} v_{3}^{2} d x,

{\tilde{F}}_{4} (v_{4}, λ) = \frac{(1 - λ) α}{2} \int_{Ω} v_{4}^{2} d x,

Therefore, defining

{\tilde{F}}_{3}^{*} : Y^{*} \times (0, 1) \to R

and

{\tilde{F}}_{4}^{*} : Y^{*} \times (0, 1) \to R

\begin{matrix} {\tilde{F}}_{3}^{*} (v_{3}^{*}, λ) & = & sup_{v_{3} \in Y} {{〈 v_{3} . v_{3}^{*} 〉}_{L^{2}} - F_{3} (v_{3}, λ) \\ = & \frac{1}{2 α λ} \int_{Ω} {(v_{3}^{*})}^{2} d x, \end{matrix}

(259)

and

\begin{matrix} {\tilde{F}}_{4}^{*} (v_{4}^{*}, λ) & = & sup_{v_{4} \in Y} {{〈 v_{4} . v_{4}^{*} 〉}_{L^{2}} - F_{4} (v_{4}, λ) \\ = & \frac{1}{2 α (1 - λ)} \int_{Ω} {(v_{4}^{*})}^{2} d x, \end{matrix}

(260)

we may also infer that

\begin{matrix} J_{1} (u, ϕ, λ) & \geq & inf_{v_{1} \in Y_{1}} {{〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} + {\tilde{F}}_{1} (v_{1}, λ)} \\ + inf_{v_{2} \in Y_{1}} {{〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} + {\tilde{F}}_{2} (v_{2}, λ)} \\ + inf_{v_{5} \in Y} \{- {〈 v_{5}, div v_{1}^{*} 〉}_{L^{2}} + \int_{Ω} (v_{5}^{2} - β) v_{3}^{*} d x - λ {〈 v_{5}, f 〉}_{L^{2}}\} \\ + inf_{v_{6} \in Y} \{- {〈 v_{6}, div v_{2}^{*} 〉}_{L^{2}} + \int_{Ω} (v_{6}^{2} - β) v_{4}^{*} d x - (1 - λ) {〈 v_{6}, f 〉}_{L^{2}}\} \\ - {\tilde{F}}_{3}^{*} (v_{3}^{*}, λ) - {\tilde{F}}_{4}^{*} (v_{4}^{*}, λ) \\ = & - {\tilde{F}}_{1}^{*} (v_{1}^{*}, λ) - {\tilde{F}}_{2}^{*} (v_{2}^{*}, λ) \\ - F_{5}^{*} (v_{1}^{*}, v_{3}^{*}, λ) - F_{6}^{*} (v_{2}^{*}, v_{4}^{*}, λ) \\ - {\tilde{F}}_{3}^{*} (v_{3}^{*}, λ) - {\tilde{F}}_{4}^{*} (v_{4}^{*}, λ), \end{matrix}

(261)

v^{*} = (v_{1}^{*}, \dots, v_{4}^{*}) \in A^{*}

where,

A^{*} = {v^{*} \in {[Y_{1}^{*}]}^{2} \times {[Y^{*}]}^{2} : v_{3}^{*} > 0 and v_{4}^{*} > 0, in Ω},

{\tilde{F}}_{1} (v_{1}, λ) = \frac{λ γ}{2} \int_{Ω} v_{1} \cdot v_{1} d x,

{\tilde{F}}_{2} (v_{2}, λ) = \frac{λ γ}{2} \int_{Ω} v_{2} \cdot v_{2} d x,

{\tilde{F}}_{5} (v_{5}, v_{3}^{*}, λ) = \int_{Ω} (v_{5}^{2} - β) v_{3}^{*} d x - λ {〈 v_{5}, f 〉}_{L^{2}},

{\tilde{F}}_{6} (v_{6}, v_{4}^{*}, λ) = \int_{Ω} (v_{6}^{2} - β) v_{4}^{*} d x - (1 - λ) {〈 v_{6}, f 〉}_{L^{2}},

and

\begin{matrix} {\tilde{F}}_{1}^{*} (v_{1}^{*}, λ) & = & sup_{v_{1} \in Y_{1}} {{〈 v_{1}, v_{1}^{*} 〉}_{L^{2}} - {\tilde{F}}_{1} (v_{1}, λ)} \\ = & \frac{1}{2 γ λ} \int_{Ω} v_{1}^{*} \cdot v_{1}^{*} d x, \end{matrix}

(262)

\begin{matrix} {\tilde{F}}_{2}^{*} (v_{2}^{*}, λ) & = & sup_{v_{2} \in Y_{1}} {{〈 v_{2}, v_{2}^{*} 〉}_{L^{2}} - {\tilde{F}}_{2} (v_{2}, λ)} \\ = & \frac{1}{2 γ (1 - λ)} \int_{Ω} v_{2}^{*} \cdot v_{2}^{*} d x, \end{matrix}

(263)

\begin{matrix} {\tilde{F}}_{5}^{*} (v_{1}^{*}, v_{3}^{*}, λ) & = & sup_{v_{5} \in Y} {{〈 v_{5}, v_{1}^{*} 〉}_{L^{2}} - {\tilde{F}}_{5} (v_{5}, v_{3}^{*}, λ)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(div v_{1}^{*} + λ f)}^{2}}{4 v_{3}^{*}} d x + β \int_{Ω} v_{3}^{*} d x, \end{matrix}

(264)

and

\begin{matrix} {\tilde{F}}_{6}^{*} (v_{2}^{*}, v_{4}^{*}, λ) & = & sup_{v_{6} \in Y} {{〈 v_{6}, v_{1}^{*} 〉}_{L^{2}} - {\tilde{F}}_{6} (v_{6}, v_{4}^{*}, λ)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(div v_{2}^{*} + (1 - λ) f)}^{2}}{4 v_{4}^{*}} d x + β \int_{Ω} v_{4}^{*} d x . \end{matrix}

(265)

Denoting, as above indicated,

v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y_{1}^{*}]}^{2} \times {[Y^{*}]}^{2}

, we define

J^{*} : {[Y_{1}^{*}]}^{2} \times {[Y^{*}]}^{2} \times (0, 1) \to R

\begin{matrix} J^{*} (v^{*}, λ) & = & - {\tilde{F}}_{1}^{*} (v_{1}^{*}, λ) - {\tilde{F}}_{2}^{*} (v_{2}^{*}, λ) \\ - F_{5}^{*} (v_{1}^{*}, v_{3}^{*}, λ) - F_{6}^{*} (v_{2}^{*}, v_{4}^{*}, λ) \\ - {\tilde{F}}_{3}^{*} (v_{3}^{*}, λ) - {\tilde{F}}_{4}^{*} (v_{4}^{*}, λ), \end{matrix}

(266)

Observe that we have got

\begin{matrix} inf_{u \in V} J (u) & \geq & inf_{(u, ϕ, λ) \in V \times V \times [0, 1]} J_{1} (u, ϕ, λ) \\ \geq & inf_{λ \in (0, 1)} \{sup_{v^{*} \in A^{*}} J^{*} (v^{*}, λ)\} . \end{matrix}

(267)

41.1. A Numerical Example

We have obtained numerical results for

γ = 0.1

α = 3.0

β = 5.0

and

f \equiv 10, in Ω,

for the special case in which

Ω = [0, 1] \subset R .

Such results have been performed through the following algorithm:

Set $n = 1$ and $λ_{n} = 1 / 2$ .
Calculate $v_{n}^{*} \in A^{*}$ such that

$J^{*} (v_{n}^{*}, λ_{n}) = sup_{v^{*} \in A^{*}} J^{*} (v^{*}, λ_{n}),$
Calculate $λ_{n + 1} \in (0, 1)$ such that

$J^{*} (v_{n}^{*}, λ_{n + 1}) = inf_{λ \in (0, 1)} J^{*} (v_{n}^{*}, λ),$
Set $n : = n + 1$ and go to step 2 until the satisfaction of an appropriate convergence criterion.

Here, we recall that for the optimal points

\frac{div v_{1}^{*} + λ f}{2 v_{3}^{*}} = u - (1 - λ) ϕ,

and

\frac{div v_{2}^{*} + (1 - λ) f}{2 v_{4}^{*}} = u + λ ϕ,

so that

u = λ (\frac{div v_{1}^{*} + λ f}{2 v_{3}^{*}}) + (1 - λ) (\frac{div v_{2}^{*} + (1 - λ) f}{2 v_{4}^{*}}) .

For such a corresponding optimal

u_{0}

please see Figure 32.

For the solution

u_{1}

of the primal problem obtained through the generalized method of lines, please see Figure 33.

We may observe the solutions

u_{0}

and

u_{1}

are qualitatively similar, as expected.

Here we present the software developed to perform such numerical results.

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

clear all

global m8 d L A3 A B yo u v e1 dv1 dv2 dv3 v5 v6 v3 v4 v1 v2 K5 e5 L1 L2 L3

m8=100;

d=1/m8;

e1=0.00001;

e5=0.001;

K5=10000.0;

A3=0.1;

A=3.0;

B=5.0;

for i=1:m8

uo(i,1)=5;

yo(i,1)=10.0;

end;

L=1/2;

for k=1:50

k

i=1;

m12= $2 + 6 * A * u o {(i, 1)}^{2} * d^{2} / A 3 - 2 * A * B / A 3 * d^{2};$

m50(i)=1/m12;

z(i)= $m 50 (i) * (y o (i, 1) * d^{2} / A 3 + 4 * A * u o {(i, 1)}^{3} * d^{2} / A 3);$

for i=2:m8-1

m12= $2 + 6 * A * u o {(i, 1)}^{2} * d^{2} / A 3 - 2 * A * B / A 3 * d^{2} - m 50 (i - 1);$

m50(i)=1/m12;

z(i)= $m 50 (i) * (y o (i, 1) * d^{2} / A 3 + 4 * A * u o {(i, 1)}^{3} * d^{2} / A 3 + z (i - 1));$

end;

w(m8,1)=0;

for i=1:m8-1

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

end;

uo=w;

uo(m8/2,1)

end;

for i=1:4*m8

xo(i,1)=3.0;

end;

for i=1:1

x1(i,1)=1/2;

end;

for k1=1:10

k1

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 50)$

k

k=k+1;

X=fminunc(’funFeb30LG’,xo);

b12=max(abs(xo-X))

xo=X;

end;

X1=fminunc(’funFeb31LG’,x1);

x1=X1;

end;

u(m8,1)=0;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

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

With the auxiliary function "funFeb30LG", where

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

function S=funFeb30LG(x)

global m8 d L A3 A B yo u v e1 dv2 dv1 dv3 v3 v4 v5 v6 v1 v2 K5 e5 L1 L2 L3

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(m8+i,1);

v3(i,1)=x(2*m8+i,1);

v4(i,1)=x(3*m8+i,1);

end; for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+ $(y o {(i, 1)}^{2} * L^{2} + 2 * y o (i, 1) * L * d v 1 (i, 1) + d v 1 {(i, 1)}^{2} + 4 * B * v 3 {(i, 1)}^{4}) / (4 * v 3 {(i, 1)}^{2});$

S=S+ $(y o {(i, 1)}^{2} * {(1 - L)}^{2} + 2 * y o (i, 1) * (1 - L) * d v 2 (i, 1) + d v 2 {(i, 1)}^{2} + 4 * B * v 4 {(i, 1)}^{4}) / (4 * v 4 {(i, 1)}^{2});$

S=S+ $v 1 {(i, 1)}^{2} / s q r t (L^{2} + e 1) / 2 / A 3 + v 2 {(i, 1)}^{2} / s q r t ({(1 - L)}^{2} + e 1) / 2 / A 3;$

S=S+ $v 3 {(i, 1)}^{4} / 2 / s q r t (L^{2} + e 1) / A + v 4 {(i, 1)}^{4} / 2 / s q r t ({(1 - L)}^{2} + e 1) / A;$

end;

for i=1:m8-1

u(i,1)= $L * (y o (i, 1) * L + d v 1 (i, 1)) / (v 3 {(i, 1)}^{2}) / 2;$

u(i,1)=u(i,1)+ $(1 - L) * ((1 - L) * y o (i, 1) + d v 2 (i, 1)) / 2 / (v 4 {(i, 1)}^{2});$

end;

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

Finally, we present the auxiliary function "funFeb31LG"

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

function S1=funFeb31LG(x)

global m8 d L L1 L2 L3 A3 A B yo u v e1 dv2 dv1 dv3 v5 v6 v3 v4 v1 v2 K5 e5

L=(sin(x(1,1))+1)/2;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+ $(y o {(i, 1)}^{2} * L^{2} + 2 * y o (i, 1) * L * d v 1 (i, 1) + d v 1 {(i, 1)}^{2} + 4 * B * v 3 {(i, 1)}^{4}) / (4 * v 3 {(i, 1)}^{2});$

S=S+ $(y o {(i, 1)}^{2} * {(1 - L)}^{2} + 2 * y o (i, 1) * (1 - L) * d v 2 (i, 1) + d v 2 {(i, 1)}^{2} + 4 * B * v 4 {(i, 1)}^{4}) / (4 * v 4 {(i, 1)}^{2});$

S=S+ $v 1 {(i, 1)}^{2} / s q r t (L^{2} + e 1) / 2 / A 3 + v 2 {(i, 1)}^{2} / s q r t ({(1 - L)}^{2} + e 1) / 2 / A 3;$

S=S+ $v 3 {(i, 1)}^{4} / 2 / s q r t (L^{2} + e 1) / A + v 4 {(i, 1)}^{4} / 2 / s q r t ({(1 - L)}^{2} + e 1) / A;$

end;

S1=-S;

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

Remark 23.

Observe that the functional

J^{*}

is convex in

A^{*}

however, the restrictions

v_{3}^{*} > 0

and

v_{4}^{*} > 0

in Ω may cause a difference between the solution obtained through

J^{*}

and the solution got through the primal formulation J, a so-called duality gap.

Anyway, through such a relaxation process, utilizing the dual functional

J^{*}

we may still obtain a good qualitative approximation of the global optimal point for the primal formulation J.

Indeed, such a global solution obtained through the dual functional

J^{*}

may be an excellent initial solution for obtaining a more accurate one through the standard Newton Method, for example.

42. A Global Existence Result for 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

𝜕 Ω = S

Define a functional

J : V \to R

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

where

γ_{i j} (u) = \frac{u_{i, j} + u_{j, i}}{2} + \frac{1}{2} u_{m, i} u_{m, j},

V = {u \in W^{1, 2} (Ω; R^{3}) : u = {\hat{v}}_{0} on S_{1} \subset 𝜕 Ω} .

We also denote

Y = Y^{*} = L^{2} (Ω; R^{3}),

so that

f = (f_{1}, f_{2}, f_{3}) \in Y .

Here

{H_{i j k l}}

is a fourth order constant, positive definite and symmetric tensor.

With such assumptions and statements in mind, we may prove the following theorem.

Theorem 11.

Assume

{H_{i j k l}}

is such that

lim_{{∥ u ∥}_{V} \to \infty} J (u) = + \infty .

Under such hypothesis, there exists

u_{0} \in V

such that

J (u_{0}) = min_{u \in V} J (u) .

Proof.

From the hypotheses, there exists

α \in R

such that

α = inf_{u \in V} J (u) .

Let

{u_{n}} \subset V

be a sequence such that

α \leq J (u_{n}) < α + \frac{1}{n}, \forall n \in N .

Suppose, to obtain contradiction, there exists a subsequence

{n_{k}} \subset N,

such that

∥ u_{n_{k}} ∥_{V} \to \infty .

From the hypotheses, we have

J (u_{n_{k}}) \to + \infty, as k \to \infty .

This contradicts

lim_{k \to \infty} J (u_{n_{k}}) = α \in R .

From such results we may infer that there exists

K > 0

such that

∥ u_{n} ∥_{V} \leq K, \forall n \in N .

Consequently, from this, the Sobolev Embedding and Rellich Kondrashov theorems, there exists

u_{0} \in V \cap L^{\infty} (Ω; R^{3})

for which, up to a not relabelled subsequence, we have

u_{n} ⇀ u_{0}, weakly in W^{1, 4} (Ω; R^{3}),

u_{n} \to u_{0}, strongly in L^{4} (Ω),

u_{n} \to u_{0}, strongly in L^{\infty} (Ω; R^{3}) .

Let

φ \in C_{c}^{\infty} (Ω) .

Thus,

\begin{matrix} |{〈\frac{𝜕 {(u_{n})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}}, φ〉}_{L^{2}}| \\ = & |{〈{(u_{n})}_{i} - {(u_{0})}_{i}, \frac{𝜕 φ}{𝜕 x_{j}}〉}_{L^{2}}| \\ \leq & ∥ {(u_{n})}_{i} - {(u_{0})}_{i} ∥_{\infty} {∥\frac{𝜕 φ}{𝜕 x_{j}}∥}_{1} \\ \to & 0, as n \to \infty . \end{matrix}

(268)

Since

φ \in C_{c}^{\infty} (Ω)

is arbitrary and

C_{c}^{\infty} (Ω)

is dense in

L^{4} (Ω)

we may infer that

\frac{𝜕 {(u_{n})}_{i}}{𝜕 x_{j}} ⇀ \frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}}, weakly in L^{4} (Ω),

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

Define

W = L^{4} (Ω)

with the norm

{∥ v ∥}_{W} {= sup {〈 v, φ 〉}_{L^{2}}, φ \in C_{c}^{\infty} {(Ω), ∥ φ ∥}_{1, 2} \leq 1} .

We may easily verify that

\frac{𝜕 {(u_{n})}_{i}}{𝜕 x_{j}} \to \frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}}, strongly in W,

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

Thus,

\{\frac{𝜕 {(u_{n})}_{i}}{𝜕 x_{j}}\}

is a Cauchy sequence in

W .

Hence, for each

n \in N

there exists

n_{k} \in N

such that

m, l \geq n_{k}

, then

{∥\frac{𝜕 {(u_{m})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{l})}_{i}}{𝜕 x_{j}}∥}_{W} < \frac{1}{k^{2}} .

where

n_{k}

may be taken as an increasing subsequence in

N .

In particular, we have got

{∥\frac{𝜕 {(u_{n_{k + 1}})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{n_{k}})}_{i}}{𝜕 x_{j}}∥}_{W} < \frac{1}{k^{2}} .

Define now

g_{l} = |\frac{𝜕 {(u_{n_{1}})}_{i}}{𝜕 x_{j}}| + \sum_{k = 1}^{l - 1} |\frac{𝜕 {(u_{n_{k + 1}})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{n_{k}})}_{i}}{𝜕 x_{j}}|,

and

g = |\frac{𝜕 {(u_{n_{1}})}_{i}}{𝜕 x_{j}}| + \sum_{k = 1}^{\infty} |\frac{𝜕 {(u_{n_{k + 1}})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{n_{k}})}_{i}}{𝜕 x_{j}}| .

Observe that

\begin{matrix} {∥ g ∥}_{W} & \leq & {∥\frac{𝜕 {(u_{n_{1}})}_{i}}{𝜕 x_{j}}∥}_{W} + \sum_{k = 1}^{\infty} {∥\frac{𝜕 {(u_{n_{k + 1}})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{n_{k}})}_{i}}{𝜕 x_{j}}∥}_{W} \\ \leq & {∥\frac{𝜕 {(u_{n_{1}})}_{i}}{𝜕 x_{j}}∥}_{W} + \sum_{k = 1}^{\infty} \frac{1}{k^{2}} \\ < & + \infty . \end{matrix}

(269)

From such results we may infer that

g (x) \in R, a . e . in Ω .

Moreover, since an absolutely convergent series is also convergent, we may infer that

\frac{𝜕 {(u_{n_{l}})}_{i}}{𝜕 x_{j}} = \frac{𝜕 {(u_{n_{1}})}_{i}}{𝜕 x_{j}} + \sum_{k = 1}^{l - 1} (\frac{𝜕 {(u_{n_{k + 1}})}_{i}}{𝜕 x_{j}} - \frac{𝜕 {(u_{n_{k}})}_{i}}{𝜕 x_{j}}) \to h_{i j}, a . e . in Ω,

for some

h_{i j} \in L^{4} (Ω), \forall i, j \in {1, 2, 3} .

From such results, we have

\frac{𝜕 {(u_{n_{l}})}_{i}}{𝜕 x_{j}} \to h_{i j}, a . e . in Ω

and

\frac{𝜕 {(u_{n_{l}})}_{i}}{𝜕 x_{j}} ⇀ \frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}}, weakly in L^{4} (Ω),

so that

\frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}} = h_{i j}, a . e . in Ω .

Consequently, we have got

\frac{𝜕 {(u_{n_{l}})}_{i}}{𝜕 x_{j}} \to \frac{𝜕 {(u_{0})}_{i}}{𝜕 x_{j}}, a . e . in Ω .

Now fix

i, j, m \in {1, 2, 3} .

Observe that from the Cauchy-Schwarz inequality, we have

\begin{matrix} \int_{Ω} {(\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}} \frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}})}^{2} d x \\ \leq & {∥\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{i}}∥}_{4}^{2} {∥\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}}∥}_{4}^{2} \\ \leq & K_{1}, \forall l \in N \end{matrix}

(270)

for some appropriate real constant

K_{1} > 0

Therefore, up to a not relabeled subsequence there exists

v_{0} \in L^{2} (Ω)

such that

\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}} ⇀ v_{0}, weakly in L^{2} (Ω),

Since

\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}} \to \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{j}}, a . e . in Ω,

we obtain

v_{0} = \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{j}}, a . e . in Ω,

so that

\frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{n_{l}})}_{m}}{𝜕 x_{j}} ⇀ \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{i}} \frac{𝜕 {(u_{0})}_{m}}{𝜕 x_{j}}, weakly in L^{2} (Ω),

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

Therefore, from such results we may infer that

γ_{i j} (u_{n_{l}}) ⇀ γ_{i j} (u_{0}), weakly in L^{2} (Ω), \forall i, j \in {1, 2, 3} .

Moreover, since J is convex in

{γ_{i j}}

we finally obtain

α = \underset{l \to \infty}{lim inf} J (u_{n_{l}}) \geq J (u_{0}),

so that

J (u_{0}) = min_{u \in V} J (u) .

The proof is complete.

□

43. A Note on a General Relaxation Procedure for the Vectorial Case in the Calculus of Variation

Let

Ω \subset R^{n}

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

𝜕 Ω .

Consider a continuous and bounded below functional

F : V \to R

where

V = {u \in W^{1, 2} (Ω; R^{N}) : u = u_{0} on 𝜕 Ω} .

Define

H_{1} : V \to R

H_{1} (u) = inf {λ_{1} F (v_{1}) + (1 - λ_{1}) F (w_{1}); 0 \leq λ_{1} \leq 1, v_{1}, w_{1} \in V, λ_{1} v_{1} + (1 - λ_{1}) w_{1} = u} .

Observe that as it has been shown in a previous section, we have

F^{* *} (u) \leq H_{1} (u) \leq F (u), \forall u \in V .

Moreover, also as indicated in a previous section, we may obtain

H_{1} (u) = inf_{(ϕ_{1}, λ_{1}) \in V_{0} \times [0, 1]} \{λ_{1} F (u - (1 - λ_{1}) ϕ_{1}) + (1 - λ_{1}) F (u + λ_{1} ϕ_{1})\},

where

V_{0} = W_{0}^{1, 2} (Ω; R^{N}) .

Reasoning inductively, having

H_{k} : V \to R

, define

H_{k + 1} : V \to R

\begin{matrix} H_{k + 1} (u) & = & inf {λ_{k + 1} H_{k} (v_{k + 1}) + (1 - λ_{k + 1}) H_{k} (w_{k + 1}); \\ 0 \leq λ_{k + 1} \leq 1, v_{k + 1}, w_{k + 1} \in V, λ_{k + 1} v_{k + 1} + (1 - λ_{k + 1}) w_{k + 1} = u} . \end{matrix}

(271)

Thus

H_{k + 1} (u) = inf_{(ϕ_{k + 1}, λ_{k + 1}) \in V_{0} \times [0, 1]} \{λ_{k + 1} H_{k} (u - (1 - λ_{k + 1}) ϕ_{k + 1}) + (1 - λ_{k + 1}) H_{k} (u + λ_{k + 1} ϕ_{k + 1})\} .

Observe that

F^{* *} (u) \leq H_{k + 1} (u) \leq H_{k} (u) \leq F (u), \forall k \in N .

Define

H_{0} : V \to R

H_{0} (u) = lim_{k \to + \infty} H_{k} (u) = inf_{k \in N} H_{k} (u), \forall u \in V .

Suppose, to obtain contradiction, that

H_{0}

is not convex.

Hence, there exists

\hat{u} \in V

such that

{(H_{0})}_{1} (\hat{u}) < H_{0} (\hat{u}),

where

{(H_{0})}_{1} (u) = inf {λ_{1} H_{0} (v_{1}) + (1 - λ_{1}) H_{0} (w_{1}); 0 \leq λ_{1} \leq 1, v_{1}, w_{1} \in V, λ_{1} v_{1} + (1 - λ_{1}) w_{1} = u} .

This contradicts

H_{0} (u) = lim_{k \to + \infty} H_{k} (u) = inf_{k \in N} H_{k} (u), \forall u \in V .

Therefore

H_{0}

is convex on V so that from this and

F^{* *} (u) \leq H_{0} (u) \leq F (u), \forall u \in V

we may infer that

H_{0} (u) = F^{* *} (u), \forall u \in V .

44. A Note on Another General Relaxation Procedure for the Vectorial Case in the Calculus of Variation

Let

Ω \subset R^{n}

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

𝜕 Ω .

Consider a continuous and bounded below functional

F : V \to R

where

V = {u \in W^{1, 2} (Ω; R^{N}) : u = u_{0} on 𝜕 Ω} .

Fix

k \in N .

Define

{(H_{1})}_{k} : V \to R

\begin{matrix} {(H_{1})}_{k} (u) & = & inf \{\sum_{j = 1}^{k} λ_{j} F (v_{j}) : 0 \leq λ_{j} \leq 1 and v_{j} \in V, \forall j \in {1, \dots, k}, \\ \sum_{j = 1}^{k} λ_{j} = 1 and \sum_{j = 1}^{k} λ_{j} v_{j} = u\} . \end{matrix}

(272)

Observe that

F^{* *} (u) \leq {(H_{1})}_{k + 1} (u) \leq {(H_{1})}_{k} (u) \leq F (u), \forall u \in V .

Define

H_{2} : V \to R

H_{2} (u) = lim_{k \to \infty} {(H_{1})}_{k} (u) = inf_{k \in N} {{(H_{1})}_{k} (u)}, \forall u \in V .

Reasoning inductively, having

H_{m} : V \to R

, we may obtain

{(H_{m})}_{k} : V \to R

\begin{matrix} {(H_{m})}_{k} (u) & = & inf \{\sum_{j = 1}^{k} λ_{j} H_{m} (v_{j}) : 0 \leq λ_{j} \leq 1 and v_{j} \in V, \forall j \in {1, \dots, k}, \\ \sum_{j = 1}^{k} λ_{j} = 1 and \sum_{j = 1}^{k} λ_{j} v_{j} = u\} . \end{matrix}

(273)

Observe that

F^{* *} (u) \leq {(H_{m})}_{k + 1} (u) \leq {(H_{m})}_{k} (u) \leq F (u), \forall u \in V .

Now we define

H_{m + 1} (u) = lim_{k \to \infty} {(H_{m})}_{k} (u) = inf_{k \in N} {{(H_{m})}_{k} (u)}, \forall u \in V,

\forall m \in N .

Therefore, we have obtained a sequence

{H_{m} : V \to R}

such that

F^{* *} (u) \leq H_{m + 1} (u) \leq H_{m} (u) \leq F (u), \forall u \in V .

Thus, we may define

H^{0} : V \to R

H^{0} (u) = lim_{m \to \infty} H_{m} (u) = inf_{m \in N} {H_{m} (u)}, \forall u \in V .

Suppose, to obtain contradiction, that

H^{0} : V \to R

is not convex on

V .

Hence, there exists

\hat{u} \in V

such that

{(H^{0})}_{1} (\hat{u}) < H^{0} (\hat{u}),

where

{(H^{0})}_{1} (u) = inf {λ_{1} H^{0} (v_{1}) + (1 - λ_{1}) H^{0} (w_{1}) : 0 \leq λ_{1} \leq 1, v_{1}, w_{1} \in V, λ_{1} v_{1} + (1 - λ_{1}) w_{1} = u},

\forall u \in V .

This contradicts

H^{0} (u) = lim_{m \to \infty} H_{m} (u) = inf_{m \in N} {H_{m} (u)}, \forall u \in V .

Therefore,

H^{0}

is convex on V so that from this and

F^{* *} (u) \leq H^{0} (u) \leq F (u), \forall u \in V,

we may infer that

H^{0} (u) = F^{* *} (u), \forall u \in V .

45. A Proximal Relaxed General Approach also Suitable for the Vectorial Case in the Calculus of Variations

Let

Ω = [0, 1] \subset R

and consider a proximal relaxed functional

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

where

\begin{matrix} J_{1} (u, ϕ, λ, z^{*}) & = & \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 + \frac{K}{2} \int_{Ω} {(u - f)}^{2} d x \\ - \int_{Ω} z^{*} (u - f) d x + \frac{1}{2 K} \int_{Ω} {(z^{*})}^{2} d x, \end{matrix}

(274)

where

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

V_{0} = W_{0}^{1, 2} (Ω), and Y = Y^{*} = L^{2} (Ω) .

In order to obtain a critical point of such a proximal relaxed primal formulation, we propose the following algorithm:

Set $n = 1$ , $ε = 10^{- 4}$ and $z_{n}^{*} \equiv 0 .$
Calculate $(u_{n}, ϕ_{n}, λ_{n}) \in V \times V_{0} \times [0, 1]$ such that

$J_{1} (u_{n}, ϕ_{n}, λ_{n}, z_{n}^{*}) = inf_{(u, ϕ, λ) \in V \times V_{0} \times [0, 1]} J_{1} (u, ϕ, λ, z_{n}^{*}) .$
Calculate $z_{n + 1}^{*} \in Y^{*}$ such that

$J_{1} (u_{n}, ϕ_{n}, λ_{n}, z_{n + 1}^{*}) = inf_{z^{*} \in Y^{*}} J_{1} (u_{n}, ϕ_{n}, λ_{n}, z^{*}),$

so that indeed,

$z_{n + 1}^{*} = K (u_{n} - f) .$
If $∥ z_{n + 1}^{*} - z_{n}^{*} ∥_{\infty} < ε,$ then stop. Otherwise set $n : = n + 1$ and go to item 2.

We have obtained numerical results for

K = 100

and

f (x) = sin (π x) / 2 .

For the optimal solution

u (x)

obtained, please see Figure 34.

At this point we present the software in MAT-LAB we have developed to obtain such numerical results.

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

clear all

global m8 d u v yo e1 K z

m8=100;

d=1/m8;

e1=0.0005;

K=100.0;

for i=1:m8

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

z(i,1)=0;

end;

for i=1:2*m8+1

xo(i,1)=0.3;

x1(i,1)=0.3;

end;

k1=1;

b14=1.0;

while $(b 14 > 10^{- 4})$ && $(k 1 < 11)$

k1

k1=k1+1;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 16)$

k

k=k+1;

X=fminunc(’funMarch24PhaseT’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

b14=max(abs(x1-xo));

z=K*(u-yo);

x1=xo;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u)

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

Here the auxiliary function "funMarch24PhaseT"

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

function S=funMarch24PhaseT(x)

global m8 d u v L yo e1 K z

for i=1:m8

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

v(i,1)=x(i+m8,1);

end;

L=(sin(x(2*m8+1,1))+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+ $L * {({(d u (i, 1) - (1 - L) * d v (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ $(1 - L) * {({(d u (i, 1) + L * d v (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ ${(u (i, 1) - y o (i, 1))}^{2} / 2;$

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

end;

for i=1:m8-1

S=S+ $e 1 * d 2 u {(i, 1)}^{2};$

end;

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

46. Another Proximal Relaxed General Approach also Suitable for the Vectorial Case in the Calculus of Variations

Let

Ω = [0, 1] \subset R

and consider a proximal relaxed functional

J_{1} : V \times {[V_{0}]}^{3} \times B \times Y^{*} \to R

where

\begin{matrix} J_{1} (u, ϕ, λ, z^{*}) & = & \frac{λ_{1}}{2} \int_{Ω} {({(u^{'} + λ_{1} ϕ_{1}^{'} + λ_{2} ϕ_{2}^{'} + λ_{3} ϕ_{3}^{'} - ϕ_{1}^{'})}^{2} - 1)}^{2} d x \\ + \frac{λ_{2}}{2} \int_{Ω} {({(u^{'} + λ_{1} ϕ_{1}^{'} + λ_{2} ϕ_{2}^{'} + λ_{3} ϕ_{3}^{'} - ϕ_{2}^{'})}^{2} - 1)}^{2} d x \\ + \frac{λ_{3}}{2} \int_{Ω} {({(u^{'} + λ_{1} ϕ_{1}^{'} + λ_{2} ϕ_{2}^{'} + λ_{3} ϕ_{3}^{'} - ϕ_{3}^{'})}^{2} - 1)}^{2} d x \\ + \frac{λ_{4}}{2} \int_{Ω} {({(u^{'} + λ_{1} ϕ_{1}^{'} + λ_{2} ϕ_{2}^{'} + λ_{3} ϕ_{3}^{'})}^{2} - 1)}^{2} d x \\ + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x + \frac{K}{2} \int_{Ω} {(u - f)}^{2} d x \\ - \int_{Ω} z^{*} (u - f) d x + \frac{1}{2 K} \int_{Ω} {(z^{*})}^{2} d x, \end{matrix}

(275)

where

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

V_{0} = W_{0}^{1, 2} (Ω), Y = Y^{*} = L^{2} (Ω),

f \in L^{2} (Ω)

and

B = \{λ = (λ_{1}, \dots, λ_{4}) \in R^{4} : λ_{j} \geq 0, \forall j \in {1, \dots, 4} and \sum_{j = 1}^{4} λ_{j} = 1\} .

In order to obtain a critical point of such a proximal relaxed primal formulation, we propose the following algorithm:

Set $n = 1$ , $ε = 10^{- 4}$ and $z_{n}^{*} \equiv 0 .$
Calculate $(u_{n}, ϕ_{n}, λ_{n}) \in V \times {[V_{0}]}^{3} \times B$ such that

$J_{1} (u_{n}, ϕ_{n}, λ_{n}, z_{n}^{*}) = inf_{(u, ϕ, λ) \in V \times {[V_{0}]}^{3} \times B} J_{1} (u, ϕ, λ, z_{n}^{*}) .$
Calculate $z_{n + 1}^{*} \in Y^{*}$ such that

$J_{1} (u_{n}, ϕ_{n}, λ_{n}, z_{n + 1}^{*}) = inf_{z^{*} \in Y^{*}} J_{1} (u_{n}, ϕ_{n}, λ_{n}, z^{*}),$

so that indeed,

$z_{n + 1}^{*} = K (u_{n} - f) .$
If $∥ z_{n + 1}^{*} - z_{n}^{*} ∥_{\infty} < ε,$ then stop. Otherwise set $n : = n + 1$ and go to item 2.

We have obtained numerical results for

K = 100

and

f (x) = 0.0 .

For the optimal solution

u (x)

obtained, please see Figure 35.

At this point we present the software in MAT-LAB we have developed to obtain such numerical results.

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

clear all

global m8 d u v yo e1 K z

m8=100;

d=1/m8;

e1=0.0007;

K=100.0;

for i=1:m8

yo(i,1)=0.0*sin(pi*i*d)/2;

z(i,1)=0;

end;

for i=1:4*m8+3

xo(i,1)=0.3;

x1(i,1)=0.3;

end;

k1=1;

b14=1.0;

while $(b 14 > 10^{- 4})$ && $(k 1 < 11)$

k1

k1=k1+1;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 16)$

k

k=k+1;

X=fminunc(’funMarch24PhaseTC’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

b14=max(abs(x1-xo));

z=K*(u-yo);

x1=xo;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u)

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

With the auxiliary function "funMarch24PhaseTC"

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

function S=funMarch24PhaseTC(x)

global m8 d u v L yo e1 K z

for i=1:m8

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

v(i,1)=x(i+m8,1);

v1(i,1)=x(i+2*m8,1);

v2(i,1)=x(i+3*m8,1);

end;

L1=(sin(x(4*m8+1,1))+1)/2;

L2=min((sin(x(4*m8+2,1))+1)/2,1-L1);

L3=min((sin(x(4*m8+3,1))+1)/2,1-L1-L2);

L4=1-L1-L2-L3;

u(m8,1)=1/2;

v(m8,1)=0.0;

v1(m8,1)=0.0;

v2(m8,1)=0.0;

du(1,1)=u(1,1)/d;

dv(1,1)=v(1,1)/d;

dv1(1,1)=v1(1,1)/d;

dv2(1,1)=v2(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;

dv1(i,1)=(v1(i,1)-v1(i-1,1))/d;

dv2(i,1)=(v2(i,1)-v2(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+ $L 1 * {({(d u (i, 1) + L 1 * d v (i, 1) + L 2 * d v 1 (i, 1) + L 3 * d v 2 (i, 1) - d v (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ $L 2 * {({(d u (i, 1) + L 1 * d v (i, 1) + L 2 * d v 1 (i, 1) + L 3 * d v 2 (i, 1) - d v 1 (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ $L 3 * {({(d u (i, 1) + L 1 * d v (i, 1) + L 2 * d v 1 (i, 1) + L 3 * d v 2 (i, 1) - d v 2 (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ $L 4 * {({(d u (i, 1) + L 1 * d v (i, 1) + L 2 * d v 1 (i, 1) + L 3 * d v 2 (i, 1))}^{2} - 1)}^{2} / 2;$

S=S+ ${(u (i, 1) - y o (i, 1))}^{2} / 2;$

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

end;

for i=1:m8-1

S=S+ $e 1 * d 2 u {(i, 1)}^{2};$

end;

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

47. A Dual Variational Formulation for a Non-Convex Primal One

Let

Ω \subset R^{3}

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

𝜕 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}

(276)

Here

V = W_{0}^{1, 2} (Ω),

α > 0

β > 0

γ > 0

, and

f \in L^{2} (Ω) \equiv Y = Y^{*} .

Denoting

Y_{1} = Y_{1}^{*} = L^{2} (Ω; R^{3}),

define

F_{1} : Y_{1} \to R

F_{2} : V \times Y \to R

and

F_{3} : Y \to R

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

F_{2} (u, v) = \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}},

and

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

Define also,

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

{\tilde{F}}_{2} : Y_{1}^{*} \times Y^{*} \times Y^{*} \to R

and

F_{3} : Y^{*} \to R

, by

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

(277)

\begin{matrix} {\tilde{F}}_{2}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & = & sup_{(u, v) \in V \times Y} {- {〈 \nabla u, v_{1}^{*} 〉}_{L^{2}} + {〈 u, z^{*} 〉}_{L^{2}} \\ + {〈 v, v_{0}^{*} 〉}_{L^{2}} - F_{2} (u, v)} \\ = & \frac{1}{2} \int_{Ω} \frac{{(div v_{1}^{*} + z^{*} + f)}^{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}

(278)

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

where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} \leq K / 2} .

Moreover,

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

(279)

At this point we define

J^{*} : Y_{1}^{*} \times B^{*} \times Y^{*} \to R

J_{1}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) = - F_{1}^{*} (v_{1}^{*}) - {\tilde{F}}_{2}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) + F_{3}^{*} (z^{*}) .

Assume

({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) \in Y_{1}^{*} \times B^{*} \times Y^{*}

is such that

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

Observe that

\begin{matrix} J^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) & = & - F_{1}^{*} (v_{1}^{*}) - {\tilde{F}}_{2}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) + F_{3}^{*} (z^{*}) \\ \leq & - {〈 \nabla u, {\hat{v}}_{1}^{*} 〉}_{L^{2}} + F_{1} (\nabla u) \\ + {〈 \nabla u, {\hat{v}}_{1}^{*} 〉}_{L^{2}} + {〈 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}^{*} d x - β \int_{Ω} {\hat{v}}_{0}^{*} d x \\ - {〈 u, f 〉}_{L 2} - {〈 u, {\hat{z}}^{*} 〉}_{L^{2}} + \frac{1}{2 K} \int_{Ω} {({\hat{z}}^{*})}^{2} d x \\ \leq & F_{1} (\nabla u) + sup_{v_{0}^{*} \in Y^{*}} \{{〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - \frac{1}{2 α} \int_{Ω} v_{0}^{*} d x - β \int_{Ω} v_{0}^{*} d x\} \\ - {〈 u, f 〉}_{L 2} + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, {\hat{z}}^{*} 〉}_{L^{2}} + \frac{1}{2 K} \int_{Ω} {({\hat{z}}^{*})}^{2} d x \\ = & F_{1} (\nabla u) + \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x - {〈 u, f 〉}_{L 2} \\ + \frac{K}{2} \int_{Ω} {(u - \frac{{\hat{z}}^{*}}{K})}^{2} d x \\ = & J (u) + \frac{K}{2} \int_{Ω} {(u - \frac{{\hat{z}}^{*}}{K})}^{2} d x, \forall u \in V . \end{matrix}

(280)

Define now

u_{0} \in V

u_{0} = \frac{{\hat{z}}^{*}}{K} .

From this and (280) we have

J^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) \leq inf_{u \in V} \{J (u) + \frac{K}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} .

Furthermore, from the variation of

J^{*}

v_{1}^{*}

we obtain

- \frac{{\hat{v}}_{1}^{*}}{γ} + \nabla (\frac{div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f}{2 {\hat{v}}_{0}^{*} + K}) = 0,

so that

{\hat{v}}_{1}^{*} = γ \nabla (\frac{div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f}{2 {\hat{v}}_{0}^{*} + K}) .

From the variation of

J^{*}

z^{*}

, we get

\frac{{\hat{z}}^{*}}{K} - (\frac{div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f}{2 {\hat{v}}_{0}^{*} + K}) = 0

so that

u_{0} = \frac{{\hat{z}}^{*}}{K} = (\frac{div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f}{2 {\hat{v}}_{0}^{*} + K}) .

From the variation of

J^{*}

v_{0}^{*}

, we obtain

\frac{{\hat{v}}_{0}^{*}}{α} - {(\frac{div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f}{2 {\hat{v}}_{0}^{*} + K})}^{2} + β = 0

so that

{\hat{v}}_{0}^{*} = α (u_{0}^{2} - β) .

Joining the pieces, we have also

{\hat{v}}_{1} = γ \nabla u_{0},

\hat{z} = K u_{0},

so that from this and

div {\hat{v}}_{1}^{*} + {\hat{z}}^{*} + f = (2 {\hat{v}}_{0}^{*} + K) u_{0},

we obtain

γ \nabla^{2} u_{0} + K u_{0} + f = α (u_{0}^{2} - β) 2 u_{0} + K u_{0},

so that

- γ \nabla^{2} u_{0} + α (u_{0}^{2} - β) 2 u_{0} - f = 0,

that is,

δ J (u_{0}) = 0 .

Finally, from the Legendre transform proprieties, we also obtain

F_{1}^{*} ({\hat{v}}_{1}^{*}) = {〈 \nabla u_{0}, {\hat{v}}_{1}^{*} 〉}_{L^{2}} - F_{1} (\nabla u_{0}),

\begin{matrix} {\tilde{F}}_{2} ({\hat{v}}_{1}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) & = & - {〈 \nabla u_{0}, {\hat{v}}_{1}^{*} 〉}_{L^{2}} + {〈 u_{0}, {\hat{z}}^{*} 〉}_{L^{2}} \\ + {〈 0, {\hat{v}}_{0}^{*} 〉}_{L^{2}} - F_{2} (u_{0}, 0) \end{matrix}

(281)

and

F_{3}^{*} (z^{*}) = {〈 u_{0}, {\hat{z}}^{*} 〉}_{L^{2}} - F_{3} (u_{0}) .

Therefore

\begin{matrix} J^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) & = & - F_{1}^{*} ({\hat{v}}_{1}^{*}) - {\tilde{F}}_{2}^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) + F_{3}^{*} ({\hat{z}}^{*}) \\ = & F_{1} (\nabla u_{0}) + F_{2} (u_{0}, 0) - F_{3} (u_{0}) \\ = & J (u_{0}) . \end{matrix}

(282)

Observe now that from

δ J (u_{0}) = 0

, for

K > 0

sufficiently large, we have

J (u_{0}) = inf_{u \in V} \{J (u) + \frac{K}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} .

Joining the pieces we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{K}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} \\ = & J^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) . \end{matrix}

(283)

We have obtained numerical results for the case A, where

γ = 0.1

α = 3.0

β = 5.0

f (x) = 10.0

and

K = 120 .

For the optimal solution

u (x)

, where

u (x) = \frac{{(v_{1} *)}^{'} + z^{*} + f}{2 v_{0}^{*} + K},

please see Figure 36.

Here we present the software in MATLAB through which we have obtained such results.

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

clear all

global m8 d yo z1 K e1 dv1 dv2 v3 v4 v1 v2 A A3 B L u

m8=100;

d=1/m8;

A3=0.1;

A=3.0;

B=5.0;

K=120;

e1=0.0007;

for i=1:m8

yo(i,1)=10.0;

z1(i,1)=0.0;

end;

L=1/2;

for i=1:2*m8

xo(i,1)=3.0;

end;

for k1=1:30

k1

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 15)$

k

k=k+1;

X=fminunc(’funMarch24LGA7’,xo);

b12=max(abs(X-xo))

xo=X;

u(m8/2,1)

end;

for i=1:m8-1

z1(i,1)=K*(dv1(i,1)+z1(i,1)+yo(i,1))/(2*v2(i,1)+K);

end;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

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

With the auxiliary function "funMarch24LGA7"

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

function S=funMarch24LGA7(x)

global m8 d yo z1 z2 K e1 dv1 dv2 v3 v4 v1 v2 A A3 B L u

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(i+m8,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+ $v 1 {(i, 1)}^{2} / 2 / A 3 + 1 / 2 * {(d v 1 (i, 1) + z 1 (i, 1) + y o (i, 1))}^{2} / (2 * v 2 (i, 1) + K);$

S=S+ $v 2 (i, 1) * B + v 2 {(i, 1)}^{2} / 2 / A;$

end;

for i=1:m8-1

u(i,1)=(dv1(i,1)+z1(i,1)+yo(i,1))/(2*v2(i,1)+K);

end;

u(m8,1)=0;

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

48. A Convex Dual Variational Formulation for a Relaxed Non-Convex Primal One

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 - f)}^{2} d x,

where

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

Denoting

V_{0} = W_{0}^{1, 2} (Ω)

, we define

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

where

\begin{matrix} J_{1} (u, ϕ, λ) & = & \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 . \end{matrix}

(284)

Observe that

\begin{matrix} J_{1} (u, ϕ, λ) & = & - {〈 {(u^{'} - (1 - λ) ϕ^{'})}^{2} - 1, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} {({(u^{'} - (1 - λ) ϕ^{'})}^{2} - 1)}^{2} d x \\ - {〈 {(u^{'} + λ ϕ^{'})}^{2} - 1, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} {({(u^{'} + λ ϕ^{'})}^{2} - 1)}^{2} d x \\ + {〈 {(u^{'} - (1 - λ) ϕ^{'})}^{2} - 1, v_{3}^{*} 〉}_{L^{2}} - {〈 u^{'} - (1 - λ) ϕ^{'}, v_{1}^{*} 〉}_{L^{2}} \\ + {〈 {(u^{'} + λ ϕ^{'})}^{2} - 1, v_{4}^{*} 〉}_{L^{2}} - {〈 u^{'} + λ ϕ^{'}, v_{2}^{*} 〉}_{L^{2}} \\ + {〈 u^{'} - (1 - λ) ϕ^{'}, v_{1}^{*} 〉}_{L^{2}} + {〈 u^{'} + λ ϕ^{'}, v_{2}^{*} 〉}_{L^{2}} \\ + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x . \end{matrix}

(285)

Therefore

\begin{matrix} J_{1} (u, ϕ, λ) & \geq & inf_{v_{3} \in Y} \{- {〈 v_{3}, v_{3}^{*} 〉}_{L^{2}} + \frac{λ}{2} \int_{Ω} {(v_{3})}^{2} d x\} \\ + inf_{v_{4} \in Y} \{- {〈 v_{4}, v_{4}^{*} 〉}_{L^{2}} + \frac{(1 - λ)}{2} \int_{Ω} {(v_{4})}^{2} d x\} \\ + inf_{{\tilde{v}}_{3} \in Y} {- {〈 {\tilde{v}}_{3}, v_{1}^{*} 〉}_{L^{2}} + {〈 {\tilde{v}}_{3}^{2} - 1, v_{3}^{*} 〉}_{L^{2}} \\ + inf_{{\tilde{v}}_{4} \in Y} {- {〈 {\tilde{v}}_{4}, v_{1}^{*} 〉}_{L^{2}} + {〈 {\tilde{v}}_{4}^{2} - 1, v_{3}^{*} 〉}_{L^{2}} \\ + inf_{(u, ϕ) \in V \times V_{0}} \{- {〈 u - (1 - λ) ϕ, {(v_{1}^{*})}^{'} 〉}_{L^{2}} - {〈 u + λ ϕ, {(v_{2}^{*})}^{'} 〉}_{L^{2}} \\ + \frac{1}{2} \int_{Ω} {(u - f)}^{2} d x + v_{1}^{*} (1) u (1) + v_{2}^{*} (1) u (1)\} \\ = & - \frac{1}{2 λ} \int_{Ω} {(v_{3}^{*})}^{2} d x - \frac{1}{2 (1 - λ)} \int_{Ω} {(v_{4}^{*})}^{2} d x \\ - \int_{Ω} v_{3}^{*} d x - \int_{Ω} v_{4}^{*} d x \\ - \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{4 v_{3}^{*}} d x - \int_{Ω} \frac{{(v_{2}^{*})}^{2}}{4 v_{4}^{*}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + λ f)}^{2} d x - \frac{1}{2} \int_{Ω} {({(v_{2}^{*})}^{'} + (1 - λ) f)}^{2} d x \\ \equiv & J^{*} (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}, λ), \end{matrix}

(286)

\forall (u, ϕ, λ) \in V \times V_{0} \times [0, 1], \forall (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in {[Y^{*}]}^{2} \times B^{*},

where

B^{*} = {(v_{3}^{*}, v_{4}^{*}) \in Y^{*} \times Y^{*} : v_{3}^{*} > 0 and v_{4}^{*} > 0, in Ω},

and

\begin{matrix} J^{*} (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}, λ) & = & - \frac{1}{2 λ} \int_{Ω} {(v_{3}^{*})}^{2} d x - \frac{1}{2 (1 - λ)} \int_{Ω} {(v_{4}^{*})}^{2} d x \\ - \int_{Ω} v_{3}^{*} d x - \int_{Ω} v_{4}^{*} d x \\ - \int_{Ω} \frac{{(v_{1}^{*})}^{2}}{4 v_{3}^{*}} d x - \int_{Ω} \frac{{(v_{2}^{*})}^{2}}{4 v_{4}^{*}} d x \\ - \frac{1}{2} \int_{Ω} {({(v_{1}^{*})}^{'} + λ f)}^{2} d x - \frac{1}{2} \int_{Ω} {({(v_{2}^{*})}^{'} + (1 - λ) f)}^{2} d x . \end{matrix}

(287)

From such results, we may infer that

inf_{(u, ϕ, λ) \in V \times V_{0} \times [0, 1]} J_{1} (u, ϕ, λ) \geq inf_{λ \in [0, 1]} \{sup_{(v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}) \in [Y^{*}] \times B^{*}} J^{*} (v_{1}^{*}, v_{2}^{*}, v_{3}^{*}, v_{4}^{*}, λ)\} .

We have developed numerical results for the cases

f (x) = sin (π x) / 2

and

f (x) = 0

For the corresponding optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

, please see Figure 37.

For the corresponding optimal solution

u (x)

for the case

f (x) = 0

, please see Figure 38.

Here we present the software in MATLAB through which we have obtained such numerical results.

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

clear all

global m8 d yo u L v1 v2 v3 v4 dv1 dv2 K dz1 z1 e1

m8=100;

d=1/m8;

K=1.0;

e1=0.0007;

L=1/2;

for i=1:m8

yo(i,1)=0.0*sin(pi*i*d)/2;

end;

for i=1:4*m8

xo(i,1)=0.8;

end;

x1(1,1)=1/2;

for k1=1:12

k1

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 15)$

k

k=k+1;

X=fminunc(’funMarch24A18’,xo);

b12=max(abs(X-xo))

u(m8/2,1)

xo=X;

end;

X1=fminunc(’funMarch24A19’,x1);

x1=X1;

u(m8/2,1)

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

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

With the auxiliary functions "funMarch24A18" and "funMarch24A19":

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

function S=funMarch24A18(x)

global m8 d yo u e1 v1 v2 v3 v4 dv1 dv2 L

for i=1:m8

v1(i,1)=x(i,1);

v2(i,1)=x(i+m8,1);

v3(i,1)=x(i+2*m8,1);

v4(i,1)=x(i+3*m8,1);

end;

for i=1:m8-1

dv1(i,1)=(v1(i+1,1)-v1(i,1))/d;

dv2(i,1)=(v2(i+1,1)-v2(i,1))/d;

end;

S=0;

for i=1:m8-1

S=S+ ${(v 1 (i, 1))}^{2} / (2 * v 3 {(i, 1)}^{2}) / 2 + v 3 {(i, 1)}^{4} / 2 / (L + e 1) + v 3 {(i, 1)}^{2} + {(d v 1 (i, 1) + L * y o (i, 1))}^{2} / 2 + v 1 {(i, 1)}^{2} / 2 / (L + e 1);$

S=S+ ${(v 2 (i, 1))}^{2} / (2 * v 4 {(i, 1)}^{2}) / 2 + v 4 {(i, 1)}^{4} / 2 / ((1 - L) + e 1) + v 4 {(i, 1)}^{2}$ ;

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

end;

S=S-v1(m8,1)/2/d-v2(m8,1)/2/d;

for i=1:m8-1

u(i,1)=L*(dv1(i,1)+L*yo(i,1))+(1-L)*(dv2(i,1)+(1-L)*yo(i,1));

end;

u(m8,1)=1/2;

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

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

function S1=funMarch24A19(x)

global m8 d yo e1 v1 v2 v3 v4 dv1 dv2 L u

L=(sin(x(1,1))+1)/2;

S=0;

for i=1:m8-1

S=S+ ${(v 1 (i, 1))}^{2} / (2 * v 3 {(i, 1)}^{2}) / 2 + v 3 {(i, 1)}^{4} / 2 / (L + e 1) + v 3 {(i, 1)}^{2} + {(d v 1 (i, 1) + L * y o (i, 1))}^{2} / 2 + v 1 {(i, 1)}^{2} / 2 / (L + e 1);$

S=S+ ${(v 2 (i, 1))}^{2} / (2 * v 4 {(i, 1)}^{2}) / 2 + v 4 {(i, 1)}^{4} / 2 / ((1 - L) + e 1) + v 4 {(i, 1)}^{2}$

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

end;

S=S-v1(m8,1)/2/d-v2(m8,1)/2/d;

S1=-S;

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

49. A Dual Variational Formulation for the Shape Optimization of a Beam Model

Let

Ω \subset [0, 1] \subset R

be the horizontal axis of a straight beam with a variable thickness

H (x)

Consider the problem of minimizing a relaxed functional

J : V \times [0, 1] \times B \to R

, where

\begin{matrix} J (w, λ, L_{1}, L_{2}) & = & \frac{λ E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x}^{2} d x \\ + \frac{(1 - λ) E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x}^{2} d x, \end{matrix}

(288)

subject to

\begin{matrix} {(λ E_{0} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x})}_{x x} \\ + {((1 - λ) E_{0} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x})}_{x x} - P \\ = 0, in Ω . \end{matrix}

(289)

Here

H (x) = L_{1} (x) h_{0},

H_{1} (x) = L_{2} (x) h_{0},

h_{0} = 0.2 m

b = 0.15 m

E_{0} = 10^{7} Pa

P = 400 N

Also, for a simply supported beam,

V = {w \in W^{2, 2} (Ω) : w (0) = w_{x x} (0) = w (1) = w_{x x} (1) = 0},

\begin{matrix} B & = & \{(L_{1}, L_{2}) : Ω \to R^{2} measurable : 0.3 \leq L_{1} \leq 1, \\ - 0.7 \leq L_{2} \leq 0.7, in Ω, \int_{Ω} L_{1} (x) d x = 0.61 and \int_{Ω} L_{2} (x) d x = 0\} . \end{matrix}

(290)

Moreover, we define

Y = Y^{*} = L^{2} (Ω),

and

\begin{matrix} A & = & \{(w, λ, L_{1}, L_{2}) \in V \times [0, 1] \times B : \\ {(λ E_{0} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x})}_{x x} \\ + {((1 - λ) E_{0} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x})}_{x x} - P \\ = 0, in Ω\} . \end{matrix}

(291)

Observe that

\begin{matrix} inf_{(w, λ, L_{1}, L_{2}) \in A} J (w, λ, L_{1}, L_{2}) \\ = & inf_{(λ, L_{1}, L_{2}) \in [0, 1] \times B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{J (w, λ, L_{1}, L_{2}) \\ - 〈\hat{w}, {(λ E_{0} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x})}_{x x} \\ {+ {((1 - λ) E_{0} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x})}_{x x} - P〉}_{L^{2}}\}\}\} \\ = & inf_{(λ, L_{1}, L_{2}) \in [0, 1] \times B} \{sup_{\hat{w} \in V} \{inf_{w \in V} \{\frac{λ E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x}^{2} d x \\ + \frac{(1 - λ) E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x}^{2} d x \\ - 〈\hat{w}, {(λ E_{0} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} w_{x x})}_{x x} \\ {+ {((1 - λ) E_{0} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} w_{x x})}_{x x} - P〉}_{L^{2}}\}\}\} \\ = & inf_{(λ, L_{1}, L_{2}) \in [0, 1] \times B} \{sup_{\hat{w} \in V} \{- \frac{λ E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3} {\hat{w}}_{x x}^{2} d x \\ - \frac{(1 - λ) E_{0}}{2} \int_{Ω} \frac{b}{12} {(H (L_{1}) + λ H_{1} (L_{2}))}^{3} {\hat{w}}_{x x}^{2} d x + {〈 \hat{w}, P 〉}_{L^{2}}\}\} \\ = & inf_{(λ, L_{1}, L_{2}) \in [0, 1] \times B} \{inf_{(M_{1}, M_{2}) \in C^{*}} \{\frac{1}{2 λ E_{0} b / 12} \int_{Ω} \frac{{(M_{1})}^{2}}{{(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3}} d x \\ + \frac{1}{2 (1 - λ) E_{0} b / 12} \int_{Ω} \frac{{(M_{1})}^{2}}{{(H (L_{1}) + λ H_{1} (L_{2}))}^{3}} d x\}\}, \end{matrix}

(292)

where

C^{*} = {(M_{1}, M_{2}) \in Y^{*} \times Y^{*} : {(M_{1})}_{x x} + {(M_{2})}_{x x} + P = 0, in Ω} .

We have obtained numerical results through the following algorithm. It is worth highlighting the convergence criterion in this software slightly differs from the one in the algorithm.

Set $n = 1$ , $ε = 10^{- 4}$ and ${(L_{1})}_{n} \equiv 1 / 2$ , ${(L_{2})}_{n} \equiv 0.1$ , $λ_{n} = 1 / 2$ .
Calculate $w_{n} \in V$ such that

$\begin{matrix} {(λ_{n} E_{0} \frac{b}{12} {(H ({(L_{1})}_{n}) - (1 - λ) H_{1} ({(L_{2})}_{n}))}^{3} {(w_{n})}_{x x})}_{x x} \\ + {((1 - λ_{n}) E_{0} \frac{b}{12} {(H ({(L_{1})}_{n}) + λ H_{1} ({(L_{2})}_{n}))}^{3} {(w_{n})}_{x x})}_{x x} - P \\ = 0, in Ω, \end{matrix}$

(293)
Calculate $λ_{n + 1} \in [0, 1]$ such that

$J (w_{n}, λ_{n + 1}, {(L_{1})}_{n}, {(L_{2})}_{n}) = inf_{λ \in [0, 1]} J ((w_{n}, λ, {(L_{1})}_{n}, {(L_{2})}_{n}) .$
Calculate $({(L_{1})}_{n + 1}, {(L_{2})}_{n + 1}) \in B$ such that

$J^{*} ({(M_{1})}_{n}, {(M_{2})}_{n}, λ_{n + 1}, {(L_{1})}_{n + 1}, {(L_{2})}_{n + 1}) = inf_{(L_{1}, L_{2}) \in B} J^{*} ({(M_{1})}_{n}, {(M_{2})}_{n}, λ_{n + 1}, L_{1}, L_{2}),$

where

${(M_{1})}_{n} = - λ_{n + 1} E_{0} \frac{b}{12} {(H ({(L_{1})}_{n}) - (1 - λ_{n + 1}) {(L_{2})}_{n})}^{3} {(w_{n})}_{x x},$

${(M_{2})}_{n} = - (1 - λ_{n + 1}) E_{0} \frac{b}{12} {(H ({(L_{1})}_{n}) + λ_{n + 1} {(L_{2})}_{n})}^{3} {(w_{n})}_{x x},$

and

$\begin{matrix} J^{*} (M_{1}, M_{2}) & = & \frac{1}{2 λ E_{0} b / 12} \int_{Ω} \frac{{(M_{1})}^{2}}{{(H (L_{1}) - (1 - λ) H_{1} (L_{2}))}^{3}} d x \\ + \frac{1}{2 (1 - λ) E_{0} b / 12} \int_{Ω} \frac{{(M_{2})}^{2}}{{(H (L_{1}) + λ H_{1} (L_{2}))}^{3}} d x . \end{matrix}$

(294)
If

$∥ ({(L_{1})}_{n + 1}, {(L_{2})}_{n + 1}) - {({(L_{1})}_{n}, {(L_{2})}_{n} ∥}_{\infty} < ε,$

then stop, otherwise $n : = n + 1$ and go to item 2.

We have obtained numerical results for a case A with the constant values above specified.

For the optimal solution

L_{1} (x)

, please see Figure 39.

Here we present the software in MATLAB through which we have obtained such results.

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

clear all

global m8 d yo u L1 L2 ho Eo B L H H1 Ho Ho1

m8=100;

d=1/m8;

P=400;

Eo= $10^{7};$

for i=1:m8 yo(i,1)=P; end;

ho=0.20;

B=0.15;

for i=1:m8

L1(i,1)=1/2;

L2(i,1)=0.3;

uo(i,1)=0.1;

Ho(i,1)=L1(i,1)*ho;

Ho1(i,1)=0.1*L2(i,1)*ho;

end;

L=1/2;

for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

for i=1:2*m8

xo(i,1)=0.3;

end;

x1(1,1)=1/2;

A=zeros(2*m8,2*m8);

for i=1:m8

A(1,i)=1.0;

A(2,i+m8)=1.0;

end;

b=zeros(2*m8,1);

b(1,1)=m8*0.61;

for i=1:m8

lb(i,1)=0.3;

lb(i+m8,1)=-0.7;

end;

for i=1:m8

ub(i,1)=1;

ub(i+m8,1)=0.7;

end;

i=1;

m12=2;

m50(i)=1/m12;

z(i)= $m 50 (i) * (- y o (i, 1) * d^{2});$

for i=2:m8-1

m12=2-m50(i-1);

m50(i)=1/m12;

z(i)= $m 50 (i) * (- y o (i, 1) * d^{2} + z (i - 1));$

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;

k1=1;

b14=1.0;

while $(b 14 > 10^{- 4})$ && $(k 1 < 15)$

k1

k1=k1+1;

for i=1:m8

y1(i,1)= $v (i, 1) / (E o * L * B / 12 * {(H (i, 1) - (1 - L) * H 1 (i, 1))}^{3} + E o * (1 - L) * B / 12 * {(H (i, 1) + L * H 1 (i, 1))}^{3});$

end;

i=1;

m12=2;

m60(i)=1/m12;

z1(i)= $m 60 (i) * (- y 1 (i, 1) * d^{2});$

for i=2:m8-1

m12=2-m60(i-1);

m60(i)=1/m12;

z1(i)= $m 60 (i) * (- y 1 (i, 1) * d^{2} + z 1 (i - 1));$

end;

u(m8,1)=0;

for i=1:m8-1

u(m8-i,1)=m60(m8-i)*u(m8-i+1)+z1(m8-i);

end;

k=1;

b12=1.0;

while $(b 12 > 10^{- 4})$ && $(k < 100)$

k

k=k+1;

X=fmincon(’funMarch2024Beam1’,xo,[],[],A,b,lb,ub);

b12=abs(max(xo-X))

xo=X;

L1(m8/2,1)

end;

Ho=H;

Ho1=H1;

X1=fminunc(’funMarch2024Beam2’,x1);

x1=X1;

b14=max(abs(u-uo))

uo=u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,L1);

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

With the auxiliary function "funMarch2024Beam1"

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

function S1=funMarch2024Beam1(x)

global m8 d yo u L1 L2 ho Eo B L Ho Ho1

for i=1:m8

L1(i,1)=x(i,1);

L2(i,1)=x(i+m8,1);

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;

for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

S=0;

for i=1:m8-1

S=S+ $L * {(E o * B / 12 * {(H o (i, 1) - (1 - L) * H o 1 (i, 1))}^{3} * d 2 u (i, 1))}^{2} / (E o * B / 12 * {(H (i, 1) - (1 - L) * H 1 (i, 1))}^{3});$

S=S+ $(1 - L) * {(E o * B / 12 * {(H o (i, 1) + L * H o 1 (i, 1))}^{3} * d 2 u (i, 1))}^{2} / (E o * B / 12 * {(H (i, 1) + L * H 1 (i, 1))}^{3});$

end;

S1=S;

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

And the auxiliary function "funMarch2024Beam2"

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

function S=funMarch2024Beam2(x)

global m8 d yo u L1 L2 ho Eo B L Ho Ho1

L=(sin(x(1,1))+1)/2;

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; for i=1:m8

H(i,1)=L1(i,1)*ho;

H1(i,1)=L2(i,1)*ho;

end;

S=0;

for i=1:m8-1

S=S+ $L * E o * B / 12 * {(H (i, 1) - (1 - L) * H 1 (i, 1))}^{3} * d 2 u {(i, 1)}^{2};$

S=S + $(1 - L) * E o * B / 12 * {(H (i, 1) + L * H 1 (i, 1))}^{3} * d 2 u {(i, 1)}^{2};$

end;

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

50. A Dual Variational Formulation for a Relaxed Primal Formulation Related to a Shape Optimization Model in Elasticity

Let

Ω \subset R^{3}

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

𝜕 Ω .

Consider the problem of minimizing a relaxed functional

J : V \times [0, 1] \times B \to R

where

J (u, λ, λ_{1}, λ_{2}) = \frac{1}{2} \int_{Ω} H_{i j k l} (λ, λ_{1} (x), λ_{2} (x)) e_{i j} (u) e_{k l} (u) d x,

subject to

{(H_{i j k l} (λ, λ_{1} (x), λ_{2} (x)) e_{k l} (u))}_{, j} + f_{i} = 0, in Ω, \forall i {1, 2, 3} .

Here for simplicity

V = W_{0}^{1, 2} (Ω; R^{3}),

Y = Y^{*} = L^{2} (Ω; R^{3}),

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

and

f \in L^{2} (Ω; R^{3}) .

Also,

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

denotes the field of displacements resulting from the action of f,

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

and

E_{b} \leq E (λ, λ_{1}, λ_{2}) \leq E_{a}

E_{a} ≫ E_{b} > 0

, where

λ_{1} (x) = 1

corresponds to the presence of a stronger material with Young modulus

E_{a}

at the point

x \in Ω

. Moreover,

λ_{1} (x) = 0

corresponds to the presence of a much weaker material with elasticity model

E_{b}

, simulating a void space at the point

x \in Ω .

On the other hand,

λ

and

λ_{2} (x)

are a real parameter and a function related to the relaxation process for the minimization of J in

λ_{1}

Furthermore,

\begin{matrix} E (λ, λ_{1} (x), λ_{2} (x)) & = & λ [{(λ_{1} (x) - (1 - λ) λ_{2} (x))}^{3} E_{a} + {(1 - (λ_{1} (x) - (1 - λ) λ_{2} (x)))}^{3} E_{b}] \\ + (1 - λ) [{(λ_{1} (x) + λ λ_{2} (x))}^{3} E_{a} + {(1 - (λ_{1} (x) + λ λ_{2} (x)))}^{3} E_{b}], \end{matrix}

(295)

H_{i j k l} (λ, λ_{1} (x), λ_{2} (x)) = E (λ, λ_{1} (x), λ_{2} (x)) A_{i j k l},

where

A_{i j k l} = \tilde{λ} δ_{i j} δ_{k l} + \tilde{μ} (δ_{i k} δ_{j l} + δ_{i l} δ_{j k}),

\forall i, j, k, l \in {1, 2, 3} .

Here

{δ_{i j}}

is the Kronecker delta and

\tilde{λ} > 0

\tilde{μ} > 0

are appropriate real constants.

At this point we define

\begin{matrix} B & = & \{(λ_{1}, λ_{2}) : Ω \to R^{2} measurable : 0 \leq λ_{1} (x) \leq 1, \\ - 0.8 \leq λ_{2} (x) \leq 0.8, in Ω, \int_{Ω} λ_{1} (x) d x = c_{0} V o l (Ω), \int_{Ω} λ_{2} (x) d x = 0\}, \end{matrix}

(296)

and

\begin{matrix} A & = & {(u, λ, λ_{1}, λ_{2}) \in V \times [0, 1] \times B : \\ {(H_{i j k l} (λ, λ_{1}, λ_{2}) e_{k l} (u))}_{, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3}} . \end{matrix}

(297)

Observe that

\begin{matrix} inf_{(u, λ, λ_{1}, λ_{2}) \in A} J (u, λ, λ_{1}, λ_{2}) \\ = & inf_{(λ, λ_{1}, λ_{2}) \in [0, 1] \times B} \{sup_{\hat{u} \in V} \{inf_{u \in V} \{J (u, λ, λ_{1}, λ_{2}) + {〈 {\hat{u}}_{i}, {(H_{i j k l} (λ, λ_{1}, λ_{2}) e_{k l} (u))}_{j} + f_{i} 〉}_{L^{2}}\}\}\} \\ = & inf_{(λ, λ_{1}, λ_{2}) \in [0, 1] \times B} \{sup_{\hat{u} \in V} \{inf_{u \in V} \{\frac{1}{2} \int_{Ω} H_{i j k l} (λ, λ_{1}, λ_{2}) e_{i j} (u) e_{k l} (u) d x \\ + {〈 {\hat{u}}_{i}, {(H_{i j k l} (λ, λ_{1}, λ_{2}) e_{k l} (u))}_{j} + f_{i} 〉}_{L^{2}}\}\}\} \\ = & inf_{(λ, λ_{1}, λ_{2}) \in [0, 1] \times B} \{sup_{\hat{u} \in V} \{- \frac{1}{2} \int_{Ω} H_{i j k l} (λ, λ_{1}, λ_{2}) e_{i j} (\hat{u}) e_{k l} (\hat{u}) d x + {〈 {\hat{u}}_{i}, f_{i} 〉}_{L^{2}}\}\} \\ = & inf_{(λ, λ_{1}, λ_{2}) \in [0, 1] \times B} \{inf_{σ \in C^{*}} \{\frac{1}{2} \int_{Ω} \bar{H_{i j k l} (λ, λ_{1}, λ_{2})} σ_{i j} σ_{k l} d x\}\}, \end{matrix}

(298)

where

{\bar{H_{i j k l} (λ, λ_{1}, λ_{2})}} = {H_{i j k l} (λ, λ_{1}, λ_{2})}^{- 1}

in an appropriate tensor sense and

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

We have obtained numerical results concerning the optimal shape of a two-dimensional beam though the following algorithm:

Set $n = 1$ , $ε = 10^{- 4}$ , $λ_{n} = 1 / 2$ , ${(λ_{1})}_{n} (x) \equiv 1 / 2$ , ${(λ_{2})}_{n} (x) \equiv 0 .$
Calculate $u_{n} \in V$ such that

${(H_{i j k l} (λ_{n}, {(λ_{1})}_{n}, {(λ_{2})}_{n}) e_{k l} (u_{n}))}_{, j} + f_{i} = 0, in Ω, \forall i \in {1, 2, 3} .$
Calculate $λ_{n + 1} \in [0, 1]$ such that

$J (u_{n}, λ_{n + 1}, {(λ_{1})}_{n}, 0) = inf_{λ \in [0, 1]} {J (u_{n}, λ, {(λ_{1})}_{n}, 0)} .$
Calculate $({(λ_{1})}_{n + 1}, {(λ_{2})}_{n + 1}) \in B$ such that

$- J (u_{n}, λ_{n + 1}, {(λ_{1})}_{n + 1}, {(λ_{2})}_{n + 2}) = inf_{(λ_{1}, λ_{2}) \in B} {- J (u_{n}, λ_{n + 1}, λ_{1}, λ_{2})} .$
Set ${(λ_{2})}_{n + 1} \equiv 0$ .
If $∥ {(λ_{1})}_{n + 1} - {(λ_{1})}_{n} ∥_{\infty} \leq ε,$ then stop. Otherwise $n : = n + 1$ and go to item 2.

We developed numerical results for a two-dimensional beam, in a two-dimensional elasticity context for two cases, namely, case A and case B.

For the case A we consider a two-dimensional beam of dimensions

1 m \times 0.5 m

, clamped at

x = 0

and with a vertical load of

P = - 42000000 (4) 500 j

applied to the point

(x_{0}, y_{0}) = (1, 0.25)

For the case B, we consider a a two-dimensional beam of dimensions

1 m \times 0.5 m

, simply supported at

(x, y) = (0, 0)

and

(x, y) = (1, 0)

, with a vertical load

P = - 42000000 500 j

applied to the point

(x_{0}, y_{0}) = (1 / 3, 0.5)

Denoting

u = (u, v)

, for both cases we define the strain tensor as

e (u) = {(e_{x} (u), e_{y} (u), e_{x y} (u))}^{T},

where

e_{x} (u) = u_{x},

e_{y} (u) = v_{y},

and

e_{x y} (u) = \frac{1}{2} (u_{y} + v_{x}) .

We also set

E_{a} = 205 10^{9} P_{a}

and

E_{b} = 300 P_{a}

ν = 0.33

and

c_{0} = 0.6091

for both the cases.

Moreover the stress tensor

σ

is given by

σ = H (e (u)),

where

H = \frac{E (λ, λ_{1} (x), λ_{2} (x))}{1 - ν^{2}} \{\begin{matrix} 1 & ν & 0 \\ ν & 1 & 0 \\ 0 & 0 & \frac{1}{2} (1 - ν) \end{matrix}\} .

(299)

For the optimal shape obtained through

λ_{1}

for the case A, please see Figure 40.

For the optimal shape obtained through

λ_{1}

for the case B, please see Figure 41.

Here we present the software through which we have obtained such results, in a finite differences context for the case B.

We highlight the convergence criterion in the software is a little different from the one in the algorithm above described.

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

clear all

global P m8 d w Ea Eb Lo d1 z1 m9 du1 du2 dv1 dv2 c3 Lo1 L u v

m8=24;

m9=22;

c3=0.95;

d=1.0/m8;

d1=0.50/m9;

Ea= $410 * 10^{6} * 500;$

Eb=300;

w=0.30;

P=-42000000*500;

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

A3=zeros(2*z1,2*z1);

for i=1:z1

A3(1,i)=1.0;

A3(2,i+z1)=1.0;

end;

L=1/2;

b=zeros(2*z1,1);

b(1,1)=c3*z1;

for i=1:z1

uo(i,1)=0.0;

uo(i+z1,1)=-0.80;

end;

for i=1:z1

u1(i,1)=1.0;

u1(i+z1,1)=0.80;

end;

for i=1:m9-1

for j=1:m8-1

Lo(i,j)=c3;

Lo1(i,j)=0.1*c3;

end;

end;

for i=1:z1*2

x1(i,1)=c3*z1;

end;

x3(1,1)=1/2;

for i=1:4*m8*m9

xo(i,1)=0.000;

end;

xw=xo;

xv=Lo;

for k2=1:22

c3=0.98*c3;

b(1,1)=c3*z1;

k2

b14=1.0;

k3=0;

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

k3=k3+1;

b12=1.0;

k=0;

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

k=k+1;

k2

k3

k

X=fminunc(’funbeamMarch24’,xo); xo=X;

b12=max(abs(xw-xo))

xw=X;

end;

X1=fminunc(’funbeamMarch24A1’,x3);

x3=X1;

for i=1:m9-1

for j=1:m8-1

E1= $3 * L * ({(L o (i, j) - (1 - L) * L o 1 (i, j))}^{2} * E a - {(1 - (L o (i, j) - (1 - L) * L o 1 (i, j)))}^{2} * E b);$

E1=E1+ $3 * (1 - L) * ({(L o (i, j) + L * L o 1 (i, j))}^{2} * E a - {(1 - (L o (i, j) + L * L o 1 (i, j)))}^{2} * E b);$

E2= $3 * L * {(L o (i, j) - (1 - L) * L o 1 (i, j))}^{2} * E a * (- (1 - L)) - {(1 - (L o (i, j) - (1 - L) * L o 1 (i, j)))}^{2} * E b * (- (1 - L));$

E2=E2+ $3 * (1 - L) * ({(L o (i, j) + L * L o 1 (i, j))}^{2} * E a * L - {(1 - (L o (i, j) + L * L o 1 (i, j)))}^{2} * E b * L);$

ex=du1(i,j);

ey=dv2(i,j);

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

Sx1= $E 1 * (e x + w * e y) / (1 - w^{2});$

Sy1= $E 1 * (w * e x + e y) / (1 - w^{2});$

Sxy1= $E 1 / (2 * (1 + w)) * e x y;$

Sx2= $E 2 * (e x + w * e y) / (1 - w^{2});$

Sy2= $E 2 * (w * e x + e y) / (1 - w^{2});$

Sxy2= $E 2 / (2 * (1 + w)) * e x y;$

dc31(i,j)=-(Sx1*ex+Sy1*ey+2*Sxy1*exy);

dc32(i,j)=-(Sx2*ex+Sy2*ey+2*Sxy2*exy);

end;

end;

for i=1:m9-1

for j=1:m8-1

f(j+(i-1)*(m8-1))=dc31(i,j);

f((m9-1)*(m8-1)+j+(i-1)*(m8-1))=dc32(i,j);

end;

end;

for k1=1:1

k1

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

x1=X1;

end;

for i=1:z1

x1(i+z1,1)=0;

end;

for i=1:m9-1

for j=1:m8-1

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

Lo1(i,j)=X1((m8-1)*(m9-1)+j+(m8-1)*(i-1))*0.0;

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;

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

With the auxiliary function "funbeamMarch24"

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

function S=funbeamMarch24(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2 Lo1 L

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;

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;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E1= $L * ({(L o (i, j) - (1 - L) * L o 1 (i, j))}^{3} * E a + {(1 - (L o (i, j) - (1 - L) * L o 1 (i, j)))}^{3} * E b);$

E2= $(1 - L) * ({(L o (i, j) + L * L o 1 (i, j))}^{3} * E a + {(1 - (L o (i, j) + L * L o 1 (i, j)))}^{3} * E b);$

ex=du1(i,j);

ey=dv2(i,j);

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

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

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

Sxy= $(E 1 + E 2) / (2 * (1 + w)) * e x y;$

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

end;

end;

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

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

And the auxiliary function "funbeamMarch24A1"

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

function S1=funbeamMarch24A1(x)

global P m8 d w u v Ea Eb Lo d1 m9 du1 du2 dv1 dv2 L Lo1

L=(sin(x(1,1))+1)/2;

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;

end;

S=0;

for i=1:m9-1

for j=1:m8-1

E1= $L * ({(L o (i, j) - (1 - L) * L o 1 (i, j))}^{3} * E a + {(1 - (L o (i, j) - (1 - L) * L o 1 (i, j)))}^{3} * E b);$

E2= $(1 - L) * ({(L o (i, j) + L * L o 1 (i, j))}^{3} * E a + {(1 - (L o (i, j) + L * L o 1 (i, j)))}^{3} * E b);$

ex=du1(i,j);

ey=dv2(i,j);

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

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

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

Sxy= $(E 1 + E 2) / (2 * (1 + w)) * e x y;$

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

end;

end;

S1=S;

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

51. An Existence Result for a General Parabolic Non-Linear Equation

Let

Ω \subset R^{m}

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

𝜕 Ω .

Consider the parabolic non-linear equation

\{\begin{matrix} \frac{𝜕 u}{𝜕 t} = ε \nabla^{2} u + g (u) + \sum_{j = 1}^{m} g_{j} (u) \frac{𝜕 u}{𝜕 x_{j}} + f, & in Ω \times (0, T), \\ u (x, 0) = {\hat{u}}_{0}, in Ω, \\ u = 0, on 𝜕 Ω \times [0, T] . \end{matrix}

(300)

Here

ε > 0

f \in L^{2} ([0, T], W^{1, 2} (Ω)) \cap L^{\infty} (Ω \times [0, T]),

{\hat{u}}_{0} \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω),

where t denotes time and

[0, T]

is a time interval.

Also

g : R \to R

and

g_{j} : R \to R

are continuous functions neither necessarily linear nor convex,

\forall j \in {1, \dots, m}

We assume there exist

K_{33} > 0

and

K_{1} > 0

such that

{∥ g ∥}_{\infty} \leq \frac{K_{33}}{m {(Ω)}^{1 / 2}},

∥ g_{j} ∥_{\infty} \leq \frac{K_{1}}{m},

\forall j \in {1, \dots, n} .

At this point, we recall that fixing

γ > 0

{(I_{d} - γ \nabla^{2})}^{- 1} : L^{2} (Ω) \to H_{0}^{1} (Ω)

is a bounded and linear operator, so that for each

h \in L^{2} (Ω)

there exists a unique

u \in H_{0}^{1} (Ω)

such that

(I_{d} - γ \nabla^{2}) u = h .

In such a case we denote

u = {(I_{d} - γ \nabla^{2})}^{- 1} h,

so that

{∥ u ∥}_{1, 2, Ω} \leq ∥ {(I_{d} - γ \nabla^{2})}^{- 1} {∥ ∥ h ∥}_{0, 2, Ω} .

Moreover, fixing

N \in N

and defining

Δ t_{N} = \frac{T}{N},

in a partial finite differences context, discretizing in t consider the approximate equation system

\frac{u_{n + 1} - u_{n}}{Δ t_{N}} = ε \nabla^{2} u_{n + 1} + g (u_{n}) + \sum_{j = 1}^{m} g_{j} (u_{n}) {(u_{n})}_{x_{j}} + f_{n}, in Ω,

\forall n \in {0, 1, \dots, N - 1} .

From such a system, for

n = 0

, we obtain

u_{1} - {\hat{u}}_{0} = ε \nabla^{2} (u_{1}) Δ t_{N} + g ({\hat{u}}_{0}) Δ t_{N} + \sum_{j = 1}^{m} g_{j} ({\hat{u}}_{0}) {({\hat{u}}_{0})}_{x_{j}} Δ t_{N} + f_{0} Δ t_{N} .

Hence

u_{1} = {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ({\hat{u}}_{0} + g ({\hat{u}}_{0}) Δ t_{N} + \sum_{j = 1}^{m} g_{j} ({\hat{u}}_{0}) {({\hat{u}}_{0})}_{x_{j}} Δ t_{N} + f_{0} Δ t_{N}),

so that

\begin{matrix} ∥ u_{1} ∥_{1, 2, Ω} \\ \leq & ∥ {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ \\ \times (∥ {\hat{u}}_{0} ∥_{0, 2, Ω} + ∥ g ({\hat{u}}_{0}) ∥_{0, 2, Ω} Δ t_{N} + \sum_{j = 1}^{m} ∥ g_{j} ({\hat{u}}_{0}) {({\hat{u}}_{0})}_{x_{j}} ∥_{0, 2, Ω} Δ t_{N} + {∥ f_{0} ∥}_{0, 2, Ω} Δ t_{N}) . \end{matrix}

(301)

Observe that there exists

K_{2} > 0

such that

{∥ f ∥}_{\infty, Ω \times [0, T]} \leq K_{2}

so that

∥ f_{n} ∥_{1, 2, Ω} \leq K_{36}, \forall n \in {0, 1, \dots, N - 1},

for some appropriate

K_{36} > 0 .

From such results and the hypotheses, we may infer that

\begin{matrix} ∥ u_{1} ∥_{1, 2, Ω} \leq ∥ {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ (∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{33} Δ t_{N} + K_{1} ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} Δ t_{N} + K_{36} Δ t_{N}) \\ \leq & ∥ {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ (∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{1} ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} Δ t_{N} + K_{3} Δ t_{N}), \end{matrix}

(302)

where

K_{3} = K_{33} + K_{36},

so that

∥ u_{1} ∥_{1, 2, Ω} \leq α_{1} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + α_{2},

where

α_{1} = ∥ {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ (1 + K_{1} Δ t_{N}),

and

α_{2} = ∥ {(I_{d} - ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ K_{3} Δ t_{N} .

In fact, generically we may similarly obtain

∥ u_{n + 1} ∥_{1, 2, Ω} \leq α_{1} {∥ u_{n} ∥}_{1, 2, Ω} + α_{2},

\forall n \in {0, 1, \dots, N - 1} .

From such a result, inductively we may obtain

∥ u_{j} ∥_{1, 2, Ω} \leq {(α_{1})}^{j} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + \sum_{k = 0}^{j - 1} α_{1}^{k} α_{2} .

In particular for

j = N

, we get

\begin{matrix} ∥ u_{N} ∥_{1, 2, Ω} \\ \leq & {(α_{1})}^{N} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + \sum_{k = 0}^{N - 1} α_{1}^{k} α_{2} \\ = & {(α_{1})}^{N} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + \frac{1 - α_{1}^{N}}{1 - α_{1}} α_{2} \\ = & {∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥}^{N} {(1 + K_{1} \frac{T}{N})}^{N} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} \\ + \frac{1 - α_{1}^{N}}{1 - α_{1}} α_{2} . \end{matrix}

(303)

Observe that

\begin{matrix} {∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥}^{N} {(1 + K_{1} \frac{T}{N})}^{N} \\ \leq & {(1 + K_{1} \frac{T}{N})}^{N} \\ \to & e^{K_{1} T}, as N \to \infty . \end{matrix}

(304)

Also,

\begin{matrix} |\frac{α_{2}}{1 - α_{1}}| \\ = & \frac{∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ K_{3} \frac{T}{N}}{|1 - α_{1}|} \\ \leq & \frac{K_{3} \frac{T}{N}}{| 1 - α_{1} |} \\ = & \frac{K_{3} \frac{T}{N}}{|1 - ∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ (1 + K_{1} \frac{T}{N})|} \\ = & \frac{K_{3}}{\frac{N}{T} |1 - ∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ (1 + K_{1} \frac{T}{N})|} \\ = & \frac{K_{3}}{|\frac{N}{T} - (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1 + 1) (\frac{N}{T} + K_{1})|} \\ = & \frac{K_{3}}{|\frac{N}{T} - (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) (\frac{N}{T} + K_{1}) - \frac{N}{T} - K_{1}|} \\ = & ​ \\ = & \frac{K_{3}}{|- (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) (\frac{N}{T}) - K_{1} - (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ = & ​ \\ = & \frac{K_{3}}{|- (∥I_{d} + \sum_{j = 1}^{\infty} {(ε (\nabla^{2}) \frac{T}{N})}^{j}∥ - 1) (\frac{N}{T}) - K_{1} - (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ = & \frac{K_{3}}{|K_{1} + (∥I_{d} + \sum_{j = 1}^{\infty} {(ε (\nabla^{2}) \frac{T}{N})}^{j}∥ - 1) (\frac{N}{T}) + (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ \leq & \frac{K_{3}}{|K_{1} + (∥ I_{d} ∥ - ∥\sum_{j = 1}^{\infty} {(ε (\nabla^{2}) \frac{T}{N})}^{j}∥ - 1) (\frac{N}{T}) + (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ \leq & \frac{K_{3}}{|K_{1} + (∥ I_{d} ∥ - \sum_{j = 1}^{\infty} {∥ε (\nabla^{2}) \frac{T}{N}∥}^{j} - 1) (\frac{N}{T}) + (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ \leq & \frac{K_{3}}{|K_{1} - (\sum_{j = 1}^{\infty} {∥ε (\nabla^{2}) \frac{T}{N}∥}^{j}) (\frac{N}{T}) + (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ \leq & \frac{K_{3}}{|K_{1} - \frac{∥ ε (\nabla^{2}) ∥}{1 - ∥ ε \nabla^{2} ∥ \frac{T}{N}} + (∥{(I_{d} - ε (\nabla^{2}) \frac{T}{N})}^{- 1}∥ - 1) K_{1}|} \\ \to & \frac{K_{3}}{| K_{1} - ∥ ε (\nabla^{2}) ∥ |}, as N \to \infty . \end{matrix}

(305)

From such results we may infer that

|\frac{(1 - α_{1}^{N}) α_{2}}{1 - α_{1}}| \leq (1 + α_{1}^{N}) |\frac{α_{2}}{1 - α_{1}}|,

so that

\underset{N \to \infty}{lim sup} |\frac{(1 - α_{1}^{N}) α_{2}}{1 - α_{1}}| \leq \frac{(1 + e^{K_{1} T}) K_{3}}{| K_{1} - ∥ ε \nabla^{2} ∥ |} .

From these results, denoting now more generically

u_{n} \equiv u_{n}^{N}

joining the pieces, we have got

\underset{N \to \infty}{lim sup} ∥ u_{N}^{N} ∥_{1, 2, Ω} \leq e^{K_{1} T} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + \frac{(1 + e^{K_{1} T}) K_{3}}{| K_{1} - ∥ ε \nabla^{2} ∥ |} .

Consequently, we may infer that there exists

K_{4} > 0

such that

∥ u_{j}^{N} ∥_{1, 2, Ω} \leq K_{4}, \forall j \in {0, 1, \dots, N}, \forall N \in N .

Define now

u_{0}^{N} (x, t) = u_{n}^{N} (x) (n + 1 - \frac{t}{Δ t_{N}}) + u_{n + 1}^{N} (x) (\frac{t}{Δ t_{N}} - n),

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Observe that

u_{0}^{N} (x, t) = u_{n}^{N} (x), if t = n Δ t_{N}, \forall n \in {0, 1, \dots, N},

and

\begin{matrix} \frac{𝜕 u_{0}^{N} (x, t)}{𝜕 t} & = & \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} \\ = & ε \nabla^{2} u_{n + 1}^{N} + g (u_{n}^{N}) + \sum_{j = 1}^{m} g_{j} (u_{n}^{N}) {(u_{n}^{N})}_{x_{j}} + f_{n}, \end{matrix}

(306)

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Fix

φ \in C_{c}^{\infty} (Ω) .

Thus, fixing

t \in [n Δ t_{N}, (n + 1) Δ t_{N}],

we have

\begin{matrix} |{〈\frac{𝜕 u_{0}^{N}}{𝜕 t}, φ〉}_{L^{2}}| & \leq & ε | {〈 \nabla u_{n + 1}^{N}, \nabla φ 〉}_{L^{2}} | + | {〈 g (u_{n}^{N}), φ 〉}_{L^{2}} | \\ + \int_{Ω} |\sum_{j = 1}^{m} g_{j} (u_{n}^{N}) {(u_{n}^{N})}_{x_{j}} φ| d x + | {〈 φ, f_{n} 〉}_{L^{2}} | \\ \leq & ε ∥ u_{n + 1}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} + K_{1} ∥ u_{n}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} + K_{3} {∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{5} {∥ φ ∥}_{1, 2, Ω}, \forall φ \in C_{c}^{\infty} (Ω), \end{matrix}

(307)

for some appropriate

K_{5} > 0 .

Since

φ \in C_{c}^{\infty} (Ω)

is arbitrary, we may conclude that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω)} \leq K_{6}, \forall N \in N,

uniformly in t on

[0, T],

for some appropriate constant

K_{6} > 0 .

Also, from the definition of

u_{0}^{N}

we have that there exists

K_{7} > 0

such that

∥ u_{0}^{N} ∥_{1, 2, Ω} \leq K_{7}, \forall N \in N

also uniformly in t on

[0, T] .

From such results, there exist

u_{0} \in L^{2} ([0, T], H_{0}^{1} (Ω))

and

v_{0} \in L^{2} ([0, T]; H^{- 1} (Ω))

such that

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} ((0, T); W^{1, 2} (Ω)),

and

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in L^{2} ([0, T], H^{- 1} (Ω)),

so that we may easily obtain

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense.

At this point, we provide more details about this last result.

Fix

t \in (0, T)

. Thus, there exists

n \in {0, 1, \dots, N - 1}

such that

t \in [n Δ t_{N}, (n + 1) Δ t_{N}]

Let

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

From this, we may infer that

\begin{matrix} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x \\ = & \int_{Ω} \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} φ (x, t) d x \\ \leq & ε \int_{Ω} | \nabla u_{n + 1}^{N} \cdot \nabla φ | d x \\ + \int_{Ω} | g (u_{n}^{N}) φ (x, t) | d x + \int_{Ω} |\sum_{j = 1}^{m} g_{j} (u_{n}^{N}) {(u_{n}^{N})}_{x_{j}} φ| d x \\ + \int_{Ω} | f_{n} φ | d x \\ \leq & (K_{8} ∥ u_{n}^{N} ∥_{1, 2, Ω} + K_{20} {) ∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{9} {∥ φ ∥}_{1, 2, Ω}, \end{matrix}

(308)

for some appropriate constants

K_{8} > 0, K_{9} > 0, K_{20} > 0 .

Hence,

\begin{matrix} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x d x \\ \leq & K_{9} \int_{Ω} {∥ φ ∥}_{1, 2, Ω} d t \\ \leq & K_{19} {∥ φ ∥}_{1, 2, Ω \times (0, T)}, \end{matrix}

(309)

for some appropriate

K_{19} > 0

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω \times (0, T))} \leq K_{15},

for

N \in N

, for some

K_{15} > 0 .

From such a result and from the Banach-Alaoglu Theorem, there exists

v_{0} \in H^{- 1} (Ω \times (0, T))

such that, up to a not relabeled subsequence

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in H^{- 1} (Ω \times (0, T)) .

Therefore,

\int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \to \int_{0}^{T} \int_{Ω} v_{0} φ d x d t,

N \to \infty,

\forall φ \in H_{0}^{1} (Ω \times (0, T))

On the other hand

∥ u_{0}^{N} ∥_{0, 2, Ω \times (0, T)} \leq K_{16},

\forall N \in N

, for some

K_{16} > 0 .

From this and the Kakutani Theorem, there exists

u_{0} \in L^{2} (Ω \times (0, T))

such that, up to a not relabeled subsequence,

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T)) .

Now fix again

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

Observe that

\begin{matrix} \int_{0}^{T} \int_{Ω} u_{0} φ_{t} d x d t & = & lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} φ_{t} d x d t \\ = & - lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & - \int_{0}^{T} \int_{Ω} v_{0} φ d x d t, \end{matrix}

(310)

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

lim_{N \to \infty} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x = \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x,

\forall φ \in H_{0}^{1} (Ω) .

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

u_{0}^{N_{k} (t)} \to u_{0} (x, t), strongly in L^{2} (Ω), for almost all t \in [0, T] .

so that, up to subsequences,

u_{0}^{N_{k} (t)} (x, t) \to u_{0} (x, t), a . e . in Ω, for almost all t \in [0, T] .

Here we emphasise the sequence

{N_{k} (t)} \subset N

may depends on t.

Since g is continuous we have that

g (u_{0}^{N_{k} (t)} (x, t)) \to g (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix

t \in (0, T) .

Let

ε > 0

. From the Egorov Theorem, there exists a closed set F such that

m (Ω ∖ F) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | \leq ε, for almost all x \in F .

Let

φ \in C_{c}^{\infty} (Ω) .

Observe now that

\begin{matrix} |\int_{Ω} (g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t))) φ d x| \\ \leq & \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ = & \int_{F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x + \int_{Ω ∖ F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ \leq & \int_{F} {ε ∥ φ ∥}_{\infty} d x + \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | χ_{Ω ∖ F} d x \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + (∥ g (u_{0}^{N_{k} (t)}) ∥_{0, 2, Ω} + ∥ g (u_{0}) ∥_{0, 2, Ω} {) ∥ φ ∥}_{0, 4, Ω} {∥ χ_{Ω ∖ F} ∥}_{0, 4, Ω} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} m {(Ω ∖ F)}^{1 / 4} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} ε^{1 / 4}, \forall k > k_{0}, \end{matrix}

(311)

for some appropriate constant

K_{21} > 0

which does not depend on t.

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x \to \int_{Ω} g (u_{0}) φ d x, as k \to \infty,

\forall φ \in C_{c}^{\infty} (Ω) .

Similarly, fixing

j \in {1, \dots, n},

since

g_{j}

is continuous we have that

g_{j} (u_{0}^{N_{k} (t)} (x, t)) \to g_{j} (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix again

t \in (0, T)

Let

ε > 0

(a new value). From the Egorov Theorem, there exists a closed set

F_{1}

such that

m (Ω ∖ F_{1}) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | \leq ε, for almost all x \in F_{1} .

Observe now that

\begin{matrix} \int_{Ω} {| g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) |}^{2} d x \\ \leq & \int_{F_{1}} | g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) |^{2} d x + \int_{Ω ∖ F_{1}} {| g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) |}^{2} d x \\ \leq & \int_{F_{1}} ε^{2} d x + \int_{Ω} {| g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) |}^{2} χ_{Ω ∖ F_{1}} d x \\ \leq & ε^{2} m (Ω) + 2 K_{1}^{2} \int_{Ω} χ_{Ω ∖ F_{1}} d x \\ \leq & ε^{2} m (Ω) + 2 K_{1}^{2} ε, \forall k > k_{0} . \end{matrix}

(312)

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} {| g_{j} (u_{0}^{N_{k} (t)}) - g_{j} (u_{0}) |}^{2} d x \to 0, as k \to \infty,

\forall j \in {1, \dots, n} .

Select again

φ \in C_{c}^{\infty} (Ω) .

Since

∥ g_{j} (u_{0}^{N_{k} (t)}) - g_{j} (u_{0}) ∥_{0, 2, Ω} \to 0, as k \to \infty

and

\nabla u_{0}^{N_{k} (t)} ⇀ \nabla u_{0}, weakly in L^{2} (Ω; R^{m}),

we obtain,

\begin{matrix} |\int_{Ω} g_{j} (u_{0}^{N_{k} (t)}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x - \int_{Ω} g_{j} (u_{0}) {(u_{0})}_{x_{j}} φ d x| \\ \leq & |\int_{Ω} g_{j} (u_{0}^{N_{k} (t)}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x - \int_{Ω} g_{j} (u_{0}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x| \\ + |\int_{Ω} g_{j} (u_{0}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x - \int_{Ω} g_{j} (u_{0}) {(u_{0})}_{x_{j}} φ d x| \\ \leq & ∥ g_{j} (u_{0}^{N_{k} (t)}) - g_{j} (u_{0}) ∥_{0, 2, Ω} K_{7} {∥ φ ∥}_{\infty} \\ + |\int_{Ω} g_{j} (u_{0}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x - \int_{Ω} g_{j} (u_{0}) {(u_{0})}_{x_{j}} φ d x| \\ \to & 0, as k \to \infty, \end{matrix}

(313)

\forall j \in {1, \dots, n} .

From such results, we have

\begin{matrix} 0 & = & lim_{k \to \infty} (\int_{Ω} \frac{𝜕 u_{0}^{N_{k} (t)}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0}^{N_{k} (t)} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x - \sum_{j = 1}^{m} \int_{Ω} g_{j} (u_{0}^{N_{k} (t)}) {(u_{0}^{N_{k} (t)})}_{x_{j}} φ d x \\ - \int_{Ω} f^{N_{k} (t)} φ d x) \\ = & \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x - \sum_{j = 1}^{m} \int_{Ω} g_{j} (u_{0}) {(u_{0})}_{x_{j}} φ d x \\ - \int_{Ω} f φ d x . \end{matrix}

(314)

so that, from this and by the density of

C_{c}^{\infty} (Ω)

H_{0}^{1} (Ω)

, we have got

\begin{matrix} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x - \sum_{j = 1}^{m} \int_{Ω} g_{j} (u_{0}) {(u_{0})}_{x_{j}} φ d x \\ - \int_{Ω} f φ d x = 0, \forall φ \in H_{0}^{1} (Ω), \end{matrix}

(315)

a . e . on [0, T] .

Observe now that

𝜕 (Ω \times (0, T)) = (𝜕 Ω \times [0, T]) \cup (𝜕 [0, T] \times \bar{Ω}) .

Let

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

Hence

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t .

From this, since

C_{c}^{\infty} (Ω \times (0, T))

is dense

L^{2} (Ω \times (0, T))

we may infer that

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t,

\forall φ \in L^{2} (Ω \times (0, T)) .

Let

φ \in C^{\infty} (Ω \times [0, T])

such that

φ (x, T) = 0, in Ω .

From such results, we may obtain

\begin{matrix} lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & lim_{N \to \infty} (- \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x) \\ = & - \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0} (x, 0) φ (x, 0) d x . \end{matrix}

(316)

However, since

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T))

, we obtain

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t = \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t .

From these last results, we may infer that

\begin{matrix} \int_{Ω} {\hat{u}}_{0} φ (x, 0) d x & = & lim_{N \to \infty} \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x \\ = & \int_{Ω} u_{0} (x, 0) φ (x, 0) d x, \end{matrix}

(317)

so that

\int_{Ω} {\hat{u}}_{0} (x) φ (x, 0) d x = \int_{Ω} u_{0} (x, 0) φ (x, 0) d x,

\forall φ \in C^{\infty} (Ω \times [0, T])

such that

φ (x, T) = 0, in Ω .

Therefore, we may infer that

u_{0} (x, 0) = {\hat{u}}_{0} (x)

in this specified weak sense.

Similarly, it may be proven that

u_{0} = 0, on 𝜕 Ω \times [0, T],

in an appropriate weak sense.

Hence, we have obtained that

u_{0}

is a solution, in a weak sense, of the parabolic non-linear equation in question.

52. An Existence Result for a General Hyperbolic Non-Linear Equation

Let

Ω \subset R^{m}

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

𝜕 Ω .

Consider the hyperbolic non-linear equation

\{\begin{matrix} \frac{𝜕^{2} u}{𝜕 t^{2}} = ε \nabla^{2} u + g (u) + f, & in Ω \times (0, T), \\ u (x, 0) = {\hat{u}}_{0}, in Ω, \\ u (x, T) = u_{f}, in Ω, \\ u = 0, on 𝜕 Ω \times [0, T] . \end{matrix}

(318)

Here

ε > 0

f \in L^{2} ([0, T], W^{1, 2} (Ω)) \cap L^{\infty} (Ω \times [0, T]),

{\hat{u}}_{0}, u_{f} \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω),

where t denotes time and

[0, T]

is a time interval.

Also

g : R \to R

is a continuous function neither necessarily linear nor convex.

We assume there exists

K_{33} > 0

such that

{∥ g ∥}_{\infty} \leq \frac{K_{33}}{m {(Ω)}^{1 / 2}},

Fixing

N \in N

and defining

Δ t_{N} = \frac{T}{N},

in a partial finite differences context, discretizing in t consider the approximate equation system

\frac{u_{n + 1} - 2 u_{n} + u_{n - 1}}{Δ t_{N}^{2}} = ε \nabla^{2} u_{n} + g (u_{n}) + f_{n}, in Ω,

\forall n \in {1, \dots, N - 1} .

From such a system, for

n = 1

, we obtain

u_{2} - 2 u_{1} + {\hat{u}}_{0} = ε \nabla^{2} (u_{1}) Δ t_{N}^{2} + g (u_{1}) Δ t_{N}^{2} + f_{1} Δ t_{N}^{2} .

Hence

(2 I_{d} u + ε \nabla^{2} Δ t_{N}^{2}) u_{1} = (u_{2} + {\hat{u}}_{0} - g (u_{1}) Δ t_{N}^{2} - f_{1} Δ t_{N}^{2}),

so that

\begin{matrix} ∥ u_{1} ∥_{1, 2, Ω} \\ \leq & ∥ {(2 I_{d} + ε (\nabla^{2}) Δ t_{N}^{2})}^{- 1} ∥ \\ \times (∥ u_{2} ∥_{0, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{0, 2, Ω} + ∥ g (u_{1}) ∥_{0, 2, Ω} Δ t_{N}^{2} + {∥ f_{1} ∥}_{0, 2, Ω} Δ t_{N}^{2}) . \end{matrix}

(319)

Observe that there exists

K_{2} > 0

such that

{∥ f ∥}_{\infty, Ω \times [0, T]} \leq K_{2}

so that

∥ f_{n} ∥_{1, 2, Ω} \leq K_{3}, \forall n \in {0, 1, \dots, N - 1},

for some appropriate

K_{3} > 0 .

From such results and the hypotheses, we may infer that

\begin{matrix} ∥ u_{1} ∥_{1, 2, Ω} \leq ∥ {(2 I_{d} + ε (\nabla^{2}) Δ t_{N}^{2})}^{- 1} ∥ (∥ u_{2} ∥_{1, 2, Ω} + K_{33} Δ t_{N}^{2} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{3} Δ t_{N}^{2}) \\ \leq & ∥ {(2 I_{d} + ε (\nabla^{2}) Δ t_{N}^{2})}^{- 1} ∥ (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}), \end{matrix}

(320)

where

K_{85} = K_{33} + K_{3} .

On the other hand, through a symbolic auxiliary notation, we have

\begin{matrix} {(2 I_{d} + ε (\nabla^{2}) Δ t_{N})}^{- 1} & = & \frac{1}{(2 I_{d} + ε (\nabla^{2}) Δ t_{N}^{2})} \\ = & \frac{1}{2 (I_{d} + ε (\nabla^{2}) Δ t_{N}^{2} / 2)} \\ = & \frac{1}{2} (I_{d} - \frac{ε (\nabla^{2}) Δ t_{N}^{2} / 2}{I_{d} + ε (\nabla^{2}) Δ t_{N}^{2} / 2}), \end{matrix}

(321)

so that

∥ {(2 I_{d} + ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ \leq \frac{1}{2} + ∥\frac{ε (\nabla^{2}) Δ t_{N}^{2} / 4}{(I_{d} + ε (\nabla^{2}) Δ t_{N}^{2} / 2)}∥ .

Now denote

θ_{N} = ∥\frac{ε (\nabla^{2})}{(I_{d} + ε (\nabla^{2}) Δ t_{N}^{2} / 2)}∥ .

Thus,

∥ {(2 I_{d} + ε (\nabla^{2}) Δ t_{N})}^{- 1} ∥ \leq \frac{1}{2} + \frac{θ_{N} Δ t_{N}^{2}}{4},

so that

∥ u_{1} ∥_{1, 2, Ω} \leq (\frac{1}{2} + \frac{θ_{N} Δ t_{N}^{2}}{4}) (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}) .

Consequently, from such results, we may infer that

{(\frac{1}{2} + \frac{θ_{N} Δ t_{N}^{2}}{4})}^{- 1} ∥ u_{1} ∥_{1, 2, Ω} \leq (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}),

so that

2 (1 - \frac{θ_{N} Δ t_{N}^{2} / 2}{(1 + θ_{N} Δ t_{N}^{2} / 2)}) ∥ u_{1} ∥_{1, 2, Ω} \leq (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}) .

Therefore,

(2 - \frac{θ_{N} Δ t_{N}^{2}}{(1 + θ_{N} Δ t_{N}^{2} / 2)}) ∥ u_{1} ∥_{1, 2, Ω} \leq (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}) .

Let

ε_{1} \in R

be such that

0 < ε_{1} ≪ max {ε, 1} .

Define

\hat{α} = ε ∥ \nabla^{2} ∥

and observe that

\frac{θ_{N}}{(1 + θ_{N} Δ t_{N}^{2} / 2)} \to \hat{α}, as N \to \infty .

Hence, there exists

N_{0} \in N

such that if

N > N_{0},

then

|\frac{θ_{N}}{(1 + θ_{N} Δ t_{N}^{2} / 2)} - \hat{α}| < ε_{1} .

From these results, if

N > N_{0}

, we have

(2 - \frac{θ_{N} Δ t_{N}^{2}}{(1 + θ_{N} Δ t_{N}^{2} / 2)}) \geq (2 - (\hat{α} + ε_{1}) Δ t_{N}^{2}) > 0 .

Therefore, defining

α = \hat{α} + ε_{1},

we have got,

(2 - α Δ t_{N}^{2}) ∥ u_{1} ∥_{1, 2, Ω} \leq (∥ u_{2} ∥_{1, 2, Ω} + ∥ {\hat{u}}_{0} ∥_{1, 2, Ω} + K_{85} Δ t_{N}^{2}) .

so that

∥ u_{1} ∥_{1, 2, Ω} \leq α_{1} ∥ u_{2} ∥_{1, 2, Ω} + β_{1} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{1},

where

α_{1} = {(2 - α Δ t_{N}^{2})}^{- 1},

β_{1} = α_{1}

and

γ_{1} = α_{1} K_{85} Δ t_{N}^{2} .

Reasoning inductively, for

n \geq 2

having

∥ u_{n - 1} ∥_{1, 2, Ω} \leq α_{n - 1} ∥ u_{n} ∥_{1, 2, Ω} + β_{n - 1} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{n - 1},

we are going to obtain

α_{n}, β_{n}

and

γ_{n}

Similarly as above, we may obtain

\begin{matrix} (2 - α Δ t_{N}^{2}) {∥ u_{n} ∥}_{1, 2, Ω} \\ \leq & ∥ u_{n + 1} ∥_{1, 2, Ω} + {∥ u_{n - 1} ∥}_{1, 2, Ω} + K_{85} Δ t_{n}^{2}, \\ \leq & ∥ u_{n + 1} ∥_{1, 2, Ω} + α_{n - 1} ∥ u_{n} ∥_{1, 2, Ω} + β_{n - 1} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{n - 1} + K_{85} Δ t_{n}^{2} . \end{matrix}

(322)

Thus,

\begin{matrix} (2 - α Δ t_{N}^{2} - α_{n - 1}) {∥ u_{n} ∥}_{1, 2, Ω} \\ \leq & ∥ u_{n + 1} ∥_{1, 2, Ω} + β_{n - 1} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{n - 1} + K_{85} Δ t_{N}^{2} . \end{matrix}

(323)

Consequently,

∥ u_{n} ∥_{1, 2, Ω} \leq α_{n} ∥ u_{n + 1} ∥_{1, 2, Ω} + β_{n} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{n},

where

α_{n} = \frac{1}{2 - α Δ t_{N}^{2} - α_{n - 1}},

β_{n} = α_{n} β_{n - 1},

and

γ_{n} = α_{n} (γ_{n - 1} + K_{85} Δ t_{N}^{2}) .

We recall that

α = ε ∥ \nabla^{2} ∥ + ε_{1} .

Here we assume

T \geq 1

and

α T^{2} \leq \frac{1}{2} .

Consider the sequence

{b_{n}} \subset R

such that

b_{1} = 1 / 2,

and

b_{n + 1} = \frac{1}{2 - b_{n}}, \forall n \in N .

We may easily obtain by induction that

b_{n} = \frac{n}{n + 1} .

Define

a_{n} = b_{n - 1} = \frac{n - 1}{n}, \forall n \geq 2 .

Observe that

\begin{matrix} a_{n} + \frac{2 α T^{2}}{N} & \leq & \frac{N - 1}{N} + \frac{2 α T^{2}}{N} \\ \leq & \frac{N - 1}{N} + \frac{1}{N} \\ = & 1, \forall n \in {1, \dots, N - 1} . \end{matrix}

(324)

Observe that

\begin{matrix} α_{1} & \leq & \frac{1}{2 - α \frac{T^{2}}{N^{2}}} \\ = & \frac{1}{2} + (\frac{1}{2 - α \frac{T^{2}}{N^{2}}} - \frac{1}{2}) \\ \leq & \frac{1}{2} + \frac{α T^{2}}{N^{2}} \\ \leq & a_{1} + \frac{α T}{N} + \frac{α T^{2}}{N^{2}} . \end{matrix}

(325)

At this point we shall prove by induction that

α_{n} \leq a_{n} + \frac{α T}{N} + n \frac{α T^{2}}{N^{2}}, \forall n \in {1, \dots, N - 1} .

For

n = 1

we have already proved it above.

Suppose now that for

n \geq 1

, we have

α_{n} \leq a_{n} + \frac{α T}{N} + n \frac{α T^{2}}{N^{2}} .

Observe that

\begin{matrix} α_{n + 1} & = & \frac{1}{2 - α \frac{T^{2}}{N^{2}} - α_{n - 1}} \\ = & \frac{1}{2 - a_{n}} + (\frac{1}{2 - α \frac{T^{2}}{N^{2}} - α_{n - 1}} - \frac{1}{2 - a_{n}}) \\ = & a_{n + 1} + (\frac{1}{2 - α \frac{T^{2}}{N^{2}} - α_{n}} - \frac{1}{2 - a_{n}}) \\ \leq & a_{n + 1} + (- a_{n} + α_{n} + α \frac{T^{2}}{N^{2}}) \\ \leq & a_{n + 1} + \frac{α T}{N} + n \frac{α T^{2}}{N^{2}} + α \frac{T^{2}}{N^{2}} \\ = & a_{n + 1} + \frac{α T}{N} + (n + 1) \frac{α T^{2}}{N^{2}} . \end{matrix}

(326)

The induction is complete, indeed we have proven that

α_{n} \leq a_{n} + \frac{α T}{N} + n \frac{α T^{2}}{N^{2}}, \forall n \in {1, \dots, N - 1} .

Thus, we have obtained

\begin{matrix} α_{n} & \leq & a_{n} + \frac{α T}{N} + n \frac{α T^{2}}{N^{2}} \\ \leq & a_{n} + \frac{α T^{2}}{N} + \frac{α T^{2}}{N} \\ \leq & a_{n} + \frac{2 α T^{2}}{N} \\ \leq & 1, \forall n \in {1, \dots N - 1} . \end{matrix}

(327)

Summarizing,

0 \leq α_{n} \leq 1, \forall n \in {1, \dots, N - 1} .

Now denoting more generically

α_{n}^{N} = α_{n}

we may infer that

0 \leq α_{n}^{N} \leq 1, \forall n \in {1, \dots, N - 1}, \forall N > N_{0} .

From such results we may also obtain that there exist

K_{15} > 0

and

K_{16} > 0

such that

| β_{n}^{N} | \leq K_{15},

and

| γ_{n}^{N} | \leq K_{16},

\forall n \in {1, \dots N - 1}, \forall N > N_{0} .

We recall that

u_{N}^{N} = u_{f},

so that since

∥ u_{N - 1}^{N} ∥_{1, 2, Ω} \leq α_{N - 1}^{N} ∥ u_{N} ∥_{1, 2, Ω} + β_{N - 1}^{N} {∥ {\hat{u}}_{0} ∥}_{1, 2, Ω} + γ_{N - 1}^{N},

from this and these last results we may infer that

∥ u_{n}^{N} ∥_{1, 2, Ω} \leq K_{18},

\forall n \in {0, \dots, N - 1}, \forall N > N_{0},

for some appropriate real constant

K_{18} > 0 .

Define now

W^{N} (x, t) = \frac{u_{n + 1}^{N} - 2 u_{n}^{N} + u_{n - 1}^{N}}{Δ t_{N}^{2}},

(x, t) \in Ω \times (n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T],

and

u_{0}^{N} (x, t) = {\hat{u}}_{0} (x) + {(u_{1}^{N})}^{'} (x) t + \int_{0}^{t} \int_{0}^{τ} W^{N} (x, τ_{1}) d τ_{1} d τ,

where

{(u_{1}^{N})}^{'} (x)

is such that

u_{0}^{N} (x, T) = u_{f} (x) .

Here we highlight that

\begin{matrix} \frac{𝜕^{2} u_{0}^{N} (x, t)}{𝜕 t^{2}} & = & W^{N} (x, t) \\ = & \frac{u_{n + 1}^{N} - 2 u_{n}^{N} + u_{n - 1}^{N}}{Δ t_{N}^{2}}, \end{matrix}

(328)

(x, t) \in Ω \times (n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Observe that

\begin{matrix} \frac{𝜕^{2} u_{0}^{N} (x, t)}{𝜕 t^{2}} & = & \frac{u_{n + 1}^{N} - 2 u_{n}^{N} + u_{n - 1}^{N}}{Δ t_{N}^{2}} \\ = & ε \nabla^{2} u_{n}^{N} + g (u_{n}^{N}) + f_{n}, \end{matrix}

(329)

t \in (n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Fix

φ \in C_{c}^{\infty} (Ω) .

Thus, fixing

t \in (n Δ t_{N}, (n + 1) Δ t_{N}],

we have

\begin{matrix} |{〈\frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}}, φ〉}_{L^{2}}| & \leq & ε | {〈 \nabla u_{n}^{N}, \nabla φ 〉}_{L^{2}} | + | {〈 g (u_{n}^{N}), φ 〉}_{L^{2}} | \\ + | {〈 φ, f_{n} 〉}_{L^{2}} | \\ \leq & ε ∥ u_{n}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} + K_{19} ∥ u_{n}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} \\ + K_{25} {∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{26} {∥ φ ∥}_{1, 2, Ω}, \forall φ \in C_{c}^{\infty} (Ω), \end{matrix}

(330)

for some appropriate

K_{26} > 0 .

Since

φ \in C_{c}^{\infty} (Ω)

is arbitrary, we may conclude that

{∥\frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}}∥}_{H^{- 1} (Ω)} \leq K_{6}, \forall N > N_{0},

uniformly in t on

[0, T],

for some appropriate constant

K_{6} > 0 .

Also, from the definition of

u_{0}^{N}

we have that there exists

K_{7} > 0

such that

∥ u_{0}^{N} ∥_{1, 2, Ω} \leq K_{7}, \forall N > N_{0}

also uniformly in t on

[0, T] .

From such results, there exist

u_{0} \in L^{2} ([0, T], H_{0}^{1} (Ω))

and

v_{0} \in L^{2} ([0, T]; H^{- 1} (Ω))

such that

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} ((0, T); W^{1, 2} (Ω)),

and

\frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} ⇀ v_{0}, weakly - star in L^{2} ([0, T], H^{- 1} (Ω)),

so that we may easily obtain

v_{0} = \frac{𝜕^{2} u_{0}}{𝜕 t^{2}}

in a distributional sense.

At this point, we provide more details about this last result.

Fix

t \in (0, T)

. Thus, there exists

n \in {0, 1, \dots, N - 1}

such that

t \in (n Δ t_{N}, (n + 1) Δ t_{N}]

Let

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

From this, we may infer that

\begin{matrix} \int_{Ω} \frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} φ (x, t) d x \\ = & \int_{Ω} (\frac{u_{n + 1}^{N} - 2 u_{n}^{N} + u_{n - 1}^{N}}{Δ t_{N}^{2}}) φ (x, t) d x \\ \leq & ε \int_{Ω} | \nabla u_{n}^{N} \cdot \nabla φ | d x \\ + \int_{Ω} | g (u_{n}^{N}) φ (x, t) | d x \\ + \int_{Ω} | f_{n} φ | d x \\ \leq & (K_{8} (∥ u_{n}^{N} ∥_{1, 2, Ω} + K_{20} {) ∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{9} {∥ φ ∥}_{1, 2, Ω}, \end{matrix}

(331)

for some appropriate constants

K_{8} > 0, K_{9} > 0, K_{20} > 0 .

Hence,

\begin{matrix} \int_{0}^{T} \int_{Ω} \frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} φ (x, t) d x d x \\ \leq & K_{9} \int_{Ω} {∥ φ ∥}_{1, 2, Ω} d t \\ \leq & K_{19} {∥ φ ∥}_{1, 2, Ω \times (0, T)}, \end{matrix}

(332)

for some appropriate

K_{19} > 0

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

{∥\frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}}∥}_{H^{- 1} (Ω \times (0, T))} \leq K_{15},

for

N > N_{0}

, for some

K_{15} > 0 .

From such a result and from the Banach-Alaoglu Theorem, there exists

v_{0} \in H^{- 1} (Ω \times (0, T))

such that, up to a not relabeled subsequence

\frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} ⇀ v_{0}, weakly - star in H^{- 1} (Ω \times (0, T)) .

Therefore,

\int_{0}^{T} \int_{Ω} \frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} φ d x d t \to \int_{0}^{T} \int_{Ω} v_{0} φ d x d t,

N \to \infty,

\forall φ \in H_{0}^{1} (Ω \times (0, T))

On the other hand

∥ u_{0}^{N} ∥_{0, 2, Ω \times (0, T)} \leq K_{16},

\forall N > N_{0}

, for some

K_{16} > 0 .

From this and the Kakutani Theorem, there exists

u_{0} \in L^{2} (Ω \times (0, T))

such that, up to a not relabeled subsequence,

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T)) .

Now fix again

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

Observe that

\begin{matrix} \int_{0}^{T} \int_{Ω} u_{0} φ_{t t} d x d t & = & lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} φ_{t t} d x d t \\ = & lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} φ d x d t \\ = & \int_{0}^{T} \int_{Ω} v_{0} φ d x d t, \end{matrix}

(333)

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

v_{0} = \frac{𝜕^{2} u_{0}}{𝜕 t^{2}}

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

lim_{N \to \infty} \int_{Ω} \frac{𝜕^{2} u_{0}^{N}}{𝜕 t^{2}} φ d x = \int_{Ω} \frac{𝜕^{2} u_{0}}{𝜕 t^{2}} φ d x,

\forall φ \in H_{0}^{1} (Ω) .

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

u_{0}^{N_{k} (t)} \to u_{0} (x, t), strongly in L^{2} (Ω), for almost all t \in [0, T] .

so that, up to subsequences,

u_{0}^{N_{k} (t)} (x, t) \to u_{0} (x, t), a . e . in Ω, for almost all t \in [0, T] .

Here we emphasise the sequence

{N_{k} (t)} \subset N

may depends on t.

Since g is continuous we have that

g (u_{0}^{N_{k} (t)} (x, t)) \to g (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix

t \in (0, T) .

Let

ε > 0

. From the Egorov Theorem, there exists a closed set F such that

m (Ω ∖ F) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | \leq ε, for almost all x \in F .

Let

φ \in C_{c}^{\infty} (Ω) .

Observe now that

\begin{matrix} |\int_{Ω} (g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t))) φ d x| \\ \leq & \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ = & \int_{F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x + \int_{Ω ∖ F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ \leq & \int_{F} {ε ∥ φ ∥}_{\infty} d x + \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | χ_{Ω ∖ F} d x \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + (∥ g (u_{0}^{N_{k} (t)}) ∥_{0, 2, Ω} + ∥ g (u_{0}) ∥_{0, 2, Ω} {) ∥ φ ∥}_{0, 4, Ω} {∥ χ_{Ω ∖ F} ∥}_{0, 4, Ω} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} m {(Ω ∖ F)}^{1 / 4} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} ε^{1 / 4}, \forall k > k_{0}, \end{matrix}

(334)

for some appropriate constant

K_{21} > 0

which does not depend on t.

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x \to \int_{Ω} g (u_{0}) φ d x, as k \to \infty,

\forall φ \in C_{c}^{\infty} (Ω) .

From such results, we have

\begin{matrix} 0 & = & lim_{k \to \infty} (\int_{Ω} \frac{𝜕^{2} u_{0}^{N_{k} (t)}}{𝜕 t^{2}} φ d x + ε \int_{Ω} \nabla u_{0}^{N_{k} (t)} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x \\ - \int_{Ω} f^{N_{k} (t)} φ d x) \\ = & \int_{Ω} \frac{𝜕^{2} u_{0}}{𝜕 t^{2}} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x \\ - \int_{Ω} f φ d x . \end{matrix}

(335)

so that, from this and by the density of

C_{c}^{\infty} (Ω)

H_{0}^{1} (Ω)

, we have got

\begin{matrix} \int_{Ω} \frac{𝜕^{2} u_{0}}{𝜕 t^{2}} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x \\ - \int_{Ω} f φ d x = 0, \forall φ \in H_{0}^{1} (Ω), \end{matrix}

(336)

a . e . on [0, T] .

Hence, we have obtained that

u_{0}

is a solution, in a weak sense, of the hyperbolic non-linear equation in question.

53. A Numerical Procedure Combining the Euler Method and the Hyper-Finite Differences Approach

Let

Ω = [0, 1] \subset R

and consider the equation

\{\begin{matrix} ε u^{''} (x) - A u^{3} (x) + B u (x) + 1 = 0, & in Ω, \\ u (0) = 0, u (1) = 0 . \end{matrix}

(337)

Here

A > 0

B > 0

and

u \in W_{0}^{1, 2} (Ω) .

We may represent such an equation, as a first order system

\{\begin{matrix} v^{'} - A u^{3} / ε + B u) / ε + 1 / ε = 0, & in Ω, \\ u^{'} = v, in Ω, \\ u (0) = 0, u (1) = 0 . \end{matrix}

(338)

Consider now such a system with generical unknown boundary conditions

{\hat{u}}_{0}

and

{\hat{v}}_{0}

, that is,

\{\begin{matrix} v^{'} - A u^{3} / ε + B u / ε + 1 / ε = 0, & in Ω, \\ u^{'} = v, in Ω, \\ u (0) = {\hat{u}}_{0}, v (0) = {\hat{v}}_{0} . \end{matrix}

(339)

Defining

d = 1 / m_{8}

, where

m_{8}

is total number of nodes, in finite differences we have

\{\begin{matrix} \frac{v_{n} - v_{n - 1}}{d} - A u_{n - 1}^{3} / ε + B u_{n - 1} / ε + 1 / ε = 0, \\ \frac{u_{n} - u_{n - 1}}{d} = v_{n - 1} \\ u_{0} = {\hat{u}}_{0}, v_{0} = {\hat{v}}_{0} . \end{matrix}

(340)

This is simply the explicit Euler method. We may symbolically obtain

{u_{n}}

and

{v_{n}}

as functions of

{\hat{u}}_{0}

and

{\hat{v}}_{0}

(by using the MATHEMATICA or MAPLE software and by truncating the concerning polynomial solutions), through the iterations

\{\begin{matrix} v_{n} = v_{n - 1} + A u_{n - 1}^{3} \frac{d}{ε} - B u_{n - 1} \frac{d}{ε} - \frac{d}{ε}, \\ u_{n} = u_{n - 1} + v_{n - 1} d \\ u_{0} = {\hat{u}}_{0}, v_{0} = {\hat{v}}_{0} . \end{matrix}

(341)

However, it is well known the error in this process could be big. In order to minimize such an error, we use the hyper-finite differences approach for the one-dimensional analogous of the generalized method of lines. More specifically, we will subdivide the interval

[0.1]

into

N_{1}

sub-interval of same measure, and redefine a not relabeled d as

d = \frac{1}{m_{8} N_{1}} .

Hence, on each sub-interval

[\frac{k - 1}{N_{1}}, \frac{k}{N_{1}}]

, using the MATHEMATICA or MAPLE software we may obtain an approximate solution

{u_{i, k}, v_{i, k}}

as functions of the initial conditions

{u_{0, k}, v_{0, k}}

where

i \in {0, \dots, m_{8}}

\forall k \in {1 \dots, N_{1}} .

In order to obtain such a solution,

{u_{i, k}, v_{i, k}}

we use following interactions

\{\begin{matrix} v_{n, k} = v_{n - 1, k} + A u_{n - 1, k}^{3} \frac{d}{ε} - B u_{n - 1, k} \frac{d}{ε} - \frac{d}{ε}, \\ u_{n, k} = u_{n - 1, k} + v_{n - 1, k} d \\ u_{0, k} = {\hat{u}}_{0, k}, v_{0, k} = {\hat{v}}_{0, k} . \end{matrix}

(342)

Observe that for obtaining an approximate solution for the original equation in question, we must calculate

{{\hat{u}}_{0, k}, {\hat{v}}_{0, k}}

though the solution of the system:

For the boundary conditions:

u_{0, 1} = 0, u_{m_{8}, N_{1}} = 0 .

For the solution and its derivative continuity on the nodes related to the

N_{1}

sub-intervals,

u_{m 8, k} = u_{0, k + 1}, v_{m 8, k} = v_{0, k + 1}, \forall k \in {1, \dots N_{1}} .

Having obtained

{{\hat{u}}_{0, k}, {\hat{v}}_{0, k}}, \forall k \in {1, \dots, N_{1}}

we may obtain

{u_{n, k}, v_{n, k}} \forall n \in {0, \dots, m_{8}}, \forall k \in {1, \dots, N_{1}} .

Here we present the software in Mathematica through which we have obtained the numerical results, for the case

ε = 0.01,

A = B = 1

and

N_{1} = 16

subintervals.

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

m8 = 100;

N1 = 16;

d = 1.0/m8/N1;

e1 = 0.01;

A = 1.0;

B = 1.0;

For[k = 1, k < N1 + 1, k++,

Print[k];

u[0, k] = uo[k];

v[0, k] = vo[k];

For[i = 1, i < m8 + 1, i++,

z1 = $(v [i - 1, k] + A * d / e 1 * u {[i - 1, k]}^{3} - B * u [i - 1, k] * d / e 1 - 1.0 * d / e 1);$

z2 = u[i - 1, k] + v[i - 1, k]*d;

z1 = Series[z1, { uo[k], 0, 8}, { vo[k], 0, 8 }];

z2 = Series[z2, { uo[k], 0, 8 }, {vo[k], 0, 8 }];

z1 = Normal[z1];

z2 = Normal[z2];

v[i, k] = Expand[z1];

u[i, k] = Expand[z2]]];

S = $u {[0, 1]}^{2} + u {[m 8, N 1]}^{2};$

For[k = 1, k < N1, k++,

S = S + ${(u [m 8, k] - u [0, k + 1])}^{2}$ ;

S = S + ${(v [m 8, k] - v [0, k + 1])}^{2}$ ];

sol = FindMinimum[

S, {uo[1], uo[2], uo[3], uo[4], uo[5], uo[6], uo[7], uo[8], uo[9],

uo[10], uo[11], uo[12], uo[13], uo[14], uo[15], uo[16], vo[1],

vo[2], vo[3], vo[4], vo[5], vo[6], vo[7], vo[8], vo[9], vo[10],

vo[11], vo[12], vo[13], vo[14], vo[15], vo[16]}]

Clear[U];

For [k = 1, k < N1 + 1, k++,

w[k] = uo[k] $/ .$ sol[[2, k]]]

For[i = 1, i < N1 + 1, i++,

U[i - 1] = w[i]]

U[N1] = u[m8,N1];

For[i = 0, i < N1 + 1, i++,

Print["uo[", i + 1, "]=", U[i]]]

uo[1]=1.14453* $10^{- 25}$ , in fact $u (0) = 0$

uo[2]=0.817448

uo[3]=1.17018

uo[4]=1.28552

uo[5]=1.32107

uo[6]=1.33205

uo[7]=1.33546

uo[8]=1.3365

uo[9]=1.33677

uo[10]=1.33667

uo[11]=1.33596

uo[12]=1.33331

uo[13]=1.32382

uo[14]=1.2902

uo[15]=1.175

uo[16]=0.820243

uo[17]=0, in fact $u (1) = 0 .$

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

Remark 24.

Observe that along the domain the solution is approximately

1.33

which is close to

1.3247

, which is an approximate solution of equation

u^{3} - u - 1 = 0 .

This is expected since

ε = 0.01

is a relatively small value.

54. A Proximal Numerical Procedure Combined with the Euler Method

Let

Ω = [0, 1] \subset R

and consider the Ginzburg-Landau type equation

\{\begin{matrix} ε u^{''} (x) - A u^{3} (x) + B u (x) + 1 = 0, & in Ω, \\ u (0) = 0, u (1) = 0 . \end{matrix}

(343)

Here

A > 0

B > 0

and

u \in W_{0}^{1, 2} (Ω) .

We may represent such an equation, as a first order system

\{\begin{matrix} v^{'} - A u^{3} / ε + B u / ε + 1 / ε = 0, & in Ω, \\ u^{'} = v, in Ω, \\ u (0) = 0, u (1) = 0 . \end{matrix}

(344)

Consider now such a system with generical unknown boundary conditions

{\hat{u}}_{0}

and

{\hat{v}}_{0}

, that is,

\{\begin{matrix} v^{'} - A u^{3} / ε + B u / ε + 1 / ε = 0, & in Ω, \\ u^{'} = v, in Ω, \\ u (0) = {\hat{u}}_{0}, v (0) = {\hat{v}}_{0} . \end{matrix}

(345)

Defining

d = 1 / m_{8}

, where

m_{8}

is total number of nodes, in finite differences we have

\{\begin{matrix} \frac{v_{n} - v_{n - 1}}{d} - A u_{n - 1}^{3} / ε + B u_{n - 1} / ε + 1 / ε = 0, \\ \frac{u_{n} - u_{n - 1}}{d} = v_{n - 1} \\ u_{0} = {\hat{u}}_{0}, v_{0} = {\hat{v}}_{0} . \end{matrix}

(346)

This is simply the explicit Euler method. Setting

u_{0} = 0

, we may symbolically obtain

{u_{n}}

and

{v_{n}}

as functions of

v_{0} = {\hat{v}}_{0}

(by using the MATHEMATICA or MAPLE software and by truncating the concerning polynomial solutions), through the following iterations, which already include a proximal formulation about an initial fixed solution

{{(U_{0})}_{n}} .

\{\begin{matrix} v_{n} = v_{n - 1} + A u_{n - 1}^{3} \frac{d}{ε} - B u_{n - 1} \frac{d}{ε} - \frac{d}{ε}, \\ u_{n} = u_{n - 1} + v_{n - 1} d - \frac{K}{ε} (u_{n} - {(U_{0})}_{n}) d \\ u_{0} = 0, v_{0} = {\hat{v}}_{0} . \end{matrix}

(347)

\forall n \in {1, \dots, m_{8}} .

Indeed, in such a case we have

\{\begin{matrix} v_{n} = v_{n - 1} + A u_{n - 1}^{3} \frac{d}{ε} - B u_{n - 1} \frac{d}{ε} - \frac{d}{ε}, \\ u_{n} = (u_{n - 1} + v_{n - 1} d + \frac{K d}{ε} {(U_{0})}_{n}) / (1 + \frac{K d}{ε}) \\ u_{0} = 0, v_{0} = {\hat{v}}_{0} . \end{matrix}

(348)

\forall n \in {1, \dots, m_{8}} .

We emphasize such a procedure may make the error in the explicit Euler method very small, in fact proportional to

\frac{ε}{K} .

Thus, having obtained

u_{n} = u_{n} (v_{0})

, we may obtain

v_{0}

through the boundary condition

u (1) = 0

, that is, through a solution of equation

u_{m 8} (v_{0}) = 0

With such an

v_{0}

calculated, we may obtain explicitly

u_{n} = u_{n} (v_{0})

\forall n \in {1, \dots m_{8}} .

The next step is to replace

{{(U_{0})}_{n}}

{u_{n}}

and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for

ε = 0.01

A = B = 1

m 8 = 100

and

K = 10

Here we present the software through which we have obtained such results.

We highlight in this software we have fixed a total number of 800 iterations.

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

m8 = 100;

Clear[z1, z2, u, v, vo];

d = 1.0/m8;

e1 = 0.01;

A = 1.0;

B = 1.0;

K = 10.0;

For[i = 0, i < m8 + 1, i++,

uo[i] = 0.01];

For[k = 1, k < 800, k++, $(here we have fixed the number of iterations)$

Print[k];

Clear[vo];

u[0] = 0.0;

v[0] = vo;

For[i = 1, i < m8 + 1, i++,

z1 = (v[i - 1] + A*d/e1* $u {[i - 1]}^{3}$ - B*u[i - 1]*d/e1 - 1.0*d/e1);

z2 = (u[i - 1] + v[i - 1]*d + K*uo[i]*d/e1)/(K*d/e1 + 1.0);

z1 = Series[z1, {vo, 0, 9}];

z2 = Series[z2, {vo, 0, 9}];

z1 = Normal[z1];

z2 = Normal[z2];

v[i] = Expand[z1];

u[i] = Expand[z2]];

S = ${(u [m 8])}^{2};$

sol = FindMinimum[S, vo];

w = vo $/ .$ sol[[2, 1]];

vo = w;

For[i = 0, i < m8 + 1, i++,

uo[i] = u[i]];

Print[u[m8/2]]];

For[i = 0, i < m8/10 + 1, i++,

Print["u[", 10*i, "]=", u[10*i]]]

u[0]=0.

u[10]=1.09119

u[20]=1.29955

u[30]=1.32239

u[40]=1.32427

u[50]=1.3245

u[60]=1.32386

u[70]=1.31754

u[80]=1.27924

u[90]=1.04636

u[100]= $7.31252 * 10^{- 18}$

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

Remark 25.

Observe that along the domain the solution is close to

1.3247

, which is an approximate solution of equation

u^{3} - u - 1 = 0 .

This is expected since

ε = 0.01

is a relatively small value.

55. A Proximal Numerical Procedure Combined with the Euler Method for Solving Partial Differential Equations

Let

Ω = [0, 1] \times [0, 1] \subset R

and consider the Ginzburg-Landau type equation

\{\begin{matrix} ε \nabla^{2} u - A u^{3} + B u + f = 0, & in Ω, \\ u = 0, on 𝜕 Ω . \end{matrix}

(349)

Here

A > 0

B > 0

f \in L^{2} (Ω)

and

u \in W_{0}^{1, 2} (Ω) .

We may represent such an equation, as a partially first order system

\{\begin{matrix} v_{x} + u_{y y} - A u^{3} / ε + B u / ε + f / ε = 0, & in Ω, \\ u_{x} = v, in Ω, \\ u = 0, on 𝜕 Ω \end{matrix}

(350)

Defining

d = 1 / m_{8}

d_{1} = 1 / m_{9}

and denoting

m_{2} = [\begin{matrix} - 2 & 1 & 0 & 0 & \dots & 0 \\ 1 & - 2 & 1 & 0 & \dots & 0 \\ 0 & 1 & - 2 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 & - 2 & 1 \\ 0 & 0 & \dots & \dots & 1 & - 2 \end{matrix}],

(351)

where

m_{8}

is total number of nodes in x, and

m_{9}

is the number of nodes in y, in a finite differences context, we may have

\{\begin{matrix} \frac{v_{n} - v_{n - 1}}{d} + \frac{m_{2}}{d_{1}^{2}} u_{n - 1} - A u_{n - 1}^{3} / ε + B u_{n - 1} / ε + f_{n} / ε = 0, \\ \frac{u_{n} - u_{n - 1}}{d} = v_{n - 1} \\ u_{0} = 0, v_{0} = {\hat{v}}_{0} . \end{matrix}

(352)

This is simply an adaptation of the explicit Euler method. Observe that we may obtain

{u_{n}}

and

{v_{n}}

as functions of

v_{0} = {\hat{v}}_{0}

through the following iterations, which already include a proximal formulation about an initial fixed solution

{{(U_{0})}_{n}} .

\{\begin{matrix} v_{n} = v_{n - 1} - \frac{m_{2}}{d_{1}^{2}} u_{n - 1} d + A u_{n - 1}^{3} \frac{d}{ε} - B u_{n - 1} \frac{d}{ε} - \frac{f_{n} d}{ε}, \\ u_{n} = u_{n - 1} + v_{n - 1} d - \frac{K}{ε} (u_{n} - {(U_{0})}_{n}) d \\ u_{0} = 0, v_{0} = {\hat{v}}_{0} . \end{matrix}

(353)

\forall n \in {1, \dots, m_{8}} .

Indeed, in such a case we have, through a concerning linearization,

\{\begin{matrix} v_{n} = v_{n - 1} - \frac{m_{2}}{d_{1}^{2}} u_{n - 1} + 3 A {(u_{0})}_{n - 1}^{2} u_{n - 1} \frac{d}{ε} - 2 A {(u_{0})}_{n - 1}^{3} \frac{d}{ε} - B u_{n - 1} \frac{d}{ε} - \frac{f_{n} d}{ε}, \\ u_{n} = (u_{n - 1} + v_{n - 1} d + \frac{K d}{ε} {(U_{0})}_{n}) / (1 + \frac{K d}{ε}) \\ u_{0} = 0, v_{0} = {\hat{v}}_{0} . \end{matrix}

(354)

\forall n \in {1, \dots, m_{8}} .

We emphasize such a procedure may make the error in the explicit Euler method very small, in fact proportional to

\frac{ε}{K} .

Observe now that in particular for

n = 1

, we have

\begin{matrix} v_{1} & = & v_{0} - f_{1} \frac{d}{ε} \\ \equiv & {(M_{1})}_{1} v_{0} + {(y_{1})}_{1}, \end{matrix}

(355)

where

{(M_{1})}_{1} = I_{d} identity matrix (m_{9} - 1) \times (m_{9} - 1),

and

{(y_{1})}_{1} = - f_{1} \frac{d}{ε} .

Also,

\begin{matrix} u_{1} & = & (v_{0} d + K {(u_{0})}_{1} \frac{d}{ε}) / (1 + K \frac{d}{ε}) \\ \equiv & {(M_{2})}_{1} v_{0} + {(y_{2})}_{1}, \end{matrix}

(356)

where

{(M_{2})}_{1} = \frac{I_{d} d}{(1 + K \frac{d}{ε})},

and

{(y_{2})}_{1} = (\frac{K {(u_{0})}_{1} d}{ε}) / (1 + K \frac{d}{ε}) .

Reasoning inductively, having

v_{n - 1} = {(M_{1})}_{n - 1} v_{0} + {(y_{1})}_{n - 1},

and

u_{n - 1} = {(M_{2})}_{n - 1} v_{0} + {(y_{2})}_{n - 1},

and replacing such relations into the concerning system (354), we obtain

\begin{matrix} v_{n} & = & {(M_{1})}_{n - 1} + {(y_{1})}_{n - 1} - \frac{m_{2}}{d_{1}^{2}} ({(M_{2})}_{n - 1} + {(y_{2})}_{n - 1}) d \\ + 3 A {(u_{0})}_{n - 1}^{2} ({(M_{2})}_{n - 1} + {(y_{2})}_{n - 1}) \frac{d}{ε} - 2 A {(u_{0})}_{n - 1}^{3} \frac{d}{ε} \\ - B ({(M_{2})}_{n - 1} + {(y_{2})}_{n - 1}) \frac{d}{ε} - f_{n} \frac{d}{ε} \\ = & {(M_{1})}_{n} + {(y_{1})}_{n}, \end{matrix}

(357)

where

{(M_{1})}_{n} = {(M_{1})}_{n - 1} - \frac{m_{2}}{d_{1}^{2}} ({(M_{2})}_{n - 1}) d + 3 A {(u_{0})}_{n - 1}^{2} ({(M_{2})}_{n - 1}) \frac{d}{ε} - B ({(M_{2})}_{n - 1}) \frac{d}{ε},

and

\begin{matrix} {(y_{1})}_{n} & = & {(y_{1})}_{n - 1} - \frac{m_{2}}{d_{1}^{2}} ({(y_{2})}_{n - 1}) d + 3 A {(u_{0})}_{n - 1}^{2} ({(y_{2})}_{n - 1}) \frac{d}{ε} \\ - 2 A {(u_{0})}_{n - 1}^{3} \frac{d}{ε} - B ({(y_{2})}_{n - 1}) \frac{d}{ε} - f_{n} \frac{d}{ε} . \end{matrix}

(358)

Moreover,

\begin{matrix} u_{n} & = & ({(M_{2})}_{n - 1} v_{0} + {(y_{2})}_{n - 1} + {(M_{1})}_{n - 1} v_{0} d + {(y_{1})}_{n - 1} d + K {(u_{0})}_{n - 1} \frac{d}{ε}) / (1 + K \frac{d}{ε}) \\ = & {(M_{2})}_{n} v_{0} + {(y_{2})}_{n}, \end{matrix}

(359)

where

{(M_{2})}_{n} = \frac{{(M_{2})}_{n - 1} + {(M_{1})}_{n - 1} d}{(1 + K \frac{d}{ε})},

and

{(y_{2})}_{n} = ({(y_{2})}_{n - 1} + {(y_{1})}_{n - 1} d + K {(u_{0})}_{n - 1} \frac{d}{ε}) / (1 + K \frac{d}{ε}) .

Summarizing, we have obtained

v_{n} = {(M_{1})}_{n} v_{0} + {(y_{1})}_{n},

and

u_{n} = {(M_{2})}_{n} v_{0} + {(y_{2})}_{n},

\forall n \in {1, \dots, m_{8}} .

Consequently, from this and the boundary condition

u_{m_{8}} = 0,

we may have

u_{m_{8}} = 0 = {(M_{2})}_{m_{8}} v_{0} + {(y_{2})}_{m_{8}}

so that

v_{0} = - {[{(M_{2})}_{m_{8}}]}^{- 1} {(y_{2})}_{m_{8}} .

From such results we have obtained

{u_{n}}

and

{v_{n}}

\forall n \in {1, \dots, m_{8}} .

The next step is to replace

{{(u_{0})}_{n}}

{u_{n}}

and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for

ε = 0.01

A = B = 1

f \equiv 1, in Ω,

m 8 = 100

and

K = 100

For the solution

u = u (x, y)

obtained, please see Figure 42.

Here we present the software in MAT-LAB through which we have obtained such results.

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

clear all

m8=100;

m9=100;

d=1/m8;

d1=1/m9;

e1=0.01;

A=1;

B=1;

K=100;

f=ones(m9-1,1);

for i=1:m8

uo(:,i)=1.4*ones(m9-1,1);

Yo(:,i)=f;

end;

m2=zeros(m9-1,m9-1);

for i=2:m9-2

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(m9-1,m9-1)=-2.0;

m2(m9-1,m9-2)=1.0;

Id=eye(m9-1);

b12=1.0;

k=1;

while $(b 12 > 10^{- 10})$ && $(k < 9010)$

k

k=k+1;

M1(:,:,1)=Id;

y1(:,1)=-Yo(:,1)*d/e1;

M2(:,:,1)=Id*d/(K*d/e1+1);

y2(:,1)=K*uo(:,1)*(d/e1)/(K*d/e1+1);

for i=2:m8

M1(:,:,i)=M1(:,:,i-1)- $m 2 / d 1^{2}$ *d*M2(:,:,i-1)+3*A*diag(uo(:,i-1).*uo(:,i-1))*M2(:,:,i-1)*d/e1;

M1(:,:,i)=M1(:,:,i)-B*M2(:,:,i-1)*d/e1;

y1(:,i)=y1(:,i-1)- $m 2 / d 1^{2}$ *d*y2(:,i-1)+3*A*(uo(:,i-1).*uo(:,i-1)).*y2(:,i-1)*d/e1;

y1(:,i)=y1(:,i)-2*A*(uo(:,i-1).*uo(:,i-1).*uo(:,i-1))*d/e1-B*y2(:,i-1)*d/e1-Yo(:,i-1)*d/e1;

M2(:,:,i)=(M2(:,:,i-1)+d*M1(:,:,i-1))/(K*d/e1+1);

y2(:,i)=(y2(:,i-1)+d*y1(:,i-1)+K*uo(:,i)*d/e1)/(K*d/e1+1);

end;

vo(:,1)=-inv(M2(:,:,m8))*y2(:,m8);

for i=1:m8

u(:,i)=M2(:,:,i)*vo(:,1)+y2(:,i);

end;

u(m9/2,m8/2)

b12=max(max(abs(u-uo)));

uo=u;

end;

for i=1:m8

x1(i,1)=i*d;

end;

for j=1:m9-1

y3(j,1)=j*d1;

end;

mesh(x1,y3,u)

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

Remark 26.

Observe that along the domain the solution is close to

1.3247

, which is an approximate solution of equation

u^{3} - u - 1 = 0 .

This is expected since

ε = 0.01

is a relatively small value.

56. A Proximal Numerical Procedure Combined with the Euler Method for First Order Systems Applied to a Flight Mechanics Model

Let

Ω = [0, t_{f}]

be a time interval.

Consider the first order system of ordinary differential equations given by

\{\begin{matrix} \frac{d u_{j}}{d t} = f_{j} ({u_{l}}), on [0, t_{f}], \forall j \in {1, \dots, 4}, \\ u_{1} (0) = 0, u_{2} (0) = 0.12, u_{4} (0) = 0, u_{1} (t_{f}) = 11000 . \end{matrix}

(360)

Here

f_{j} : D_{j} \subset R^{4} \to R

is a smooth function on its domain

D_{j}

\forall j \in {1, \dots, 4} .

In finite differences, such a system stands for

\{\begin{matrix} \frac{{(u_{j})}_{n} - {(u_{j})}_{n - 1}}{d} = f_{j} ({u_{n - 1}}), \forall j \in {1, \dots, 4}, \\ {(u_{1})}_{0} = 0, {(u_{2})}_{0} = 0.12, {(u_{4})}_{0} = 0, {(u_{1})}_{m 8} = 11000 . \end{matrix}

(361)

\forall n \in {1, \dots, m_{8}},

where

m_{8}

is number of nodes and

d = t_{f} / m_{8} .

This is just the explicit Euler method. It is well known, at first the error in this procedure may be big.

However, such an error may be made very small by introducing a proximal formulation and related linearization about a fixed initial solution

{{(u_{0})}_{n}},

in a Newton type approach context.

In such a case the approximate system stands for

\{\begin{matrix} \frac{{(u_{1})}_{n} - {(u_{1})}_{n - 1}}{d} = f_{1} ({{(u_{0})}_{n - 1}}) \\ + \sum_{k = 1}^{4} \frac{𝜕 f_{1} ({{(u_{0})}_{n - 1}})}{𝜕 u_{k}} ({(u_{k})}_{n - 1} - {(u_{0_{k}})}_{n - 1}) - K_{5} ({(u_{1})}_{n} - ({(u_{0_{1}})}_{n}), \\ \frac{{(u_{j})}_{n} - {(u_{j})}_{n - 1}}{d} = f_{j} ({{(u_{0})}_{n - 1}}) + \sum_{k = 1}^{4} \frac{𝜕 f_{j} ({{(u_{0})}_{n - 1}})}{𝜕 u_{k}} ({(u_{k})}_{n - 1} - {(u_{0_{k}})}_{n - 1}), \\ \forall j \in {2, 3, 4}, \\ {(u_{1})}_{0} = 0, {(u_{2})}_{0} = 0.12, {(u_{4})}_{0} = 0, {(u_{1})}_{m 8} = 11000 . \end{matrix}

(362)

Indeed, setting the boundary conditions

{(u_{1})}_{0} = 0, {(u_{2})}_{0} = 0.12, {(u_{3})}_{0} = v_{0}, {(u_{4})}_{0} = 0

we will calculate

{{(u_{j})}_{n} (v_{0})}

through the following iterations

\{\begin{matrix} {(u_{1})}_{n} = ({(u_{1})}_{n - 1} + f_{1} ({{(u_{0})}_{n - 1}}) d + \\ \sum_{k = 1}^{4} \frac{𝜕 f_{1} ({{(u_{0})}_{n - 1}})}{𝜕 u_{k}} ({(u_{k})}_{n - 1} - {(u_{0_{k}})}_{n - 1}) d + K_{5} d ({(u_{0_{1}})}_{n})) / (1 + K_{5} d), \\ {(u_{j})}_{n} = {(u_{j})}_{n - 1} + f_{j} ({{(u_{0})}_{n - 1}}) d + \sum_{k = 1}^{4} \frac{𝜕 f_{j} ({{(u_{0})}_{n - 1}})}{𝜕 u_{k}} ({(u_{k})}_{n - 1} - {(u_{0_{k}})}_{n - 1}) d, \\ \forall j \in {2, 3, 4}, \\ {(u_{1})}_{0} = 0, {(u_{2})}_{0} = 0.12, {(u_{3})}_{0} = v_{0}, {(u_{4})}_{0} = 0 . \end{matrix}

(363)

Observe that the boundary condition

u_{1} (t_{f}) = 11000

corresponds to

{(u_{1})}_{m_{8}} (v_{0}) = 11000

so that, through this last equation we may obtain

v_{0} .

Having obtained

v_{0}

, we may obtain

{{(u_{j})}_{n}} = {{(u_{j})}_{n} (v_{0})}, \forall n \in {1, \dots, m_{8}},

\forall j \in {1, \dots, 4} .

The next step is to replace

{{(u_{0_{j}})}_{n}})

{{(u_{j})}_{n}}

and then to repeat the process until an appropriate convergence criterion is satisfied.

We have obtained numerical results for a model in flight mechanics.

More specifically, we model an in-plan climbing motion of an airplane AIR BUS 320, through the variables

h, γ, V, x

where h denotes the airplane altitude,

γ

is the angle between its velocity and the axis x, V is the airplane speed and x corresponds to its horizontal coordinate.

The concerning system of equations is given by

\{\begin{matrix} \dot{h} = V sin (γ) \\ \dot{γ} = \frac{1}{m_{f} V} (F sin (a + a_{F}) + L) - \frac{g}{V} cos (γ) \\ \dot{V} = \frac{1}{m_{f}} (F cos (a + a_{F}) - D) - g sin (γ) \\ \dot{x} = V cos (γ), on [0, t_{f}], \\ h (0) = 0, γ (0) = 0.12, x (0) = 0, h (t_{f}) = 11000 . \end{matrix}

(364)

Here

t_{f} = 515 s

F = 240000 N

m_{f} = 120000 K_{g}

S_{f} = 260 m^{2}

a = 0.138

g = 9.8 m / s^{2}

ρ (h) = 1.225 {(1 - \frac{0.0065 h}{288.15})}^{4.225} K_{g} / m^{3},

a_{F} = 0.0175

{(C_{L})}_{0} = 0,

{(C_{L})}_{a} = 5.0,

{(C_{D})}_{0} = 0.0175,

K_{1} = 0,

K_{2} = 0.06,

C_{L} = {(C_{L})}_{0} + {(C_{L})}_{a} a,

C_{D} = {(C_{D})}_{0} + K_{1} C_{L} + K_{2} C_{L}^{2},

L = \frac{1}{2} ρ (h) V^{2} C_{L} S_{f},

D = \frac{1}{2} ρ (h) V^{2} C_{D} S_{f} .

For numerical purposes, we define

u_{1} = h, u_{2} = γ (= b), u_{3} = V, u_{4} = x .

Here we present the software in MATHEMATICA through which we have obtained the numerical results.

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

m8 = 20000;

tf = 515.0;

d = tf/m8;

K5 = 10.0/d;

h1 = 11000.0;

Clear[h, b, V, x, u, a, c];

h = u[1];

b = u[2];

V = u[3];

x = u[4];

mf = 120000.0;

g = 9.8;

Sf = 260.0;

a = 0.138;

af = 0.0175;

CLo = 0.0;

CLa = 5.0;

CDo = 0.0175;

K1 = 0.0;

K2 = 0.06;

CL = CLo + CLa*a;

CD = CDo + K1*CL + K2* $C L^{2}$ ;

F = 240000.0;

r = $1.225 * {(1.0 - 0.0065 * h / 288.15)}^{4.225}$ ;

L = $1 / 2 * r * V^{2} * C L * S f;$

D1 = $1 / 2 * r * V^{2} * C D * S f;$

f[1] = V*Sin[b];

f[2] = 1/mf/V*(F*Sin[a + af] + L) - g/V*Cos[b];

f[3] = 1/mf*(F*Cos[a + af] - D1) - g*Sin[b];

f[4] = V*Cos[b];

For[i = 0, i < m8 + 1, i++,

uo[i, 1] = 11000*i/m8;

uo[i, 2] = 0.15;

uo[i, 3] = 120;

uo[i, 4] = 50000*i/m8];

Clear[u];

For[i = 1, i < 5, i++,

For[j = 1, j < 5, j++,

c[i, j] = D[f[i], u[j]]]];

uo[0, 1] = 0.0;

uo[0, 2] = 0.12;

uo[0, 3] = 120;

uo[0, 4] = 0.0;

For[k3 = 1, k3 < 30, k3++, (Here we have fixed a total of 30 iterations)

Print[k3];

Clear[vo, U];

U[0, 1] = 0.0;

U[0, 2] = 0.12;

U[0, 3] = vo;

U[0, 4] = 0.0;

For[i = 1, i < m8 + 1, i++,

Clear[u];

u[1] = uo[i - 1, 1];

u[2] = uo[i - 1, 2];

u[3] = uo[i - 1, 3];

u[4] = uo[i - 1, 4];

z1 = Expand[U[i - 1, 1] + K5*(uo[i, 1])*d + f[1]*d];

z2 = Expand[U[i - 1, 2] + 0.0*K5*(uo[i, 2])*d + f[2]*d];

z3 = Expand[U[i - 1, 3] + 0.0*K5*(uo[i, 3])*d + f[3]*d];

z4 = Expand[U[i - 1, 4] + 0.0*K5*(uo[i, 4])*d + f[4]*d];

For[k = 1, k < 5, k++,

z1 = z1 + c[1, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z2 = z2 + c[2, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z3 = z3 + c[3, k]*(U[i - 1, k] - uo[i - 1, k])*d;

z4 = z4 + c[4, k]*(U[i - 1, k] - uo[i - 1, k])*d;];

U[i, 1] = Expand[z1/(1.0 + K5*d)];

U[i, 2] = Expand[z2/(1.0 + 0.0*K5*d)];

U[i, 3] = Expand[z3/(1.0 + 0.0*K5*d)];

U[i, 4] = Expand[z4/(1.0 + 0.0*K5*d)]];

Print[U[m8, 1]];

S = (U[m8, 1] - h1);

sol = NSolve[S == 0, vo];

vo = vo $/ .$ sol[[1, 1]];

Print[vo];

Print[U[m8, 2]];

Print[U[m8, 3]];

Print[U[m8, 4]];

For[i = 0, i < m8 + 1, i++,

For[k = 1, k < 5, k++,

uo[i, k] = U[i, k]]];

Print[U[m8/2, 1]]];

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

For[i = 1, i < 11, i++,

Print["h(", 2000*i*d, "s)=U[", 2000*i, ",1]=", U[2000*i, 1]]]

h(51.5s)=U[2000,1]=1099.37

h(103.s)=U[4000,1]=2199.41

h(154.5s)=U[6000,1]=3299.45

h(206.s)=U[8000,1]=4399.5

h(257.5s)=U[10000,1]=5499.6

h(309.s)=U[12000,1]=6599.74

h(360.5s)=U[14000,1]=7699.8

h(412.s)=U[16000,1]=8799.76

h(463.5s)=U[18000,1]=9899.89

h(515.s)=U[20000,1]=11000.
For[i = 1, i < 11, i++,

Print["gamma(", 2000*i*d, "s)=U[", 2000*i, ",2]=", U[2000*i, 2]]]

gamma(51.5s)=U[2000,2]=0.120754

gamma(103.s)=U[4000,2]=0.120085

gamma(154.5s)=U[6000,2]=0.117905

gamma(206.s)=U[8000,2]=0.116329

gamma(257.5s)=U[10000,2]=0.119054

gamma(309.s)=U[12000,2]=0.125181

gamma(360.5s)=U[14000,2]=0.122861

gamma(412.s)=U[16000,2]=0.111435

gamma(463.5s)=U[18000,2]=0.115118

gamma(515.s)=U[20000,2]=0.115257
For[i = 1, i < 11, i++,

Print["V(", 2000*i*d, "s)=U[", 2000*i, ",3]=", U[2000*i, 3]]]

V(51.5s)=U[2000,3]=107.325

V(103.s)=U[4000,3]=113.338

V(154.5s)=U[6000,3]=119.7

V(206.s)=U[8000,3]=126.381

V(257.5s)=U[10000,3]=133.568

V(309.s)=U[12000,3]=142.044

V(360.5s)=U[14000,3]=152.19

V(412.s)=U[16000,3]=162.209

V(463.5s)=U[18000,3]=172.269

V(515.s)=U[20000,3]=185.79
For[i = 1, i < 11, i++,

Print["x(", 2000*i*d, "s)=U[", 2000*i, ",4]=", U[2000*i, 4]]]

x(51.5s)=U[2000,4]=5318.63

x(103.s)=U[4000,4]=10930.8

x(154.5s)=U[6000,4]=16860.9

x(206.s)=U[8000,4]=23137.6

x(257.5s)=U[10000,4]=29795.8

x(309.s)=U[12000,4]=36872.5

x(360.5s)=U[14000,4]=44395.

x(412.s)=U[16000,4]=52396.6

x(463.5s)=U[18000,4]=60960.3

x(515.s)=U[20000,4]=70129.5

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

57. A Review of the Convergence of Newton’s Method Combined with a Proximal Approach

Firstly we highlight similar results to those presented in this section have been presented in my book entitled "Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering", reference [8], in Chapter 25, page 488.

Let

f : R^{n} \to R

be a

C^{2}

class function and consider the problem of finding a critical point of f, there is, to find a point

{\hat{x}}_{0} \in R^{n}

such that

f^{'} ({\hat{x}}_{0}) = 0 .

Fix

k \in N

and let

x_{k} \in R^{n} .

Define

F : R^{n} \times R^{n} \to R

\begin{matrix} F (x, x_{k}) & = & f (x_{k}) + f^{'} (x_{k}) \cdot (x - x_{k}) + \frac{1}{2} [f^{''} (x_{k}) (x - x_{k})] \cdot (x - x_{k}) \\ + \frac{K}{2} {∥ x - x_{k} ∥}^{2}, \end{matrix}

(365)

for some

K > 0

to be specified.

Let

x_{k + 1} \in R^{n}

be such that

{[\frac{𝜕 F (x, x_{k})}{𝜕 x}]}_{x = x_{k + 1}} = 0,

so that

f^{'} (x_{k}) + f^{''} (x_{k}) (x_{k + 1} - x_{k}) + K (x_{k + 1} - x_{k}) = 0,

that is

x_{k + 1} = x_{k} - {(f^{''} (x_{k}) + K I_{d})}^{- 1} f^{'} (x_{k}) .

Now, assume

x_{0} \in R^{n}

is such that

∥ f^{''} (x) ∥ \leq {\hat{K}}_{1}, \forall x \in B_{r} (x_{0})

for some

r > 0 .

Assume

{\hat{K}}_{1} > 0

is such that

K - {\hat{K}}_{1} > 0 .

Suppose also

0 < α_{1} < 1

is such that

f^{''} (x) \geq α_{1} ({\hat{K}}_{1} + K) I_{d}

and

(1 - \frac{α_{1}}{4}) I_{d} \leq {(f^{''} (x) + K I_{d})}^{- 1} (f^{''} (y) + K I_{d}) \equiv H (x, y) \leq (1 + \frac{α_{1}}{4}) I_{d},

\forall x, y \in B_{r} (x_{0}) .

We recall that

∥ f^{''} (x) ∥ \leq {\hat{K}}_{1},

so that

(K - {\hat{K}}_{1}) I_{d} \leq K I_{d} + f^{''} (x),

and therefore

{(f^{''} (x) + K I_{d})}^{- 1} \leq \frac{I_{d}}{K - {\hat{k}}_{1}},

\forall x \in B_{r} (x_{0}) .

Suppose also

f^{'} (x) - f^{'} (y) = H_{5} (x, y) \cdot (x - y),

where

H_{5} (x, y)

is a symmetric matrix such that

0 \leq \frac{H_{5} (x, y)}{K - {\hat{K}}_{1}} \leq (1 - \frac{α_{1}}{2}) I_{d},

and

H_{5} (x, y) \geq α_{1} (K + {\hat{K}}_{1}) I_{d},

\forall x, y \in B_{r} (x_{0}) .

Assume also

K > 0

is such that

x_{1} \in B_{r (1 - α_{0})} (x_{0}),

where

α_{0} = (1 - \frac{3}{4} α_{1}) .

Reasoning inductively, suppose

x_{0}, x_{1}, \dots, x_{k + 1} \in B_{r} (x_{0}) .

Observe that

x_{k + 2} - x_{k + 1} = - {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} f^{'} (x_{k + 1}),

and

x_{k + 1} - x_{k} = - {(f^{''} (x_{k}) + K I_{d})}^{- 1} f^{'} (x_{k}),

so that

(f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 2} - x_{k + 1}) = - f^{'} (x_{k + 1}),

and

(f^{''} (x_{k}) + K I_{d}) (x_{k + 1} - x_{k}) = - f^{'} (x_{k}) .

Hence,

\begin{matrix} (f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 2} - x_{k + 1}) & = & (f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 1} - x_{k}) - f^{'} (x_{k + 1}) + f^{'} (x_{k}), \end{matrix}

so that

\begin{matrix} (x_{k + 2} - x_{k + 1}) & = & {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} [(f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 1} - x_{k}) - f^{'} (x_{k + 1}) + f^{'} (x_{k})] \\ = & {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} [(f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 1} - x_{k}) \\ - H_{5} (x_{k + 1}, x_{k}) (x_{k + 1} - x_{k})] \\ = & {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} [(f^{''} (x_{k + 1}) + K I_{d}) (x_{k + 1} - x_{k})] \\ - {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} H_{5} (x_{k + 1}, x_{k}) (x_{k + 1} - x_{k}) \\ = & [H (x_{k + 1}, x_{k}) - {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} H_{5} (x_{k + 1}, x_{k})] (x_{k + 1} - x_{k}) . \end{matrix}

(366)

Observe that

\begin{matrix} H_{5} (x_{k + 1}, x_{k}) & \geq & α_{1} ({\hat{K}}_{1} + K) I_{d} \\ \geq & α_{1} (f^{''} (x_{k + 1}) + K I_{d}), \end{matrix}

(367)

so that

{(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} H_{5} (x_{k + 1}, x_{k}) \geq α_{1} I_{d} .

Consequently, from such results we may infer that

\begin{matrix} I_{d} (1 - \frac{3}{4} α_{1}) \\ = & I_{d} (1 + \frac{α_{1}}{4}) - α_{1} I_{d} \\ \geq & H (x_{k + 1}, x_{k}) - {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} H_{5} (x_{k + 1}, x_{k}) \\ \geq & I_{d} (1 - \frac{α_{1}}{4}) - {(K - {\hat{K}}_{1})}^{- 1} I_{d} H_{5} (x_{k + 1}, x_{k}) \\ \geq & I_{d} (1 - \frac{α_{1}}{4}) - I_{d} (1 - \frac{α_{1}}{2}) \\ = & \frac{I_{d} α_{1}}{4} \\ \geq & 0 . \end{matrix}

(368)

from such results we may infer that

∥ H (x_{k + 1}, x_{k}) - {(f^{''} (x_{k + 1}) + K I_{d})}^{- 1} H_{5} (x_{k + 1}, x_{k}) ∥ \leq (1 - \frac{3 α_{1}}{4}) .

Defining

α_{0} = (1 - \frac{3 α_{1}}{4})

we have got

∥ x_{j + 2} - x_{j + 1} ∥ \leq α_{0} ∥ x_{j + 1} - x_{j} ∥, \forall j \in {1, \dots, k} .

Therefore

\begin{matrix} ∥ x_{j + 2} - x_{j + 1} ∥ & \leq & α_{0} ∥ x_{j + 1} - x_{j} ∥ \\ \leq & α_{0}^{2} ∥ x_{j} - x_{j - 1} ∥ \\ ⋮ & ​ \\ \leq & α_{0}^{j + 1} ∥ x_{1} - x_{0} ∥ . \end{matrix}

(369)

Thus,

\begin{matrix} ∥ x_{k + 2} - x_{1} ∥ & = & ∥ x_{k + 2} - x_{k + 1} + x_{k + 1} - \dots - x_{2} + x_{2} - x_{1} ∥ \\ \leq & ∥ x_{k + 2} - x_{k + 1} ∥ + ∥ x_{k + 1} - x_{k} ∥ + \dots + ∥ x_{2} - x_{1} ∥ \\ \leq & \sum_{j = 1}^{k + 1} α_{0}^{j} ∥ x_{1} - x_{0} ∥ \\ \leq & \sum_{j = 1}^{\infty} α_{0}^{j} ∥ x_{1} - x_{0} ∥ \\ = & \frac{α_{0}}{1 - α_{0}} ∥ x_{1} - x_{0} ∥ . \end{matrix}

(370)

Therefore

\begin{matrix} ∥ x_{k + 2} - x_{0} ∥ & \leq & ∥ x_{k + 2} - x_{1} + x_{1} - x_{0} ∥ \\ \leq & ∥ x_{k + 2} - x_{1} ∥ + ∥ x_{1} - x_{0} ∥ \\ \leq & \frac{α_{0}}{1 - α_{0}} ∥ x_{1} - x_{0} ∥ + ∥ x_{1} - x_{0} ∥ \\ = & \frac{1}{1 - α_{0}} ∥ x_{1} - x_{0} ∥ \\ \leq & \frac{1}{1 - α_{0}} (1 - α_{0}) r \\ = & r . \end{matrix}

(371)

Summarizing,

∥ x_{k + 2} - x_{0} ∥ < r,

so that

x_{k + 2} \in B_{r} (x_{0}) .

The induction is complete, so that

x_{k} \in B_{r} (x_{0}), \forall k \in N .

From such results we have also obtained

∥ x_{k + 2} - x_{k + 1} ∥ \leq α_{0} ∥ x_{k + 1} - x_{k} ∥, \forall k \in N .

Thus, from these results and the Banach fixed point theorem, there exists

{\hat{x}}_{0} \in \bar{B_{r}} (x_{0})

such that

lim_{k \to \infty} x_{k} = {\hat{x}}_{0} .

Hence,

\begin{matrix} 0 & = & lim_{k \to \infty} x_{k + 1} - x_{k} \\ = & lim_{k \to \infty} {(- f^{''} (x_{k}) + K I_{d})}^{- 1} f^{'} (x_{k}) \\ = & - {(f^{''} ({\hat{x}}_{0}) + K I_{d})}^{- 1} f^{'} ({\hat{x}}_{0}) . \end{matrix}

(372)

Since

det {(f^{''} ({\hat{x}}_{0}) + K I_{d})}^{- 1} \neq 0,

from this last equation we obtain

f^{'} ({\hat{x}}_{0}) = 0 .

The objective of this section is complete.

57.1. Applications to a Ginzburg-Landau Type Equation

Let

Ω = {[0, 1]}^{2} \subset R^{3}

and consider a functional

F : V \to R

where

\begin{matrix} F (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}

(373)

where

V = H_{0}^{1} (Ω)

f \in L^{2} (Ω)

α > 0, β > 0

and

γ > 0 .

Let

u \in H_{0}^{1} (Ω)

and

φ \in H_{0}^{1} (Ω) .

Observe that

\begin{matrix} δ F (u; φ) & = & γ \int_{Ω} \nabla u \cdot \nabla φ d x \\ α \int_{Ω} u^{3} φ d x - β \int_{Ω} u φ d x \\ - {〈 φ, f 〉}_{L^{2}} . \end{matrix}

(374)

Consider the problem of finding

u_{0} \in H_{0}^{1} (Ω)

such that

δ F (u_{0}; φ) = 0, \forall φ \in H_{0}^{1} (Ω) .

Fixing

N \in N,

consider now a mesh in finite differences for

Ω

, where we define

d = 1 / N

and the related grid

Ω_{N} = {(j / N, k / N) \forall j, k \in {0, 1, \dots, N} .

Denoting by

V_{N}

the finite-dimensional space in a finite diferences context corresponding to V and considering the functional F, we assume there exist

u_{0} \in V

, the corresponding

u_{0}^{N} \in V_{N}

and

r > 0

such that the hypotheses indicated in the last section also for the corresponding function

F_{N} : V_{N} \to R

are satisfied so that, as developed in such a previous section, we may obtain a solution

u_{N} : Ω \to R

such that

F_{N}^{'} (u_{N}) = 0

that is,

- γ \nabla_{N}^{2} u_{N} + α u_{N}^{3} - β u_{N} - f_{N} = 0,

where

\nabla_{N}^{2}

is the finite dimensional operator corresponding to the Laplace operator

\nabla^{2} .

Also,

F_{N}^{''} (u) = - γ \nabla_{N}^{2} + 3 α diag (u^{2}) - β I_{d},

so that

\begin{matrix} F_{N}^{'} (u_{1}) - F_{N}^{'} (u_{2}) & = & - γ \nabla_{N}^{2} u_{1} + α u_{1}^{3} - β u_{1} - f_{N} \\ - (- γ \nabla_{N}^{2} u_{2} + α u_{2}^{3} - β u_{2} - f_{N}) \\ = & - γ \nabla_{N}^{2} (u_{1} - u_{2}) + α (u_{1}^{3} - u_{2}^{3}) - β (u_{1} - u_{2}) \\ = & - γ \nabla_{N}^{2} (u_{1} - u_{2}) + 3 α ({\tilde{u}}^{2}) (u_{2} - u_{1}) - β (u_{1} - u_{2}) \\ = & (- γ \nabla_{N}^{2} + 3 α diag {\tilde{u}}^{2} - β I_{d}) (u_{2} - u_{1}) \\ = & F_{N}^{''} (\tilde{u}) (u_{2} - u_{1}) \end{matrix}

(375)

where

{(u_{1})}_{j} \leq {\tilde{u}}_{j} \leq {(u_{2})}_{j}

\forall u_{1}, u_{2} \in B_{r} (u_{0}) .

From such results, concerning the notation of the last section, we may infer that

\begin{matrix} H_{5} (u_{1}, u_{2}) & = & F_{N}^{''} (\tilde{u} (u_{1}, u_{2})) \\ = & - γ \nabla_{N}^{2} + 3 α diag {{[(\tilde{u}) (u_{1}, u_{2})]}^{2}} - β I_{d} . \end{matrix}

(376)

Now fix

M, N \in N .

Observe that

- γ \nabla_{N}^{2} u_{N} + α u_{N}^{3} - β u_{N} - f_{N} = 0,

and

- γ \nabla_{M}^{2} u_{M} + α u_{M}^{3} - β u_{M} - f_{M} = 0 .

At this point, denoting

u^{N} = {u_{j, k}^{N}},

we define

{\tilde{u}}_{0}^{N} (x, y) = \{\begin{matrix} u_{j, k}^{N}, if (x, y) \in ((j - 1) d, j d] \times ((k - 1) d, k d], \\ \forall j, k \in {1, \dots, N} . \end{matrix}

(377)

We also denote for a not relabeled operator

\nabla_{N}^{2}

\nabla_{N}^{2} ({\tilde{u}}_{0}^{N} (x, y)) = \{\begin{matrix} \frac{u_{j + 1, k}^{N} - 2 u_{j, k}^{N} + u_{j - 1, k}^{N}}{d^{2}} + \frac{u_{j, k + 1}^{N} - 2 u_{j, k}^{N} + u_{j, k - 1}^{N}}{d^{2}}, \\ if (x, y) \in ((j - 1) d, j d] \times ((k - 1) d, k d], \\ \forall j, k \in {1, \dots, N} . \end{matrix}

(378)

and

\nabla_{N}^{2} ({\tilde{u}}_{0}^{N} (x, y)) = [\nabla^{2} {\tilde{u}}_{0}^{N}] (x - d, y - d), if x \in (1 - d, 1] or y \in (1 - d, 1] .

Moreover, we define

u_{0}^{N} (x, y) = {(\nabla^{2})}^{- 1} (\nabla_{N}^{2} ({\tilde{u}}_{0}^{N} (x, y))), in Ω .

Observe that

\begin{matrix} - γ \nabla^{2} u_{0}^{N} & = & - γ \nabla_{N}^{2} {\tilde{u}}_{0}^{N} \\ = & - α {({\tilde{u}}_{0}^{N})}^{3} + β {\tilde{u}}_{0}^{N} + f_{N} \\ = & - α {(u_{0}^{N})}^{3} + β u_{0}^{N} + f_{N} \\ - α [{({\tilde{u}}_{0}^{N})}^{3} - {(u_{0}^{N})}^{3}] + β ({\tilde{u}}_{0}^{N} - u_{0}^{N}), \end{matrix}

(379)

Similarly, we may obtain

\begin{matrix} - γ \nabla^{2} u_{0}^{M} & = & - γ \nabla_{M}^{2} {\tilde{u}}_{0}^{N} \\ = & - α {(u_{0}^{M})}^{3} + β u_{0}^{M} + f_{M} \\ - α [{({\tilde{u}}_{0}^{M})}^{3} - {(u_{0}^{M})}^{3}] + β ({\tilde{u}}_{0}^{M} - u_{0}^{M}) . \end{matrix}

(380)

Consequently, from such results, we have

\begin{matrix} u_{0}^{N} - u_{0}^{M} \\ = & {(- γ \nabla^{2} + 3 α {({\hat{u}}^{N, M})}^{2} - β I_{d})}^{- 1} \\ \times [f_{N} - f_{M} - 3 α {({\hat{u}}^{N})}^{2} (u_{0}^{N} - {\tilde{u}}_{0}^{N}) + 3 α {({\hat{u}}^{M})}^{2} (u_{0}^{M} - {\tilde{u}}_{0}^{M}) \\ + β (u_{0}^{N} - {\tilde{u}}_{0}^{N}) - β (u_{0}^{M} - {\tilde{u}}_{0}^{M})] \end{matrix}

(381)

where

{\hat{u}}^{N}

is on the line connecting

u_{0}^{N}

{\tilde{u}}_{0}^{N}

and

{\hat{u}}^{M}

is on the line connecting

u_{0}^{M}

and

{\tilde{u}}_{0}^{M}

and

{\hat{u}}^{N, M}

is on the line connecting

u_{0}^{N}

and

u_{0}^{M} .

From these results, we obtain

\begin{matrix} ∥ u_{0}^{N} - u_{0}^{M} ∥_{1, 2, Ω} \\ = & ∥ {(- γ \nabla^{2} + 3 α {({\hat{u}}^{N, M})}^{2} - β I_{d})}^{- 1} ∥ \\ \times [∥ f_{N} - f_{M} ∥_{0, 2, Ω} + 3 α ∥ ({\hat{u}}^{N}) ∥_{0, 4, Ω}^{2} ∥ (u_{0}^{N} - {\tilde{u}}_{0}^{N}) ∥_{0, 2, Ω} + 3 α ∥ ({\hat{u}}^{M}) ∥_{0, 4, Ω}^{2} {∥ (u_{0}^{M} - {\tilde{u}}_{0}^{M}) ∥}_{0, 2, Ω} \\ + β ∥ (u_{0}^{N} - {\tilde{u}}_{0}^{N}) ∥_{0, 2, Ω} + β {∥ (u_{0}^{M} - {\tilde{u}}_{0}^{M}) ∥}_{0, 2, Ω}] \\ \leq & [K_{8} ∥ f_{N} - f_{M} ∥_{0, 2, Ω} + K_{9} ∥ (u_{0}^{N} - {\tilde{u}}_{0}^{N}) ∥_{0, 2, Ω} + K_{9} {∥ (u_{0}^{M} - {\tilde{u}}_{0}^{M}) ∥}_{0, 2, Ω}] \end{matrix}

(382)

for some appropriate constants

K_{8} > 0, K_{9} > 0 .

Let

ε > 0

Observe that there exists

N_{0} \in N

such that if

M, N > N_{0}

, then

∥ f_{N} - f_{M} ∥_{0, 2, Ω} < \frac{ε}{3 K_{8}},

∥ (u_{0}^{N} - {\tilde{u}}_{0}^{N}) ∥_{0, 2, Ω} \leq \frac{ε}{3 K_{9}},

and

∥ (u_{0}^{M} - {\tilde{u}}_{0}^{M}) ∥_{0, 2, Ω} \leq \frac{ε}{3 K_{9}},

so that,

∥ u_{0}^{N} - u_{0}^{M} ∥_{1, 2, Ω} < ε .

Therefore,

{u_{0}^{N}}

is a Cauchy sequence in

H_{0}^{1} (Ω)

so that there exists

{\hat{u}}_{0} \in H_{0}^{1} (Ω)

such that

u_{0}^{N} \to {\hat{u}}_{0}, strongly in H_{0}^{1} (Ω) .

Let

φ \in H_{0}^{1} (Ω) .

From such results and from the Sobolev Imbedding theorem, we may infer that

\begin{matrix} 0 & = & lim_{N \to \infty} (γ {〈 \nabla u_{0}^{N}, \nabla φ 〉}_{L^{2}} \\ + α {〈 {(u_{0}^{N})}^{3}, φ 〉}_{L^{2}} - β {〈 u_{0}^{N}, φ 〉}_{L^{2}} \\ - {〈 f_{N}, φ 〉}_{L^{2}}) \\ = & (γ {〈 \nabla {\hat{u}}_{0}, \nabla φ 〉}_{L^{2}} \\ + α {〈 {\hat{u}}_{0}^{3}, φ 〉}_{L^{2}} - β {〈 {\hat{u}}_{0}, φ 〉}_{L^{2}} \\ - {〈 f, φ 〉}_{L^{2}}) . \end{matrix}

(383)

Thus,

γ {〈 \nabla {\hat{u}}_{0}, \nabla φ 〉}_{L^{2}} + α {〈 {\hat{u}}_{0}^{3}, φ 〉}_{L^{2}} - β {〈 {\hat{u}}_{0}, φ 〉}_{L^{2}} - {〈 f, φ 〉}_{L^{2}} = 0,

\forall φ \in H_{0}^{1} (Ω) .

From this result we may infer that

{\hat{u}}_{0}

is a weak solution of equation

F^{'} ({\hat{u}}_{0}) = 0 .

58. On the Convergence of the Newton’s Method Combined with a Proximal Formulation for a General Parabolic Equation

Let

Ω \subset R^{m}

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

𝜕 Ω .

Consider the parabolic non-linear equation

\{\begin{matrix} \frac{𝜕 u}{𝜕 t} = ε \nabla^{2} u + g (u) + f, & in Ω \times (0, T), \\ u (x, 0) = {\hat{u}}_{0}, in Ω, \\ u = 0, on 𝜕 Ω \times [0, T] . \end{matrix}

(384)

Here

ε > 0

f \in L^{2} ([0, T], W^{1, 2} (Ω)) \cap L^{\infty} (Ω \times [0, T]),

{\hat{u}}_{0} \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω),

where t denotes time and

[0, T]

is a time interval.

Also

g : R \to R

is a continuous function neither necessarily linear nor convex.

We assume there exists

r > 0

such that

∥ g^{'} {(u) ∥}_{\infty} \leq K_{2},

and

{∥ g (u) ∥}_{\infty} \leq {\hat{K}}_{7}

\forall u \in B_{r} ({\hat{u}}_{0}),

for some

K_{2} > 0

and

{\hat{K}}_{7} > 0 .

Here

B_{r} ({\hat{u}}_{0}) = {u \in H_{0}^{1} (Ω) : ∥ u - {\hat{u}}_{0} ∥_{1, 2, Ω} < r} .

We assume also there exists

K_{1} > 0

such that

- K_{1} I_{d} \leq - ε \nabla^{2} - g^{'} (u) \leq K_{1} I_{d}, \forall u \in B_{r} ({\hat{u}}_{0}) .

Moreover, fixing

N \in N

and defining

Δ t_{N} = \frac{T}{N},

in a partial finite differences context, discretizing in t consider the approximate equation system

\frac{u_{n + 1} - u_{n}}{Δ t_{N}} = ε \nabla^{2} u_{n + 1} + g (u_{n + 1}) + f_{n}, in Ω,

\forall n \in {0, 1, \dots, N - 1} .

Fix

M \in N

. In a finite elements context for the variable

x \in R^{n}

, denoting

h_{M} = L_{0} / M

, for an appropriate fixed

L_{0} > 0

consider a mesh with a concerning thickness

h_{M}

and a related solution

u_{n}^{M}

of the following system

\frac{u_{n + 1}^{M} - u_{n}^{M}}{Δ t_{N}} = ε \nabla_{M}^{2} u_{n + 1}^{M} + g (u_{n + 1}^{M}) + f_{n}^{M}, in Ω,

\forall n \in {0, 1, \dots, N - 1} .

Here

\nabla_{M}^{2}

is the operator in a finite elements context corresponding to the Laplace operator

\nabla^{2}

We highlight in the next lines, as the meaning is clear, we may denote simply

\nabla_{M}^{2} = \nabla^{2} .

Observe that there exists a not relabeled

r > 0

K_{1} > 0

and

K_{2} > 0

such that

∥ g^{'} (u^{M}) ∥ \leq K_{2},

and

- K_{1} I_{d} \leq - ε \nabla_{M}^{2} - g^{'} (u^{M}) \leq K_{1} I_{d},

\forall u^{M} \in B_{r} ({\hat{u}}_{0}^{M}),

\forall M \in N .

Observe also that there exists

N_{0} \in N

such that if

N > N_{0}

, then

0 < \frac{K + 2 K_{2} Δ t_{N}}{K + 1 - K_{1} Δ t_{N}} < 1 .

Indeed, we may find

α_{0} \in R

such that

0 < \frac{K + 2 K_{2} Δ t_{N}}{K + 1 - K_{1} Δ t_{N}} < α_{0} < 1, \forall N > N_{0} .

Let

M_{N} \subset N

be a sequence such that

M_{N} < M_{N + 1}, \forall N \in N .

Fix

N > N_{0}

For

n = 0

, we are going to calculate

u_{1} = u_{1}^{M_{n}, N}

though the following iterations, which already include a proximal formulation and concerning linearization.

Set

u_{1}^{0} = {\hat{u}}_{0} = {\hat{u}}_{0}^{M_{N}, N},

Having

u_{1}^{k}

, let

u_{1}^{k + 1}

be such that

\begin{matrix} u_{1}^{k + 1} - {\hat{u}}_{0} & = & ε \nabla^{2} u_{1}^{k + 1} Δ t_{N} + g (u_{1}^{k})) Δ t_{N} \\ + g^{'} (u_{1}^{k}) (u_{1}^{k + 1} - u_{1}^{k}) Δ t_{N} + f_{1} Δ t_{N} - K (u_{1}^{k + 1} - u_{1}^{k}), \end{matrix}

(385)

Here we suppose

K \equiv K_{n}^{N} > 0

is large enough so that

u_{1}^{1} \in B_{\frac{r (1 - α_{0})}{N}} (u_{0}) .

Reasoning inductively, suppose

u_{1}^{0}, u_{1}^{1}, \dots, u_{1}^{k + 1} \in B_{\frac{r}{N}} (u_{0}),

and observe that

\begin{matrix} u_{1}^{k + 1} - {\hat{u}}_{0} - ε \nabla^{2} u_{1}^{k + 1} Δ t_{N} - g (u_{1}^{k})) Δ t_{N} \\ - g^{'} (u_{1}^{k}) (u_{1}^{k + 1} - u_{1}^{k}) Δ t_{N} - f_{1} Δ t_{N} + K (u_{1}^{k + 1} - u_{1}^{k}) \\ = & 0, \end{matrix}

(386)

and

\begin{matrix} u_{1}^{k + 2} - {\hat{u}}_{0} - ε \nabla^{2} u_{1}^{k + 2} Δ t_{N} - g (u_{1}^{k + 1})) Δ t_{N} \\ - g^{'} (u_{1}^{k + 1}) (u_{1}^{k + 2} - u_{1}^{k + 1}) Δ t_{N} - f_{1} Δ t_{N} + K (u_{1}^{k + 2} - u_{1}^{k + 1}) \\ = & 0, \end{matrix}

(387)

so that for an appropriate

{\tilde{u}}_{1}^{k}

\begin{matrix} (I_{d} - ε \nabla_{N}^{2} Δ t_{N} - g^{'} (u_{1}^{k + 1}) Δ t_{N} + K I_{d}) (u_{1}^{k + 2} - u_{1}^{k + 1}) \\ = & ((- g^{'} ({\tilde{u}}_{1}^{k}) + g^{'} (u_{1}^{k + 1})) Δ t_{N} + K I_{d}) (u_{1}^{k + 1} - u_{1}^{k}) . \end{matrix}

(388)

Hence,

\begin{matrix} ∥ u_{1}^{k + 2} - u_{1}^{k + 1} ∥ \\ \leq & ∥{(I_{d} - ε \nabla_{N}^{2} Δ t_{N} - g^{'} (u_{1}^{k + 1}) Δ t_{N} + K I_{d})}^{- 1} ((- g^{'} ({\tilde{u}}_{1}^{k}) + g^{'} (u_{1}^{k + 1})) Δ t_{N} + K I_{d})∥ \\ \times ∥u_{1}^{k + 1} - u_{1}^{k}∥ \\ \leq & \frac{K + 2 K_{2} Δ t_{N}}{K + 1 - K_{1} Δ t_{N}} ∥u_{1}^{k + 1} - u_{1}^{k}∥ \\ \leq & α_{0} ∥u_{1}^{k + 1} - u_{1}^{k}∥ . \end{matrix}

(389)

Thus, we have got

∥ u_{1}^{j + 2} - u_{1}^{j + 1} ∥ \leq α_{0} ∥ u_{1}^{j + 1} - u_{1}^{j} ∥, \forall j \in {1, \dots, k} .

Therefore

\begin{matrix} ∥ u_{1}^{j + 2} - u_{1}^{j + 1} ∥ & \leq & α_{0} ∥ u_{1}^{j + 1} - u_{1}^{j} ∥ \\ \leq & α_{0}^{2} ∥ u_{1}^{j} - u_{1}^{j - 1} ∥ \\ ⋮ & ​ \\ \leq & α_{0}^{j + 1} ∥ u_{1}^{1} - u_{1}^{0} ∥ . \end{matrix}

(390)

Thus,

\begin{matrix} ∥ u_{1}^{k + 2} - u_{1}^{1} ∥ & = & ∥ u_{1}^{k + 2} - u_{1}^{k + 1} + u_{1}^{k + 1} - \dots - u_{1}^{2} + u_{1}^{2} - u_{1}^{1} ∥ \\ \leq & ∥ u_{1}^{k + 2} - u_{1}^{k + 1} ∥ + ∥ u_{1}^{k + 1} - u_{1}^{k} ∥ + \dots + ∥ u_{1}^{2} - u_{1}^{1} ∥ \\ \leq & \sum_{j = 1}^{k + 1} α_{0}^{j} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & \sum_{j = 1}^{\infty} α_{0}^{j} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ = & \frac{α_{0}}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ . \end{matrix}

(391)

Therefore

\begin{matrix} ∥ u_{1}^{k + 2} - u_{1}^{0} ∥ & \leq & ∥ u_{1}^{k + 2} - u_{1}^{1} + u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & ∥ u_{1}^{k + 2} - u_{1}^{1} ∥ + ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & \frac{α_{0}}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ + ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ = & \frac{1}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ < & \frac{1}{1 - α_{0}} (1 - α_{0}) \frac{r}{N} \\ = & \frac{r}{N} . \end{matrix}

(392)

Summarizing,

∥ u_{1}^{k + 2} - u_{1}^{0} ∥ < \frac{r}{N},

so that

u_{1}^{k + 2} \in B_{\frac{r}{N}} (u_{0}^{1}) .

The induction is complete, so that

u_{1}^{k} \in B_{\frac{r}{N}} (u_{0}^{1}), \forall k \in N .

From such results we have also obtained

∥ u_{1}^{k + 2} - u_{1}^{k + 1} ∥ \leq α_{0} ∥ u_{1}^{k + 1} - u_{1}^{k} ∥, \forall k \in N .

Thus, from these results and the Banach fixed point theorem, there exists

u_{1} = u_{1}^{M_{N}, N} \in B_{\frac{r}{N}} (u_{0}^{1})

such that

lim_{k \to \infty} u_{1}^{k} = u_{1} = u_{1}^{M_{N}, N} .

\begin{matrix} 0 & = & lim_{k \to \infty} (u_{1}^{k + 1} - {\hat{u}}_{0} \\ - ε \nabla^{2} u_{1}^{k + 1} Δ t_{N} - g (u_{1}^{k})) Δ t_{N} \\ - g^{'} (u_{1}^{k}) (u_{1}^{k + 1} - u_{1}^{k}) Δ t_{N} - f_{1} Δ t_{N} + K (u_{1}^{k + 1} - u_{1}^{k})) \\ = & u_{1} - {\hat{u}}_{0} - ε \nabla^{2} u_{1} Δ t_{N} - g (u_{1}) Δ t_{N} - f_{1} Δ t_{N}, \end{matrix}

(393)

so that

\frac{u_{1} - {\hat{u}}_{0}}{Δ t_{N}} = ε \nabla^{2} u_{1} + g (u_{1}) + f_{1}, in Ω,

Reasoning inductively again having

u_{1} \in B_{\frac{r}{N}} ({\hat{u}}_{0})

and

u_{j} \in B_{\frac{r}{N}} (u_{j - 1}), \forall j \in {2, \dots, n}

similarly as we have obtained

u_{1}

in the last lines, we may obtain

u_{n + 1} = u_{n + 1}^{M_{N}, N} \in B_{\frac{r}{N}} (u_{n}),

such that

\frac{u_{n + 1} - u_{n}}{Δ t_{N}} = ε \nabla^{2} u_{n + 1} + g (u_{n + 1}) + f_{n}, in Ω .

The induction on n is also complete.

Fix

n \in {1, \dots, N - 1}

Observe that

\begin{matrix} ∥ u_{n} - {\hat{u}}_{0} ∥ & = & ∥ u_{n} - u_{n - 1} + u_{n - 1} - u_{n - 2} + \dots - u_{1} + u_{1} - u_{0} ∥ \\ \leq & ∥ u_{n} - u_{n - 1} ∥ + \dots + ∥ u_{1} - {\hat{u}}_{0} ∥ \\ \leq & \frac{n}{N} r \\ < & r \end{matrix}

(394)

Summarizing

u_{n} \in B_{r} ({\hat{u}}_{0}), \forall n \in {0, 1, \dots, N - 1} .

From these results, denoting now more generically

u_{n} \equiv u_{n}^{M_{N}, N} = u_{n}^{N}

, we may infer that there exists

K_{4} > 0

such that

∥ u_{j}^{N} ∥ \leq K_{4}, \forall j \in {0, 1, \dots, N}, \forall N \in N .

With a completely analogous reasoning, we may obtain that

∥ u_{j}^{N} ∥_{1, 2, Ω} \leq {\hat{K}}_{4}, \forall j \in {0, 1, \dots, N}, \forall N \in N,

for some

{\hat{K}}_{4} > 0 .

Define now

u_{0}^{N} (x, t) = u_{n}^{N} (x) (n + 1 - \frac{t}{Δ t_{N}}) + u_{n + 1}^{N} (x) (\frac{t}{Δ t_{N}} - n),

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Observe that

u_{0}^{N} (x, t) = u_{n}^{N} (x), if t = n Δ t_{N}, \forall n \in {0, 1, \dots, N},

and

\begin{matrix} \frac{𝜕 u_{0}^{N} (x, t)}{𝜕 t} & = & \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} \\ = & ε \nabla^{2} u_{n + 1}^{N} + g (u_{n + 1}^{N}) + f_{n}, \end{matrix}

(395)

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Fix

φ \in C_{c}^{\infty} (Ω) .

Thus, fixing

t \in [n Δ t_{N}, (n + 1) Δ t_{N}],

we have

\begin{matrix} |{〈\frac{𝜕 u_{0}^{N}}{𝜕 t}, φ〉}_{L^{2}}| & \leq & ε | {〈 \nabla u_{n + 1}^{N}, \nabla φ 〉}_{L^{2}} | + | {〈 g (u_{n + 1}^{N}), φ 〉}_{L^{2}} | \\ + | {〈 φ, f_{n} 〉}_{L^{2}} | \\ \leq & ε ∥ u_{n + 1}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} + K_{18} ∥ u_{n + 1}^{N} ∥_{1, 2, Ω} {∥ φ ∥}_{1, 2, Ω} + K_{3} {∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{5} {∥ φ ∥}_{1, 2, Ω}, \forall φ \in C_{c}^{\infty} (Ω), \end{matrix}

(396)

for some appropriate

K_{5} > 0 .

Since

φ \in C_{c}^{\infty} (Ω)

is arbitrary, we may conclude that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω)} \leq K_{6}, \forall N > N_{0},

uniformly in t on

[0, T],

for some appropriate constant

K_{6} > 0 .

Also, from the definition of

u_{0}^{N}

we have that there exists

K_{7} > 0

such that

∥ u_{0}^{N} ∥_{1, 2, Ω} \leq K_{7}, \forall N \in N

also uniformly in t on

[0, T] .

From such results, there exist

u_{0} \in L^{2} ([0, T], H_{0}^{1} (Ω))

and

v_{0} \in L^{2} ([0, T]; H^{- 1} (Ω))

such that

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} ((0, T); W^{1, 2} (Ω)),

and

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in L^{2} ([0, T], H^{- 1} (Ω)),

so that we may easily obtain

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense.

At this point, we provide more details about this last result.

Fix

t \in (0, T)

. Thus, there exists

n \in {0, 1, \dots, N - 1}

such that

t \in [n Δ t_{N}, (n + 1) Δ t_{N}]

Let

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

From this, we may infer that

\begin{matrix} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x \\ = & \int_{Ω} \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} φ (x, t) d x \\ \leq & ε \int_{Ω} | \nabla u_{n + 1}^{N} \cdot \nabla φ | d x \\ + \int_{Ω} | g (u_{n + 1}^{N}) φ (x, t) | d x + \int_{Ω} | f_{n} φ | d x \\ \leq & (K_{8} ∥ u_{n + 1}^{N} ∥_{1, 2, Ω} + K_{20} {) ∥ φ ∥}_{1, 2, Ω} \\ \leq & K_{9} {∥ φ ∥}_{1, 2, Ω}, \end{matrix}

(397)

for some appropriate constants

K_{8} > 0, K_{9} > 0, K_{20} > 0 .

Hence,

\begin{matrix} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x d x \\ \leq & K_{9} \int_{Ω} {∥ φ ∥}_{1, 2, Ω} d t \\ \leq & K_{19} {∥ φ ∥}_{1, 2, Ω \times (0, T)}, \end{matrix}

(398)

for some appropriate

K_{19} > 0

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω \times (0, T))} \leq K_{15},

for

N \in N

, for some

K_{15} > 0 .

From such a result and from the Banach-Alaoglu Theorem, there exists

v_{0} \in H^{- 1} (Ω \times (0, T))

such that, up to a not relabeled subsequence

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in H^{- 1} (Ω \times (0, T)) .

Therefore,

\int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \to \int_{0}^{T} \int_{Ω} v_{0} φ d x d t,

N \to \infty,

\forall φ \in H_{0}^{1} (Ω \times (0, T))

On the other hand

∥ u_{0}^{N} ∥_{0, 2, Ω \times (0, T)} \leq K_{16},

\forall N \in N

, for some

K_{16} > 0 .

From this and the Kakutani Theorem, there exists

u_{0} \in L^{2} (Ω \times (0, T))

such that, up to a not relabeled subsequence,

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T)) .

Now fix again

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

Observe that

\begin{matrix} \int_{0}^{T} \int_{Ω} u_{0} φ_{t} d x d t & = & lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} φ_{t} d x d t \\ = & - lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & - \int_{0}^{T} \int_{Ω} v_{0} φ d x d t, \end{matrix}

(399)

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T))

is arbitrary, we may infer that

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

lim_{N \to \infty} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x = \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x,

\forall φ \in H_{0}^{1} (Ω) .

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

u_{0}^{N_{k} (t)} \to u_{0} (x, t), strongly in L^{2} (Ω), for almost all t \in [0, T] .

so that, up to subsequences,

u_{0}^{N_{k} (t)} (x, t) \to u_{0} (x, t), a . e . in Ω, for almost all t \in [0, T] .

Here we emphasise the sequence

{N_{k} (t)} \subset N

may depends on t.

Since g is continuous we have that

g (u_{0}^{N_{k} (t)} (x, t)) \to g (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix

t \in (0, T) .

Let

ε > 0

. From the Egorov Theorem, there exists a closed set F such that

m (Ω ∖ F) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | \leq ε, for almost all x \in F .

Let

φ \in C_{c}^{\infty} (Ω) .

Observe now that

\begin{matrix} |\int_{Ω} (g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t))) φ d x| \\ \leq & \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ = & \int_{F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x + \int_{Ω ∖ F} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | d x \\ \leq & \int_{F} {ε ∥ φ ∥}_{\infty} d x + \int_{Ω} | g (u_{0}^{N_{k} (t)} (x, t)) - g (u_{0} (x, t)) | | φ | χ_{Ω ∖ F} d x \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + (∥ g (u_{0}^{N_{k} (t)}) ∥_{0, 2, Ω} + ∥ g (u_{0}) ∥_{0, 2, Ω} {) ∥ φ ∥}_{0, 4, Ω} {∥ χ_{Ω ∖ F} ∥}_{0, 4, Ω} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} m {(Ω ∖ F)}^{1 / 4} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} ε^{1 / 4}, \forall k > k_{0}, \end{matrix}

(400)

for some appropriate constant

K_{21} > 0

which does not depend on t.

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x \to \int_{Ω} g (u_{0}) φ d x, as k \to \infty,

\forall φ \in C_{c}^{\infty} (Ω) .

From such results, we have

\begin{matrix} 0 & = & lim_{k \to \infty} (\int_{Ω} \frac{𝜕 u_{0}^{N_{k} (t)}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0}^{N_{k} (t)} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}^{N_{k} (t)}) φ d x - \int_{Ω} f^{N_{k} (t)} φ d x) \\ = & \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x - \int_{Ω} f φ d x . \end{matrix}

(401)

so that, from this and by the density of

C_{c}^{\infty} (Ω)

H_{0}^{1} (Ω)

, we have got

\begin{matrix} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x + ε \int_{Ω} \nabla u_{0} \cdot \nabla φ d x \\ - \int_{Ω} g (u_{0}) φ d x - \int_{Ω} f φ d x = 0, \forall φ \in H_{0}^{1} (Ω), \end{matrix}

(402)

a . e . on [0, T] .

Observe now that

𝜕 (Ω \times (0, T)) = (𝜕 Ω \times [0, T]) \cup (𝜕 [0, T] \times \bar{Ω}) .

Let

φ \in C_{c}^{\infty} (Ω \times (0, T)) .

Hence

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t .

From this, since

C_{c}^{\infty} (Ω \times (0, T))

is dense

L^{2} (Ω \times (0, T))

we may infer that

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t,

\forall φ \in L^{2} (Ω \times (0, T)) .

Let

φ \in C^{\infty} (Ω \times [0, T])

such that

φ (x, T) = 0, in Ω .

From such results, we may obtain

\begin{matrix} lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & lim_{N \to \infty} (- \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x) \\ = & - \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0} (x, 0) φ (x, 0) d x . \end{matrix}

(403)

However, since

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T))

, we obtain

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t = \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t .

From these last results, we may infer that

\begin{matrix} \int_{Ω} {\hat{u}}_{0} φ (x, 0) d x & = & lim_{N \to \infty} \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x \\ = & \int_{Ω} u_{0} (x, 0) φ (x, 0) d x, \end{matrix}

(404)

so that

\int_{Ω} {\hat{u}}_{0} (x) φ (x, 0) d x = \int_{Ω} u_{0} (x, 0) φ (x, 0) d x,

\forall φ \in C^{\infty} (Ω \times [0, T])

such that

φ (x, T) = 0, in Ω .

Therefore, we may infer that

u_{0} (x, 0) = {\hat{u}}_{0} (x)

in this specified weak sense.

Similarly, it may be proven that

u_{0} = 0, on 𝜕 Ω \times [0, T],

in an appropriate weak sense.

Hence, we have obtained that

u_{0}

is a solution, in a weak sense, of the parabolic non-linear equation in question.

59. On the Convergence of Newton’s Method Combined with a Proximal Approach for an Eigenvalue Problem

Let

Ω \subset R^{3}

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

𝜕 Ω .

Consider the eigenvalue problem of finding

u \in V

and

λ \in R

such that

\{\begin{matrix} - ε \nabla^{2} u + g (u) - λ u = 0, & in Ω, \\ u = 0, on 𝜕 Ω, \\ \int_{Ω} u^{2} d x = {∥ u ∥}_{0, 2, Ω}^{2} = 1 . \end{matrix}

(405)

Here

ε > 0

V = H_{0}^{1} (Ω),

and

g : R \to R

is a

C^{1}

class function, such that either g is linear or such that

g (t u) = t g (u), \forall t > 0, \forall u \in R

In a finite differences or finite elements context, already including a proximal formulation, we shall look for a sequence

{u_{n}} \subset R^{N}

for an appropriate

N \in N

such that

- ε \nabla^{2} u_{n + 1} + g (u_{n + 1}) - \frac{u_{n}}{∥ u_{n} ∥} + K (u_{n + 1} - u_{n}) = 0,

\forall n \in N \cup {0} .

Now considering a concerning linearization of g, such an equation approximately stands for

- ε \nabla^{2} u_{n + 1} + g (u_{n}) + g^{'} (u_{n}) (u_{n + 1} - u_{n}) - \frac{u_{n}}{∥ u_{n} ∥} + K (u_{n + 1} - u_{n}) = 0,

\forall n \in N \cup {0} .

Assume

u_{0} \in R^{N}

is such that there exists

r > 0

such that

K_{1} I_{d} \leq g^{'} (u) \leq K_{2} I_{d},

\forall u \in B_{r} (u_{0}),

for some

K_{1}, K_{2} > 0 .

Suppose there exists a symmetric matrix

H_{6} (u, v)

such that

\frac{u}{∥ u ∥} - \frac{v}{∥ v ∥} = H_{6} (u, v) (u - v),

and

- K_{3} I_{d} \leq H_{6} (u, v) \leq K_{3} I_{d},

\forall u, v \in B_{r} (u_{0}),

for some

K_{3} > 0

And also there exists a symmetric matrix

H_{5} (u, v)

such that

g (u) - g (v) = H_{5} (u, v) (u - v),

and

K_{4} I_{d} \leq H_{5} (u, v) \leq K_{5} I_{d},

\forall u, v \in B_{r} (u_{0}),

for some

K_{4}, K_{5} > 0 .

Moreover, we assume that these last constants,

K > 0

and

0 < α_{1} < 1

are such that

(1 - α_{1}) (- ε \nabla^{2} + K_{1} I_{d}) + K_{4} I_{d} - K_{3} I_{d} - K_{2} I_{d}) \geq α_{1} K I_{d} = K I_{d} - (1 - α_{1}) K I_{d},

so that

{(- ε \nabla^{2} + K_{1} I_{d} + K I_{d})}^{- 1} (- K_{4} I_{d} + K_{2} I_{d} + K_{3} I_{d} + K I_{d}) \leq (1 - α_{1}) I_{d} .

Observe that

ε \nabla^{2} + g^{'} (u) + K I_{d} \geq - ε \nabla^{2} + K_{1} I_{d} + K I_{d} \geq 0

and

0 \leq - H_{5} (u, v) + H_{6} (u, v) + g^{'} (v) + K I_{d} \leq - K_{4} I_{d} + K_{2} I_{d} + K_{3} I_{d} + K I_{d},

\forall u, v \in B_{r} (u_{0})

so that

\begin{matrix} {(ε \nabla^{2} + g^{'} (u) + K I_{d})}^{- 1} (- H_{5} (u, v) + H_{6} (u, v) + g^{'} (v) + K I_{d}) \\ \leq & {(- ε \nabla^{2} + K_{1} I_{d} + K I_{d})}^{- 1} (- K_{4} I_{d} + K_{2} I_{d} + K_{3} I_{d} + K I_{d}) \\ \leq & (1 - α_{1}) I_{d}, \end{matrix}

(406)

\forall u, v \in B_{r} (u_{0}) .

Summarizing, defining

α_{0} = 1 - α_{1}

we have got

∥ {(ε \nabla^{2} + g^{'} (u) + K I_{d})}^{- 1} (- H_{5} (u, v) + H_{6} (u, v) + g^{'} (v) + K I_{d}) ∥ \leq α_{0} < 1,

\forall u, v \in B_{r} (u_{0}) .

Suppose

K > 0

and

α_{1} > 0

are such that

u_{1} \in B_{r (1 - α_{0})} (u_{0}) .

Reasoning inductively, suppose also

u_{0}, u_{1}, \dots, u_{n + 1} \in B_{r} (u_{0}) .

From the results above we have

- ε \nabla^{2} u_{n + 2} + g (u_{n + 1}) + g^{'} (u_{n + 1}) (u_{n + 2} - u_{n + 1}) - \frac{u_{n + 1}}{∥ u_{n + 1} ∥} + K (u_{n + 2} - u_{n + 1}) = 0,

and

- ε \nabla^{2} u_{n + 1} + g (u_{n}) + g^{'} (u_{n}) (u_{n + 1} - u_{n}) - \frac{u_{n}}{∥ u_{n} ∥} + K (u_{n + 1} - u_{n}) = 0,

so that

\begin{matrix} - ε (\nabla^{2} u_{n + 2} - \nabla^{2} u_{n + 1}) + g^{'} (u_{n + 1}) (u_{n + 2} - u_{n + 1}) + K (u_{n + 2} - u_{n + 1}) \\ = & - g (u_{n + 1}) + g (u_{n}) + \frac{u_{n + 1}}{∥ u_{n + 1} ∥} - \frac{u_{n}}{∥ u_{n} ∥} + g^{'} (u_{n}) (u_{n + 1} - u_{n}) + K (u_{n + 1} - u_{n}) \\ = & (- H_{5} (u_{n + 1}, u_{n}) + H_{6} (u_{n + 1}, u_{n}) + g^{'} (u_{n}) + K I_{d}) (u_{n + 1} - u_{n}) . \end{matrix}

(407)

Therefore

\begin{matrix} u_{n + 2} - u_{n + 1} \\ = & ({(- ε \nabla^{2} + g^{'} (u_{n + 1}) + K I_{d})}^{- 1} (- H_{5} (u_{n + 1}, u_{n}) + H_{6} (u_{n + 1}, u_{n}) + g^{'} (u_{n}) + K I_{d}) \\ \times (u_{n + 1} - u_{n}), \end{matrix}

(408)

so that

\begin{matrix} ∥ u_{n + 2} - u_{n + 1} ∥ \\ \leq & ∥ ({(- ε \nabla^{2} + g^{'} (u_{n + 1}) + K I_{d})}^{- 1} (- H_{5} (u_{n + 1}, u_{n}) + H_{6} (u_{n + 1}, u_{n}) + g^{'} (u_{n}) + K I_{d}) ∥ \\ \times ∥ u_{n + 1} - u_{n} ∥ \\ \leq & α_{0} ∥ u_{n + 1} - u_{n} ∥ . \end{matrix}

(409)

Summarizing, we have got

∥ u_{j + 2} - u_{j + 1} ∥ \leq α_{0} ∥ u_{j + 1} - u_{j} ∥, \forall j \in {1, \dots, n} .

Therefore

\begin{matrix} ∥ u_{j + 2} - u_{j + 1} ∥ & \leq & α_{0} ∥ u_{j + 1} - u_{j} ∥ \\ \leq & α_{0}^{2} ∥ u_{j} - u_{j - 1} ∥ \\ ⋮ & ​ \\ \leq & α_{0}^{j + 1} ∥ u_{1} - u_{0} ∥ . \end{matrix}

(410)

Thus,

\begin{matrix} ∥ u_{n + 2} - u_{1} ∥ & = & ∥ u_{n + 2} - u_{n + 1} + u_{n + 1} - \dots - u_{2} + u_{2} - u_{1} ∥ \\ \leq & ∥ u_{n + 2} - u_{n + 1} ∥ + ∥ u_{n + 1} - u_{n} ∥ + \dots + ∥ u_{2} - u_{1} ∥ \\ \leq & \sum_{j = 1}^{n + 1} α_{0}^{j} ∥ u_{1} - u_{0} ∥ \\ \leq & \sum_{j = 1}^{\infty} α_{0}^{j} ∥ u_{1} - u_{0} ∥ \\ = & \frac{α_{0}}{1 - α_{0}} ∥ u_{1} - u_{0} ∥ . \end{matrix}

(411)

Therefore

\begin{matrix} ∥ u_{n + 2} - u_{0} ∥ & \leq & ∥ u_{n + 2} - u_{1} + u_{1} - u_{0} ∥ \\ \leq & ∥ u_{n + 2} - u_{1} ∥ + ∥ u_{1} - u_{0} ∥ \\ \leq & \frac{α_{0}}{1 - α_{0}} ∥ u_{1} - u_{0} ∥ + ∥ u_{1} - u_{0} ∥ \\ = & \frac{1}{1 - α_{0}} ∥ u_{1} - u_{0} ∥ \\ < & \frac{1}{1 - α_{0}} (1 - α_{0}) r \\ = & r . \end{matrix}

(412)

Summarizing,

∥ u_{n + 2} - u_{0} ∥ < r,

so that

u_{n + 2} \in B_{r} (u_{0}) .

The induction is complete, so that

u_{n} \in B_{r} (u_{0}), \forall n \in N .

From such results we have also obtained

∥ u_{n + 2} - u_{n + 1} ∥ \leq α_{0} ∥ u_{n + 1} - u_{n} ∥, \forall n \in N .

Thus, from these results and the Banach fixed point theorem, there exists

{\hat{u}}_{0} \in {\bar{B}}_{r} (u_{0})

such that

lim_{n \to \infty} u_{n} = {\hat{u}}_{0} .

From such results we obtain

\begin{matrix} 0 & = & lim_{n \to \infty} (- ε \nabla^{2} u_{n + 1} + g (u_{n}) + g^{'} (u_{n}) (u_{n + 1} - u_{n}) \\ - \frac{u_{n}}{∥ u_{n} ∥} + K (u_{n + 1} - u_{n})) \\ = & - ε \nabla^{2} {\hat{u}}_{0} + g ({\hat{u}}_{0}) - \frac{{\hat{u}}_{0}}{∥ {\hat{u}}_{0} ∥} . \end{matrix}

(413)

Summarizing, we have got

- ε \nabla^{2} {\hat{u}}_{0} + g ({\hat{u}}_{0}) - \frac{{\hat{u}}_{0}}{∥ {\hat{u}}_{0} ∥} = 0 .

Consequently, defining

{\tilde{u}}_{0} = \frac{{\hat{u}}_{0}}{∥ {\hat{u}}_{0} ∥_{0, 2, Ω}},

λ = \frac{1}{∥ {\hat{u}}_{0} ∥}

and recalling that

g (t {\hat{u}}_{0}) = t g ({\hat{u}}_{0}), \forall t > 0,

we have obtained

- ε \nabla^{2} {\tilde{u}}_{0} + g ({\tilde{u}}_{0}) - λ {\tilde{u}}_{0} = 0,

and

∥ {\tilde{u}}_{0} ∥_{0, 2, Ω} = 1 .

The objective of this section is complete.

Remark 27.

For the general case we may drop the hypotheses of g being linear or

g (t u) = t g (u), \forall t > 0, \forall u \in R,

by defining the following iterations:

- ε \nabla^{2} u_{n + 1} + ∥ u_{n} ∥ g (\frac{u_{n}}{∥ u_{n} ∥}) - \frac{u_{n}}{∥ u_{n} ∥} + K (u_{n + 1} - u_{n}) = 0, \forall n \in N \cup {0} .

However in such a case some changes on the hypotheses are necessary in order to obtain the related theoretical results.

59.1. A Numerical Example

For

Ω = [0, 1] \subset R,

we have obtained numerical results for the following eigenvalue equation

\{\begin{matrix} - ε u^{''} + A u^{3} - λ u = 0, & in Ω, \\ u = 0, on 𝜕 Ω, \\ \int_{Ω} u^{2} d x = {∥ u ∥}_{0, 2, Ω}^{2} = 1, \end{matrix}

(414)

where

ε = 0.01,

and

A = 1.0

Observe that for a fixed

K > 0

we may obtain for this last equation

- ε u^{''} + A u^{3} + K u - K u - λ u = 0, in Ω,

so that

- ε u^{''} + A u^{3} + K u - λ_{1} u = 0, in Ω,

where

λ_{1} = K + λ .

In this example we have fixed

K = 500

In order to obtain such numerical results we have used the following algorithm:

Choose $u_{1} \in W_{0}^{1, 2}$ , set $n = 1$ , $b_{12} = 10^{- 4}$ and $n_{m a x} = 100 .$
Calculate $u_{n + 1} \in W_{0}^{1, 2}$ solution of equation

$- ε u_{n + 1}^{''} + A \frac{u_{n}^{3}}{∥ u_{n} ∥_{0, 2, Ω}^{2}} + K u_{n + 1} - \frac{u_{n}}{∥ u_{n} ∥_{0, 2, Ω}} = 0, in Ω,$
If $∥ u_{n + 1} - u_{n} ∥_{\infty} \leq b_{12}$ or $n > n_{m a x}$ , then stop. Otherwise $n : = n + 1$ and go to item 2.

For the optimal solution u obtained, please see Figure 43.

Here we present the software in MAT-LAB through which we have obtained such numerical results.

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

clear all

m8=100;

d=1/m8;

K=500;

A=1;

e1=0.01;

for i=1:m8

uo(i,1)=0.1;

end;

b12=1.0;

k=1;

while $(b 12 > 10^{- 4})$ && $(k < 100)$

k

k=k+1;

S=0;

for i=1:m8-1

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

end;

S=sqrt(S);

m12=2+K* $d^{2}$ ;

m50(1)=1/m12;

z(1)=m50(1)*( $u o (i, 1) / S * d^{2}$ -A* $u o {(i, 1)}^{3} / S^{2} * d^{2} / e 1$ );

for i=2:m8-1

m12=2+K $* d^{2}$ -m50(i-1);

m50(i)=1/m12;

z(i)=m50(i)* $(u o (i, 1) / S * d^{2} - A * u o {(i, 1)}^{3} / S^{2} * d^{2} / e 1 + z (i - 1))$ ;

end;

u(m8,1)=0;

for i=1:m8-1

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

end;

b12=max(abs(uo-u));

uo=u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,uo/S)

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

Remark 28.

With the high value

K = 500

we have obtained the following eigenvalue for this problem:

λ = (\frac{1}{S} - K) ε = 133.8090 ε = 1.338 .

60. On the Convergence of Newton’s Method Combined with a Proximal Approach for a General Parabolic Non-Linear System

Let

Ω \subset R^{m}

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

𝜕 Ω .

Consider the parabolic non-linear system

\{\begin{matrix} \frac{𝜕 u_{j}}{𝜕 t} = ε_{j} \nabla^{2} u_{j} + g_{j} (u) + \sum_{k = 1}^{r} \sum_{l = 1}^{m} g_{j k l} (u) \frac{𝜕 u_{k}}{𝜕 x_{l}} + f_{j}, & in Ω \times (0, T), \\ u_{j} (x, 0) = {({\hat{u}}_{0})}_{j}, in Ω, \\ u_{j} = 0, on 𝜕 Ω \times [0, T], \forall j \in {1, \dots, r} . \end{matrix}

(415)

Here

u = (u_{1}, \dots, u_{r}) = {u_{j}} \in H_{0}^{1} (Ω; R^{r}),

ε_{j} > 0

f = {f_{j}} \in L^{2} ([0, T], W^{1, 2} (Ω; R^{r})) \cap L^{\infty} (Ω \times [0, T]; R^{r}),

{\hat{u}}_{0} = {{({\hat{u}}_{0})}_{j}} \in H_{0}^{1} (Ω; R^{r}) \cap L^{\infty} (Ω; R^{r}),

where t denotes time and

[0, T]

is a time interval.

Also

g_{j} : R \to R

and

g_{j k l} : R \to R

are

C^{1}

class functions neither necessarily linear nor convex,

\forall j, k \in {1, \dots, r}, l \in {1, \dots, m}

We define

F_{j} (u) = ε_{j} \nabla^{2} u_{j} + g_{j} (u) + \sum_{k = 1}^{r} \sum_{l = 1}^{m} g_{j k l} (u) \frac{𝜕 u_{k}}{𝜕 x_{l}} + f_{j},

\forall j \in {1, \dots, r}

, so that the system in question stands for

\frac{𝜕 u_{j}}{d t} = F_{j} (u), \forall j \in {1, \dots, r} .

Fixing

N \in N

and defining

Δ t_{N} = T / N

, in a finite differences context we may define the following approximate system

\frac{u_{n + 1} - u_{n}}{Δ t_{N}} = F_{j} (u_{n + 1}), \forall j \in {1, \dots, r}, \forall n \in {0, \dots, N - 1} .

Fix

n = 0

. In a Newton’s method context combined with a proximal approach, we shall obtain

u_{1}

through the following iterations,

Define

u_{1}^{0} = {\hat{u}}_{0}

and having

u_{1}^{k}

let

u_{1}^{k + 1}

be such that

u_{1}^{k + 1} - u_{1}^{0} = F_{j} (u_{1}^{k}) Δ t_{N} + Δ t_{N} \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} (u_{1}^{k + 1} - u_{1}^{k}) - K (u_{1}^{k + 1} - u_{1}^{k}) .

At this point we assume there exist

r > 0

and

K_{1} > 0

, such that

- K_{1} I_{d} \leq \{\frac{𝜕 F_{j} (u)}{𝜕 u_{l}}\} \leq K_{1} I_{d},

\forall u \in B_{r} ({\hat{u}}_{0}) .

Moreover, generically denoting

F (u) = {F_{j} (u)},

we assume there exists a matrix operator

H_{5} (u, v),

such that

F (u) - F (v) = H_{5} (u, v) (u - v),

and

- K_{3} I_{d} \leq H_{5} (u, v) \leq K_{3} I_{d},

\forall u, v \in B_{r} ({\hat{u}}_{0}),

for some appropriate real constant

K_{3} > 0 .

Now suppose

K > 0

and

0 < α_{1} < 1

are such that there exists

N_{0} \in N

such that if

N > N_{0}

, then

(1 - α_{1}) (I_{d} - K_{1} I_{d} Δ t_{N}) - K_{1} I_{d} Δ t_{N} - K_{3} I_{d} Δ t_{N}) \geq α_{1} K I_{d} = K I_{d} - (1 - α_{1}) K I_{d},

so that

{(I_{d} - K_{1} I_{d} Δ t_{N} + K I_{d})}^{- 1} (K_{1} I_{d} Δ t_{N} + K_{3} I_{d} Δ t_{N} + K I_{d}) \leq (1 - α_{1}) K I_{d} .

Observe that such an

N_{0}

may be such that

I_{d} - \{\frac{𝜕 F_{j} (u)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d} \geq I_{d} - K_{1} I_{d} Δ t_{N} + K I_{d} > 0 I_{d},

and

0 \leq H_{5} (u, v) Δ t_{N} - \{\frac{𝜕 F_{j} (v)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d} \leq K_{1} I_{d} Δ t_{N} + K_{3} I_{d} Δ t_{N} + K I_{d},

so that

\begin{matrix} {(I_{d} - \{\frac{𝜕 F_{j} (u)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})}^{- 1} (H_{5} (u, v) Δ t_{N} - \{\frac{𝜕 F_{j} (v)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d}) \\ \leq & {(I_{d} - K_{1} I_{d} Δ t_{N} + K I_{d})}^{- 1} (K_{1} I_{d} Δ t_{N} + K_{3} I_{d} Δ t_{N} + K I_{d}) \\ \leq & (1 - α_{1}) I_{d}, \end{matrix}

(416)

\forall u, v \in B_{r} ({\hat{u}}_{0}), \forall N > N_{0} .

Hence, denoting

α_{0} = (1 - α_{1}),

we have got

\begin{matrix} ∥{(I_{d} - \{\frac{𝜕 F_{j} (u)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})}^{- 1} (H_{5} (u, v) Δ t_{N} - \{\frac{𝜕 F_{j} (v)}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})∥ \\ \leq & ∥{(I_{d} - K_{1} I_{d} Δ t_{N} + K I_{d})}^{- 1} (K_{1} I_{d} Δ t_{N} + K_{3} I_{d} Δ t_{N} + K I_{d})∥ \\ \leq & α_{0} . \end{matrix}

(417)

\forall u, v \in B_{r} ({\hat{u}}_{0}),

\forall N > N_{0}

Fix now a new

N > N_{0}

Suppose now

K = K_{n}^{N} > 0

and

0 < α_{1} = {(α_{1})}_{n}^{N} < 1

are such that

u_{1}^{1} \in B_{\frac{r}{N} (1 - α_{0})} ({\hat{u}}_{0}) .

Reasoning inductively, suppose

u_{1}^{0}, u_{1}^{1}, \dots, u_{1}^{k + 1} \in B_{\frac{r}{N}} (u_{0}),

and observe that

u_{1}^{k + 1} - u_{0}^{1} = F_{j} (u_{1}^{k}) Δ t_{N} + Δ t_{N} \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} (u_{1}^{k + 1} - u_{1}^{k}) - K (u_{1}^{k + 1} - u_{1}^{k}) .

and

u_{1}^{k + 2} - u_{0}^{1} = F_{j} (u_{1}^{k + 1}) Δ t_{N} + Δ t_{N} \{\frac{𝜕 F_{j} (u_{1}^{k + 1})}{𝜕 u_{l}}\} (u_{1}^{k + 2} - u_{1}^{k + 1}) - K (u_{1}^{k + 2} - u_{1}^{k + 1}),

so that

\begin{matrix} (I_{d} - \{\frac{𝜕 F_{j} (u_{1}^{k + 1})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d}) (u_{1}^{k + 2} - u_{1}^{k + 1}) \\ = & ((F (u_{1}^{k + 1}) - F (u_{1}^{k})) Δ t_{N} - \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d}) (u_{1}^{k + 1} - u_{1}^{k}) \\ = & (H_{5} (u_{1}^{k + 1}, u_{1}^{k}) Δ t_{N} - \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d}) (u_{1}^{k + 1} - u_{1}^{k}) . \end{matrix}

(418)

Thus,

\begin{matrix} u_{1}^{k + 2} - u_{1}^{k + 1} \\ = & {(I_{d} - \{\frac{𝜕 F_{j} (u_{1}^{k + 1})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})}^{- 1} (H_{5} (u_{1}^{k + 1}, u_{1}^{k}) Δ t_{N} - \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d}) \\ \times (u_{1}^{k + 1} - u_{1}^{k}) . \end{matrix}

(419)

Summarizing, we have got

\begin{matrix} ∥u_{1}^{k + 2} - u_{1}^{k + 1}∥ \\ = & ∥{(I_{d} - \{\frac{𝜕 F_{j} (u_{1}^{k + 1})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})}^{- 1} (H_{5} (u_{1}^{k + 1}, u_{1}^{k}) Δ t_{N} - \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} Δ t_{N} + K I_{d})∥ \\ \times ∥ u_{1}^{k + 1} - u_{1}^{k} ∥ \\ \leq & α_{0} ∥ u_{1}^{k + 1} - u_{1}^{k} ∥ . \end{matrix}

(420)

Thus, we have got

∥ u_{1}^{j + 2} - u_{1}^{j + 1} ∥ \leq α_{0} ∥ u_{1}^{j + 1} - u_{1}^{j} ∥, \forall j \in {1, \dots, k} .

Therefore

\begin{matrix} ∥ u_{1}^{j + 2} - u_{1}^{j + 1} ∥ & \leq & α_{0} ∥ u_{1}^{j + 1} - u_{1}^{j} ∥ \\ \leq & α_{0}^{2} ∥ u_{1}^{j} - u_{1}^{j - 1} ∥ \\ ⋮ & ​ \\ \leq & α_{0}^{j + 1} ∥ u_{1}^{1} - u_{1}^{0} ∥ . \end{matrix}

(421)

Thus,

\begin{matrix} ∥ u_{1}^{k + 2} - u_{1}^{1} ∥ & = & ∥ u_{1}^{k + 2} - u_{1}^{k + 1} + u_{1}^{k + 1} - \dots - u_{1}^{2} + u_{1}^{2} - u_{1}^{1} ∥ \\ \leq & ∥ u_{1}^{k + 2} - u_{1}^{k + 1} ∥ + ∥ u_{1}^{k + 1} - u_{1}^{k} ∥ + \dots + ∥ u_{1}^{2} - u_{1}^{1} ∥ \\ \leq & \sum_{j = 1}^{k + 1} α_{0}^{j} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & \sum_{j = 1}^{\infty} α_{0}^{j} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ = & \frac{α_{0}}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ . \end{matrix}

(422)

Therefore

\begin{matrix} ∥ u_{1}^{k + 2} - u_{1}^{0} ∥ & \leq & ∥ u_{1}^{k + 2} - u_{1}^{1} + u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & ∥ u_{1}^{k + 2} - u_{1}^{1} ∥ + ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ \leq & \frac{α_{0}}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ + ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ = & \frac{1}{1 - α_{0}} ∥ u_{1}^{1} - u_{1}^{0} ∥ \\ < & \frac{1}{1 - α_{0}} (1 - α_{0}) \frac{r}{N} \\ = & \frac{r}{N} . \end{matrix}

(423)

Summarizing,

∥ u_{1}^{k + 2} - u_{1}^{0} ∥ < \frac{r}{N},

so that

u_{1}^{k + 2} \in B_{\frac{r}{N}} (u_{0}^{1}) .

The induction is complete, so that

u_{1}^{k} \in B_{\frac{r}{N}} (u_{0}^{1}), \forall k \in N .

From such results we have also obtained

∥ u_{1}^{k + 2} - u_{1}^{k + 1} ∥ \leq α_{0} ∥ u_{1}^{k + 1} - u_{1}^{k} ∥, \forall k \in N .

Thus, from these results and the Banach fixed point theorem, there exists

u_{1} \in B_{\frac{r}{N}} (u_{0}^{1})

such that

lim_{k \to \infty} u_{1}^{k} = u_{1} .

\begin{matrix} 0 & = & lim_{k \to \infty} (u_{1}^{k + 1} - {\hat{u}}_{0} \\ - F (u_{1}^{k}) Δ t_{N} - Δ t_{N} \{\frac{𝜕 F_{j} (u_{1}^{k})}{𝜕 u_{l}}\} (u_{1}^{k + 1} - u_{1}^{k}) \\ + K (u_{1}^{k + 1} - u_{1}^{k})) \\ = & u_{1} - {\hat{u}}_{0} - F (u_{1}) Δ t_{N} \end{matrix}

(424)

so that

\frac{u_{1} - {\hat{u}}_{0}}{Δ t_{N}} = F (u_{1}) in Ω,

Reasoning inductively again having

u_{1} \in B_{\frac{r}{N}} ({\hat{u}}_{0})

and

u_{j} \in B_{\frac{r}{N}} (u_{j - 1}), \forall j \in {2, \dots, n}

similarly as we have obtained

u_{1}

in the last lines, we may obtain

u_{n + 1} \in B_{\frac{r}{N}} (u_{n}),

such that

\frac{u_{n + 1} - u_{n}}{Δ t_{N}} = F (u_{n + 1}), in Ω .

The induction on n is also complete.

Fix

n \in {1, \dots, N - 1}

Observe that

\begin{matrix} ∥ u_{n} - u_{0} ∥ & = & ∥ u_{n} - u_{n - 1} + u_{n - 1} - u_{n - 2} + \dots - u_{1} + u_{1} - u_{0} ∥ \\ \leq & ∥ u_{n} - u_{n - 1} ∥ + \dots + ∥ u_{1} - u_{0} ∥ \\ \leq & \frac{n}{N} r \\ < & r . \end{matrix}

(425)

Summarizing

u_{n} \in B_{r} ({\hat{u}}_{0}), \forall n \in {0, 1, \dots, N - 1} .

From these results, denoting now more generically

u_{n} \equiv u_{n}^{N}

, we may infer that there exists

{\hat{K}}_{4} > 0

such that

∥ u_{j}^{N} ∥_{1, 2, Ω} \leq {\hat{K}}_{4}, \forall j \in {0, 1, \dots, N}, \forall N \in N .

Define now

u_{0}^{N} (x, t) = u_{n}^{N} (x) (n + 1 - \frac{t}{Δ t_{N}}) + u_{n + 1}^{N} (x) (\frac{t}{Δ t_{N}} - n),

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Observe that

u_{0}^{N} (x, t) = u_{n}^{N} (x), if t = n Δ t_{N}, \forall n \in {0, 1, \dots, N},

and

\begin{matrix} \frac{𝜕 u_{0}^{N} (x, t)}{𝜕 t} & = & \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} \\ = & F (u_{n + 1}^{N}), \end{matrix}

(426)

t \in [n Δ t_{N}, (n + 1) Δ t_{N}], \forall (x, t) \in Ω \times [0, T] .

Fix

φ \in C_{c}^{\infty} (Ω; R^{r}) .

Thus, fixing

t \in [n Δ t_{N}, (n + 1) Δ t_{N}],

we have

\begin{matrix} |{〈\frac{𝜕 u_{0}^{N}}{𝜕 t}, φ〉}_{L^{2}}| & \leq & |{〈 F (u_{n + 1}^{N}), φ 〉}_{L^{2}}| \\ \leq & K_{5} {∥ φ ∥}_{1, 2, Ω}, \forall φ \in C_{c}^{\infty} (Ω; R^{r}), \end{matrix}

(427)

for some appropriate

K_{5} > 0 .

Since

φ \in C_{c}^{\infty} (Ω; R^{r})

is arbitrary, we may conclude that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω; R^{r})} \leq K_{6}, \forall N > N_{0},

uniformly in t on

[0, T],

for some appropriate constant

K_{6} > 0 .

Also, from the definition of

u_{0}^{N}

we have that there exists

K_{7} > 0

such that

∥ u_{0}^{N} ∥_{1, 2, Ω} \leq K_{7}, \forall N \in N

also uniformly in t on

[0, T] .

From such results, there exist

u_{0} \in L^{2} ([0, T], H_{0}^{1} (Ω; R^{r}))

and

v_{0} \in L^{2} ([0, T]; H^{- 1} (Ω; R^{r}))

such that

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} ((0, T); W^{1, 2} (Ω; R^{r})),

and

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in L^{2} ([0, T], H^{- 1} (Ω; R^{r})),

so that we may easily obtain

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense. At this point, we provide more details about this last result.

Fix

t \in (0, T)

. Thus, there exists

n \in {0, 1, \dots, N - 1}

such that

t \in [n Δ t_{N}, (n + 1) Δ t_{N}]

Let

φ \in C_{c}^{\infty} (Ω \times (0, T); R^{r}) .

From this, we may infer that

\begin{matrix} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x \\ = & \int_{Ω} \frac{u_{n + 1}^{N} - u_{n}^{N}}{Δ t_{N}} φ (x, t) d x \\ \leq & {〈 F (u_{n + 1}^{N}), φ 〉}_{L^{2}} \\ \leq & K_{9} {∥ φ ∥}_{1, 2, Ω}, \end{matrix}

(428)

for some appropriate constant

K_{9} > 0 .

Hence,

\begin{matrix} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ (x, t) d x d t \\ \leq & K_{9} \int_{Ω} {∥ φ ∥}_{1, 2, Ω} d t \\ \leq & K_{19} {∥ φ ∥}_{1, 2, Ω \times (0, T)}, \end{matrix}

(429)

for some appropriate

K_{19} > 0

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T); R^{r})

is arbitrary, we may infer that

{∥\frac{𝜕 u_{0}^{N}}{𝜕 t}∥}_{H^{- 1} (Ω \times (0, T); R^{r})} \leq K_{15},

for

N \in N

, for some

K_{15} > 0 .

From such a result and from the Banach-Alaoglu Theorem, there exists

v_{0} \in H^{- 1} (Ω \times (0, T); R^{r})

such that, up to a not relabeled subsequence

\frac{𝜕 u_{0}^{N}}{𝜕 t} ⇀ v_{0}, weakly - star in H^{- 1} (Ω \times (0, T); R^{r}) .

Therefore,

\int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \to \int_{0}^{T} \int_{Ω} v_{0} φ d x d t,

N \to \infty,

\forall φ \in H_{0}^{1} (Ω \times (0, T); R^{r})

On the other hand

∥ u_{0}^{N} ∥_{0, 2, Ω \times (0, T)} \leq K_{16},

\forall N > N_{0}

, for some

K_{16} > 0 .

From this and the Kakutani Theorem, there exists

u_{0} \in L^{2} (Ω \times (0, T); R^{r})

such that, up to a not relabeled subsequence,

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T); R^{r}) .

Now fix again

φ \in C_{c}^{\infty} (Ω \times (0, T); R^{r}) .

Observe that

\begin{matrix} \int_{0}^{T} \int_{Ω} u_{0} φ_{t} d x d t & = & lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} φ_{t} d x d t \\ = & - lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & - \int_{0}^{T} \int_{Ω} v_{0} φ d x d t, \end{matrix}

(430)

Since such a

φ \in C_{c}^{\infty} (Ω \times (0, T); R^{r})

is arbitrary, we may infer that

v_{0} = \frac{𝜕 u_{0}}{𝜕 t}

in a distributional sense.

Moreover, from such results we may also obtain, again up to a subsequence,

lim_{N \to \infty} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x = \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x,

\forall φ \in H_{0}^{1} (Ω; R^{r}) .

Observe also that, as a consequence of the Rellich-Kondrashov theorem, through appropriate subsequences, we have

u_{0}^{N_{k} (t)} \to u_{0} (x, t), strongly in L^{2} (Ω; R^{r}), for almost all t \in [0, T] .

so that, up to subsequences,

u_{0}^{N_{k} (t)} (x, t) \to u_{0} (x, t), a . e . in Ω, for almost all t \in [0, T] .

Here we emphasise the sequence

{N_{k} (t)} \subset N

may depends on t.

Fix

j \in {1, \dots, r} .

Since

g_{j}

is continuous we have that

g_{j} (u_{0}^{N_{k} (t)} (x, t)) \to g_{j} (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix

t \in (0, T) .

Let

ε > 0

. From the Egorov Theorem, there exists a closed set F such that

m (Ω ∖ F) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | \leq ε, for almost all x \in F .

Let

φ \in C_{c}^{\infty} (Ω) .

Observe now that

\begin{matrix} |\int_{Ω} (g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t))) φ d x| \\ \leq & \int_{Ω} | g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | | φ | d x \\ = & \int_{F} | g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | | φ | d x + \int_{Ω ∖ F} | g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | | φ | d x \\ \leq & \int_{F} {ε ∥ φ ∥}_{\infty} d x + \int_{Ω} | g_{j} (u_{0}^{N_{k} (t)} (x, t)) - g_{j} (u_{0} (x, t)) | | φ | χ_{Ω ∖ F} d x \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + (∥ g_{j} (u_{0}^{N_{k} (t)}) ∥_{0, 2, Ω} + ∥ g_{j} (u_{0}) ∥_{0, 2, Ω} {) ∥ φ ∥}_{0, 4, Ω} {∥ χ_{Ω ∖ F} ∥}_{0, 4, Ω} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} m {(Ω ∖ F)}^{1 / 4} \\ \leq & {ε ∥ φ ∥}_{\infty} m (Ω) + K_{21} {∥ φ ∥}_{0, 4, Ω} ε^{1 / 4}, \forall k > k_{0}, \end{matrix}

(431)

for some appropriate constant

K_{21} > 0

which does not depend on t.

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} g_{j} (u_{0}^{N_{k} (t)}) φ d x \to \int_{Ω} g_{j} (u_{0}) φ d x, as k \to \infty,

\forall φ \in C_{c}^{\infty} (Ω),

\forall j \in {1, \dots, r} .

Similarly, fixing

j, p \in {1, \dots, n},

and

l \in {1, \dots, m},

since

g_{j p l}

is continuous we have that

g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) \to g_{j p l} (u_{0} (x, t)), a . e . in Ω, for almost all t \in [0, T] .

Fix again

t \in (0, T)

Let

ε > 0

(a new value). From the Egorov Theorem, there exists a closed set

F_{1}

such that

m (Ω ∖ F_{1}) \leq ε

and

k_{0} \in N

such that if

k > k_{0}

, then

| g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) - g_{j p l} (u_{0} (x, t)) | \leq ε, for almost all x \in F_{1} .

Observe now that

\begin{matrix} \int_{Ω} {| g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) - g_{j p l} (u_{0} (x, t)) |}^{2} d x \\ \leq & \int_{F_{1}} | g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) - g_{j p l} (u_{0} (x, t)) |^{2} d x + \int_{Ω ∖ F_{1}} {| g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) - g_{j p l} (u_{0} (x, t)) |}^{2} d x \\ \leq & \int_{F_{1}} ε^{2} d x + \int_{Ω} {| g_{j p l} (u_{0}^{N_{k} (t)} (x, t)) - g_{j p l} (u_{0} (x, t)) |}^{2} χ_{Ω ∖ F_{1}} d x \\ \leq & ε^{2} m (Ω) + 2 K_{1}^{2} \int_{Ω} χ_{Ω ∖ F_{1}} d x \\ \leq & ε^{2} m (Ω) + 2 K_{1}^{2} ε, \forall k > k_{0} . \end{matrix}

(432)

Since such a

ε > 0

is arbitrary, we may infer that

\int_{Ω} {| g_{j p l} (u_{0}^{N_{k} (t)}) - g_{j p l} (u_{0}) |}^{2} d x \to 0, as k \to \infty,

\forall j, p \in {1, \dots, r},

l \in {1, \dots, m} .

Select again

φ \in C_{c}^{\infty} (Ω) .

Since

∥ g_{j p l} (u_{0}^{N_{k} (t)}) - g_{j p l} (u_{0}) ∥_{0, 2, Ω} \to 0, as k \to \infty

and

\nabla u_{0}^{N_{k} (t)} ⇀ \nabla u_{0}, weakly in L^{2} (Ω; R^{r \times m}),

we obtain,

\begin{matrix} |\int_{Ω} g_{j p l} (u_{0}^{N_{k} (t)}) {({(u_{0})}_{p}^{N_{k} (t)})}_{x_{l}} φ d x - \int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p})}_{x_{l}} φ d x| \\ \leq & |\int_{Ω} g_{j p l} (u_{0}^{N_{k} (t)}) {({(u_{0})}_{p}^{N_{k} (t)})}_{x_{l}} φ d x - \int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p}^{N_{k} (t)})}_{x_{l}} φ d x| \\ + |\int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p}^{N_{k} (t)})}_{x_{l}} φ d x - \int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p})}_{x_{l}} φ d x| \\ \leq & ∥ g_{j p l} (u_{0}^{N_{k} (t)}) - g_{j p l} (u_{0}) ∥_{0, 2, Ω} K_{7} {∥ φ ∥}_{\infty} \\ + |\int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p}^{N_{k} (t)})}_{x_{l}} φ d x - \int_{Ω} g_{j p l} (u_{0}) {({(u_{0})}_{p})}_{x_{l}} φ d x| \\ \to & 0, as k \to \infty, \end{matrix}

(433)

\forall j, p \in {1, \dots, r}, l \in {1, \dots, m} .

From such results, for an arbitrary

φ \in C_{c}^{\infty} (Ω; R^{r})

, we have

\begin{matrix} 0 & = & lim_{k \to \infty} (\int_{Ω} \frac{𝜕 {(u_{0})}_{j}^{N_{k} (t)}}{𝜕 t} φ_{j} d x - {〈 F_{j} (u_{0}^{N_{k} (t)}), φ_{j} 〉}_{L^{2}}) \\ = & \int_{Ω} \frac{𝜕 {(u_{0})}_{j}}{𝜕 t} φ_{j} d x + ε_{j} {〈 \nabla {(u_{0})}_{j}, \nabla φ_{j} 〉}_{L^{2}} - {〈 g_{j} (u_{0}), φ_{j} 〉}_{L^{2}} \\ - \sum_{p = 1}^{r} \sum_{l = 1}^{m} {〈 g_{j p l} (u_{0}) {({(u_{0})}_{p})}_{x_{l}}, φ_{j} 〉}_{L^{2}} - {〈 f_{j}, φ_{j} 〉}_{L^{2}} \end{matrix}

(434)

so that, from this and by the density of

C_{c}^{\infty} (Ω;; R^{r})

H_{0}^{1} (Ω; R^{r})

, we have got

\begin{matrix} \int_{Ω} \frac{𝜕 {(u_{0})}_{j}}{𝜕 t} φ_{j} d x \\ = & - ε_{j} {〈 \nabla {(u_{0})}_{j}, \nabla φ_{j} 〉}_{L^{2}} + {〈 g_{j} (u_{0}), φ_{j} 〉}_{L^{2}} \\ + \sum_{p = 1}^{r} \sum_{l = 1}^{m} {〈 g_{j p l} (u_{0}) {({(u_{0})}_{p})}_{x_{l}}, φ_{j} 〉}_{L^{2}} + {〈 f_{j}, φ_{j} 〉}_{L^{2}}, \end{matrix}

(435)

\forall j \in {1, \dots, r}, \forall φ \in H_{0}^{1} (Ω; R^{r}),

a . e . on [0, T],

Observe now that

𝜕 (Ω \times (0, T)) = (𝜕 Ω \times [0, T]) \cup (𝜕 [0, T] \times \bar{Ω}) .

Let

φ \in C_{c}^{\infty} (Ω \times (0, T); R^{r}) .

Hence

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t .

From this, since

C_{c}^{\infty} (Ω \times (0, T); R^{r})

is dense

L^{2} (Ω \times (0, T); R^{r})

we may infer that

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t = \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}}{𝜕 t} φ d x d t,

\forall φ \in L^{2} (Ω \times (0, T); R^{r}) .

Let

φ \in C^{\infty} (Ω \times [0, T]; R^{r})

such that

φ (x, T) = 0, in Ω .

From such results, we may obtain

\begin{matrix} lim_{N \to \infty} \int_{0}^{T} \int_{Ω} \frac{𝜕 u_{0}^{N}}{𝜕 t} φ d x d t \\ = & lim_{N \to \infty} (- \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x) \\ = & - \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t - \int_{Ω} u_{0} (x, 0) φ (x, 0) d x . \end{matrix}

(436)

However, since

u_{0}^{N} ⇀ u_{0}, weakly in L^{2} (Ω \times (0, T); R^{r})

, we obtain

lim_{N \to \infty} \int_{0}^{T} \int_{Ω} u_{0}^{N} \frac{𝜕 φ}{𝜕 t} d x d t = \int_{0}^{T} \int_{Ω} u_{0} \frac{𝜕 φ}{𝜕 t} d x d t .

From these last results, we may infer that

\begin{matrix} \int_{Ω} {\hat{u}}_{0} φ (x, 0) d x & = & lim_{N \to \infty} \int_{Ω} u_{0}^{N} (x, 0) φ (x, 0) d x \\ = & \int_{Ω} u_{0} (x, 0) φ (x, 0) d x, \end{matrix}

(437)

so that

\int_{Ω} {\hat{u}}_{0} (x) φ (x, 0) d x = \int_{Ω} u_{0} (x, 0) φ (x, 0) d x,

\forall φ \in C^{\infty} (Ω \times [0, T]; R^{r})

such that

φ (x, T) = 0, in Ω .

Therefore, we may infer that

u_{0} (x, 0) = {\hat{u}}_{0} (x)

in this specified weak sense.

Similarly, it may be proven that

u_{0} = 0, on 𝜕 Ω \times [0, T],

in an appropriate weak sense.

Hence, we have obtained that

u_{0}

is a solution, in a weak sense, of the parabolic non-linear system in question.

61. A Note on the Convergence of the Finite Elements Method

In this section we develop some remarks on the convergence of the finite elements method.

This section is based on reference [18], Chapter 7.

For the proofs not presented here and for more details please see reference [18], Chapter 7.

We start by recalling the following classical result.

Theorem 12

(Lax-Milgram). Let V be a separable Hilbert space with a inner product

〈 \cdot, \cdot 〉 : V \times V \to R,

and related norm

∥ \cdot ∥ : V \to R^{+}

where

∥ u ∥ = \sqrt{〈 u, u 〉}, \forall u \in V .

Let

a : V \times V \to R

be a bilinear form such that

a is continuous, that is, there exists $M > 0$ such that

$| a (u, v) | \leq M ∥ u ∥ ∥ v ∥, \forall u, v \in V,$
a is coercive, that is, there exists $α > 0$ such that

$a (v, v) \geq {α ∥ v ∥}^{2}, \forall u \in V .$

Moreover, let

L : V \to R

be a linear and continuous functional.

Under such hypotheses, there exists a unique

u \in V

such that

a (u, v) = L (v), \forall v \in V .

Definition 2.

Let V be a Banach space. We say that a sequence

{V_{n}}

of finite dimensional subspaces of V is a Galerkin scheme for V if for each

v \in V

, there exists a sequence

{v_{k}} \subset \cup_{n = 1}^{\infty} V_{n}

where

v_{k} \in V_{k}, \forall k \in N,

such that

v_{k} \to v, strongly in norm, as k \to \infty .

Remark 29.

Let

Ω \subset R^{2}

be a polygonal set. A triangulation T of Ω is a finite union of subsets of

\bar{Ω},

such that

$\bar{Ω} = \cup_{K \in T} K,$
Each set $K \in T$ is a triangle,
For each pair $K_{1}, K_{2} \in T$ , such sets are quasi-disjoints, that is, their interiors are disjoint.

We define

h (T) = max_{K \in T} d i a m (K) \equiv h,

where

d i a m (K) = sup {| x, y | : x, y \in K} .

In such a case we also denote

T = T_{h} .

Moreover, we define

V_{h} = {v \in C (\bar{Ω}) : v is affine on each K \in T_{h} and v = 0, on 𝜕 Ω} .

We denote by

a_{j}

the vertices in the triangulation

T_{h}

, where

j \in {1, \dots, I (h)} .

Let

{φ_{j}} \subset V_{h}

be such that

φ_{j} (a_{k}) = δ_{j k}, \forall 1 \leq j, k \leq I (h) .

Here

δ_{j k} = \{\begin{matrix} 1, if j = k, \\ 0, if j \neq k . \end{matrix}

(438)

Observe that

{φ_{1}, \dots, φ_{I (h)}}

is a basis for

V_{h} .

At this point we define

P_{h} (v) = \sum_{j = 1}^{I (h)} v (a_{j}) φ_{j}, \forall v \in V .

Here we assume

{T_{h}}_{h > 0}

be a regular family of triangulations of

Ω .

Let

{h_{n}} \subset R^{+}

be a sequence such that

0 < h_{n + 1} < h_{n}, \forall n \in N,

and

lim_{n \to \infty} h_{n} = 0 .

We denote

V_{n} = V_{h_{n}}

and

P_{n} = P_{h_{n}}, \forall n \in N .

Consider the Ginzburg-Landau type equation

\{\begin{matrix} - γ \nabla^{2} u + α u^{3} - β u - f = 0, & in Ω, \\ u = 0, on 𝜕 Ω . \end{matrix}

(439)

Here

γ > 0,

α > 0,

β > 0

and

f \in L^{2} (Ω) .

Assume

u_{n} \in V_{n}

is a weak solution of this last equation, in the following sense,

\begin{matrix} γ {〈 \nabla u_{n}, \nabla φ 〉}_{L^{2}} + α {〈 u_{n}^{3}, φ 〉}_{L^{2}} \\ - β {〈 u_{n}, φ 〉}_{L^{2}} - {〈 f, φ 〉}_{L^{2}}, \\ = & 0, \forall φ \in V_{n} . \end{matrix}

(440)

We assume there exist

u_{0} \in H_{0}^{1} \cap W^{1, \infty} (Ω)

r > 0

α_{1} > 0

and

M > 0

such that

\begin{matrix} α_{1} {∥ u - v ∥}_{1, 2, Ω}^{2} \\ \leq & γ {〈 \nabla (u - v), \nabla (u - v) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}^{2} (u, v) (u - v), (u - v) 〉}_{L^{2}} - β {〈 (u - v), (u - v) 〉}_{L^{2}}, \end{matrix}

and

\begin{matrix} |γ {〈 \nabla (u - v), \nabla (u - w) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}^{2} (u, v) (u - v), (u - w) 〉}_{L^{2}} - β {〈 (u - v), (u - w) 〉}_{L^{2}}| \\ \leq & {M ∥ u - v ∥}_{1, 2, Ω} {∥ u - w ∥}_{1, 2, Ω}, \end{matrix}

(441)

\forall u, v, w \in B_{r} (u_{0}) .

Here

\tilde{u} (u, v)

is on the line connecting u and v so that

u^{3} - v^{3} = 3 {\tilde{u}}^{2} (u - v) .

Similarly as we have done in previous sections, we assume

u_{0}

and

r > 0

are such that we may obtain

u_{n} \in B_{r} (u_{0})

\forall n \in N .

Also similarly as in the previous section, we may consider such a ball either related to the

H_{0}^{1} (Ω)

norm or the

W^{1, \infty} (Ω)

one.

Let

m, n \in N

be such that

m > n .

Observe that

u_{n}, u_{m} \in V_{m}

so that

\begin{matrix} γ {〈 \nabla u_{m}, \nabla (u_{n} - u_{m}) 〉}_{L^{2}} + α {〈 u_{m}^{3}, (u_{n} - u_{m}) 〉}_{L^{2}} \\ - β {〈 u_{m}, (u_{n} - u_{m}) 〉}_{L^{2}} - {〈 f, (u_{n} - u_{m}) 〉}_{L^{2}} \\ = & 0, \end{matrix}

(442)

so that, for

φ \in V_{n}

, we obtain

\begin{matrix} γ {〈 \nabla (u_{n} - u_{m}), \nabla (u_{n} - u_{m}) 〉}_{L^{2}} + α {〈 (u_{n}^{3} - u_{m}^{3}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ = & γ {〈 \nabla (u_{n} - u_{m}), \nabla (u_{n} - φ) 〉}_{L^{2}} + α {〈 (u_{n}^{3} - u_{m}^{3}), (u_{n} - φ) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (u_{n} - φ) 〉}_{L^{2}} \\ + γ {〈 \nabla (u_{n} - u_{m}), \nabla (φ - u_{m}) 〉}_{L^{2}} + α {〈 (u_{n}^{3} - u_{m}^{3}), (φ - u_{n}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (φ - u_{m}) 〉}_{L^{2}} \\ = & γ {〈 \nabla (u_{n} - u_{m}), \nabla (φ - u_{m}) 〉}_{L^{2}} + α {〈 (u_{n}^{3} - u_{m}^{3}), (φ - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (φ - u_{m}) 〉}_{L^{2}} . \end{matrix}

(443)

Summarizing, we have got

\begin{matrix} γ {〈 \nabla (u_{n} - u_{m}), \nabla (u_{n} - u_{m}) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}_{n}^{2} (u_{n} - u_{m}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ = & γ {〈 \nabla (u_{n} - u_{m}), \nabla (φ - u_{m}) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}_{n}^{2} (u_{n} - u_{m}), (φ - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (φ - u_{m}) 〉}_{L^{2}} \end{matrix}

(444)

\forall φ \in V_{n},

where

{\tilde{u}}_{n}

is on the line connecting

u_{m}

and

u_{n}

Here we recall that

α_{1} > 0

and

M > 0

are such that

\begin{matrix} γ {〈 \nabla (u_{n} - u_{m}), \nabla (u_{n} - u_{m}) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}_{n}^{2} (u_{n} - u_{m}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (u_{n} - u_{m}) 〉}_{L^{2}} \\ \geq & α_{1} {∥ u_{n} - u_{m} ∥}_{1, 2, Ω}^{2} \end{matrix}

(445)

and

\begin{matrix} γ {〈 \nabla (u_{n} - u_{m}), \nabla (φ - u_{m}) 〉}_{L^{2}} + α {〈 3 {\tilde{u}}_{n}^{2} (u_{n} - u_{m}), (φ - u_{m}) 〉}_{L^{2}} \\ - β {〈 (u_{n} - u_{m}), (u_{n} - φ) 〉}_{L^{2}} \\ \leq & M ∥ u_{m} - u_{n} ∥_{1, 2, Ω} {∥ φ - u_{m} ∥}_{1, 2, Ω}, \end{matrix}

(446)

where

α_{1}

and M does not depend on

m, n

From such results, we may infer that

∥ u_{m} - u_{n} ∥_{1, 2, Ω} \leq \frac{M}{α_{1}} {∥ u_{m} - φ ∥}_{1, 2, Ω}, \forall φ \in V_{n}

so that

∥ u_{m} - u_{n} ∥_{1, 2, Ω} \leq \frac{M}{α_{1}} {∥ u_{m} - P_{n} (u_{m}) ∥}_{1, 2, Ω}, \forall m > n .

Moreover, since

u_{m} \in H_{0}^{1} (Ω)

, there exists a sequence

{v_{k} = v_{k}^{m}} \subset C_{c}^{\infty} (Ω)

such that

∥ v_{k}^{m} - u_{m} ∥_{1, 2, Ω} \to 0, as k \to \infty .

From such results, for a not relabeled subsequence we have

v_{k}^{m} \to u_{m}, a . e . in Ω,

\nabla v_{k}^{m} \to \nabla u_{m}, a . e . in Ω .

Let

ε > 0

From the Egorov theorem, for each

m \in N

there exists a closed set

F_{m} \subset Ω

such that

m (F_{m}) < ε / 2^{m}

and

v_{k}^{m} \to u_{m}, uniformly in Ω ∖ F_{m},

\nabla v_{k}^{m} \to \nabla u_{m}, uniformly in Ω ∖ F_{m} .

Define

F = \cup_{m = 1}^{\infty} F_{m}

so that

m (F) \leq \sum_{m = 1}^{\infty} (F_{m}) \leq \sum_{m = 1}^{\infty} ε / 2^{m} \leq ε .

Observe that there exists

k_{0} = k_{0}^{m} \in N

such that if

k > k_{0} = k_{0}^{m}

, then

∥ v_{k}^{m} - u_{m} ∥_{1, 2, Ω} < ε,

and

∥ v_{k}^{m} - u_{m} ∥_{\infty, Ω ∖ F_{m}} < ε,

and

∥ \nabla v_{k}^{m} - \nabla u_{m} ∥_{\infty, Ω ∖ F_{m}} < ε .

Fixing

m \in N

we may find

j_{0} \in N

(which does not depend on m) and

l_{m} \in N

∥ P_{j} (v_{k}^{l}) - P_{j} (u_{l}) ∥_{0, 2, Ω ∖ F_{l}} < K_{1} ε,

and

∥ \nabla P_{j} (v_{k}^{l}) - \nabla P_{j} (u_{l}) ∥_{0, 2, Ω ∖ F_{l}} < K_{5} ε, \forall j > j_{0}, \forall l > l_{m} \in N, \forall k > k_{0}^{l},

for some appropriate real constants

K_{1} > 0, K_{5} > 0 .

At this point we highlight that concerning the finite elements method

∥ u_{m} ∥_{1, \infty}

is uniformly bounded in m so that

{∥ v_{k}^{m} ∥_{1, \infty}, k > k_{0}^{m}, m \in N}

is also uniformly bounded in m and

k > k_{0}^{m} .

With such results in mind, fix

n > j_{0}

and select

m_{n} > max {n, k_{0}^{n}, l_{n}}

so that for

\forall m > m_{n}

and

k > k_{0}^{m}

, we have

\begin{matrix} ∥ \nabla u_{m} - \nabla P_{n} (v_{k}^{m}) ∥_{0, 2, Ω / F_{m}} \\ \leq & ∥ \nabla u_{m} - \nabla v_{k}^{m} + \nabla v_{k}^{m} - \nabla P_{n} (v_{k}^{m}) ∥_{0, 2, Ω / F_{m}} \\ \leq & ∥ \nabla u_{m} - \nabla v_{k}^{m} ∥_{0, 2, Ω} + {∥ \nabla v_{k}^{m} - \nabla P_{n} (v_{k}^{m}) ∥}_{0, 2, Ω / F_{m}} \\ \leq & ε + K_{7} / n, \end{matrix}

(447)

for some appropriate

K_{7} > 0 .

From such results, we may infer that

\begin{matrix} ∥ u_{m} - P_{n} (u_{m}) {) ∥}_{1, 2, Ω} \\ \leq & ∥ u_{m} - P_{n} (v_{k}^{m}) + P_{n} (v_{k}^{m}) - P_{n} (u_{m}) {)) ∥}_{1, 2, Ω} \\ \leq & ∥ u_{m} - P_{n} (v_{k}^{m}) ∥_{1, 2, Ω} + ∥ P_{n} (v_{k}^{m}) - P_{n} (u_{m}) ∥_{1, 2, Ω / F_{m}} + {∥ P_{n} (v_{k}^{m}) - P_{n} (u_{m}) ∥}_{1, 2, F_{m}} \\ \leq & K_{9} (ε + K_{7} / n + ε^{1 / 2}), \end{matrix}

(448)

for some appropriate

K_{9} > 0,

so that

\begin{matrix} ∥ u_{m} - u_{n} ∥_{1, 2, Ω} & \leq & \frac{M}{α_{1}} {∥ u_{m} - P_{n} (u_{m}) ∥}_{1, 2, Ω} \\ \leq & K_{10} (ε + K_{7} / n + ε^{1 / 2}), \forall m > m_{n}, \end{matrix}

(449)

where

K_{10} = K_{9} \frac{M}{α_{1}} .

Therefore, if

p, l > m_{n}

, then

\begin{matrix} ∥ u_{l} - u_{n} ∥_{1, 2, Ω} & \leq & \frac{M}{α_{1}} {∥ u_{l} - P_{n} (u_{l}) ∥}_{1, 2, Ω} \\ \leq & K_{10} (ε + K_{7} / n + ε^{1 / 2}) \end{matrix}

(450)

and

\begin{matrix} ∥ u_{p} - u_{n} ∥_{1, 2, Ω} & \leq & \frac{M}{α_{1}} {∥ u_{p} - P_{n} (u_{p}) ∥}_{1, 2, Ω} \\ \leq & K_{10} (ε + K_{7} / n + ε^{1 / 2}), \end{matrix}

(451)

so that

\begin{matrix} ∥ u_{l} - u_{p} ∥_{1, 2,, Ω} & = & ∥ u_{l} - u_{n} + u_{n} - u_{p} ∥_{1, 2, Ω} \\ \leq & ∥ u_{l} - u_{n} ∥_{1, 2, Ω} + {∥ u_{p} - u_{n} ∥}_{1, 2, Ω} \\ \leq & 2 K_{10} (ε + K_{7} / n + ε^{1 / 2}) . \end{matrix}

(452)

Consequently, from such results we may infer that

{u_{n}}

is a Cauchy sequence in

H_{0}^{1} (Ω)

so that there exists

u_{0} \in H_{0}^{1} (Ω)

such that

u_{n} \to u_{0}, strongly in H_{0}^{1} (Ω) .

Let

φ \in \cup_{n \in N} V_{n} .

Indeed, we have got

\begin{matrix} 0 & = & lim_{n \to \infty} (γ {〈 \nabla u_{n}, \nabla φ 〉}_{L^{2}} + α {〈 u_{n}^{3}, φ 〉}_{L^{2}} \\ - β {〈 u_{n}, φ 〉}_{L^{2}} - {〈 f, φ 〉}_{L^{2}}) \\ = & γ {〈 \nabla u_{0}, \nabla φ 〉}_{L^{2}} + α {〈 u_{0}^{3}, φ 〉}_{L^{2}} \\ - β {〈 u_{n}, φ 〉}_{L^{2}} - {〈 f, φ 〉}_{L^{2}} . \end{matrix}

(453)

Summarizing, we may infer that

\begin{matrix} γ {〈 \nabla u_{0}, \nabla φ 〉}_{L^{2}} + α {〈 u_{0}^{3}, φ 〉}_{L^{2}} \\ - β {〈 u_{n}, φ 〉}_{L^{2}} - {〈 f, φ 〉}_{L^{2}} \\ = & 0, \forall φ \in H_{0}^{1} (Ω) . \end{matrix}

(454)

Therefore

u_{0} \in H_{0}^{1} (Ω)

is a weak solution of the equation in question so that, under the indicated hypotheses, the finite element method is convergent.

62. A Dual Functional for a General Weak Primal Variational Formulation Combined with the Newton’s Method

Let

Ω = [0, 1] \subset R

and consider a weak variational formulation for a Ginzburg-Landau type equation corresponding to a functional

J : V \times V \to R

, where

J (u, v_{1}^{*}) = \int_{Ω} v_{1}^{*} (- ε u^{''} + A u^{3} - B u - f) d x,

where

ε > 0

A > 0

B > 0

and

f \in Y = Y^{*} = L^{2} (Ω) .

Moreover

u \in V = W_{0}^{1, 2} (Ω) .

Observe that the variation in

v_{1}^{*}

of J, which stands for

\frac{𝜕 J (u, v_{1}^{*})}{𝜕 v_{1}^{*}} = 0,

corresponds to the following Ginzburg-Landau type equation

- ε u^{''} + A u^{3} - B u - f = 0, in Ω .

In a Newton type approach context, we linearize such an equation about a initial solution

u_{0} \in V

, obtaining,

- ε u^{''} + 3 A u_{0}^{2} u - 2 A u_{0}^{3} - B u - f = 0, in Ω .

With such results in mind, we define the functional

J_{1} : {[V]}^{3} \to R

, where

J_{1} (u, u_{0}, v_{1}^{*}) = \int_{Ω} v_{1}^{*} (- ε u^{''} + 3 A u_{0}^{2} u - 2 A u_{0}^{3} - B u - f) d x .

We also define the functionals

F_{1} : {[V]}^{3} \to R

and

F_{2} : V \to R

, where

F_{1} (u, u_{0}, v_{1}^{*}) = J_{1} (u, u_{0}, v_{1}^{*}) + \frac{K}{2} \int_{Ω} u^{2} d x,

and

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

Moreover, we define the polar functionals

F_{1}^{*} : {[V]}^{2} \times Y^{*} \to R

and

F_{2}^{*} : Y^{*} \to R

F_{1}^{*} (u_{0}, v_{1}^{*}, z^{*}) = sup_{u \in V} {{〈 u, z^{*} 〉}_{L^{2}} - F_{1} (u, u_{0}, v_{1}^{*})},

and

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

(455)

Finally, we define the dual functional

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

J^{*} (u_{0}, v_{1}^{*}, z^{*}) = - F_{1}^{*} (u_{0}, v_{1}^{*}, z^{*}) + F_{2}^{*} (z^{*}) .

Remark 30.

Observe that

F_{1}^{*} (u_{0}, v_{1}^{*}, z^{*}) = sup_{u \in V} {{〈 u, z^{*} 〉}_{L^{2}} - F_{1} (u, u_{0}, v_{1}^{*})},

and such a supremum is attained through the equation

\frac{𝜕}{𝜕 u} ({〈 u, z^{*} 〉}_{L^{2}} - F_{1} (u, u_{0}, v_{1}^{*})) = 0,

which stands for

z^{*} - (- ε {(v_{1}^{*})}^{''} + 3 A u_{0}^{2} v_{1}^{*} - B v_{1}^{*}) - K u = 0,

so that

u = \frac{ε {(v_{1}^{*})}^{''} - 3 A u_{0}^{2} v_{1}^{*} + B v_{1}^{*} + z^{*}}{K} .

Consequently, we may obtain

\begin{matrix} F_{1}^{*} (u_{0}, v_{1}^{*}, z^{*}) & = & \frac{1}{2 K} \int_{Ω} {(ε {(v_{1}^{*})}^{''} - 3 A u_{0}^{2} v_{1}^{*} + B v_{1}^{*} + z^{*})}^{2} d x \\ + \int_{Ω} (2 A u_{0}^{3} + f) v_{1}^{*} d x . \end{matrix}

(456)

Hence, the variation in

v_{1}^{*}

J^{*}

\frac{𝜕 J^{*}}{𝜕 v_{1}^{*}} = - \frac{𝜕 F_{1}^{*} (v_{1}^{*})}{𝜕 v_{1}^{*}} = 0,

stands for

- ε u^{''} + 3 A u_{0}^{2} u - 2 A u_{0}^{3} - B u - f = 0, in Ω,

where, as above indicated,

u = \frac{ε {(v_{1}^{*})}^{''} - 3 A u_{0}^{2} v_{1}^{*} + B v_{1}^{*} + z^{*}}{K} .

We have obtained a critical of

J^{*}

through the following algorithm.

Set $n = 1$ , $b_{12} = 10^{- 4}$ , $n_{m a x} = 100$ , $z_{1}^{*} = 0$ and choose ${(u_{0})}_{1} \in V .$
Calculate ${(v_{1}^{*})}_{n} \in V$ such that

$\frac{𝜕 J^{*} ({(u_{0})}_{n}, {(v_{1}^{*})}_{n}, z_{n}^{*})}{𝜕 v_{1}^{*}} = 0,$
Calculate $u_{n} \in V$ such that

$\frac{𝜕 H (u_{n}, {(u_{0})}_{n}, {(v_{1}^{*})}_{n}, z_{n}^{*})}{𝜕 u} = 0,$

where

$H (u, (u_{0}), v_{1}^{*}, z^{*}) = 〈 u, z^{*} 〉 - F_{1} (u, u_{0}, v_{1}^{*}),$

so that

$u_{n} = \frac{ε {(v_{1}^{*})}_{n}^{''} - 3 A {(u_{0})}_{n}^{2} {(v_{1}^{*})}_{n} + B {(v_{1}^{*})}_{n} + z_{n}^{*}}{K} .$
Set ${(u_{0})}_{n + 1} = u_{n}$ and $z_{n + 1}^{*} = K u_{n} .$
If $∥ {(u_{0})}_{n + 1} - {(u_{0})}_{n} ∥_{\infty} < b_{12}$ or $n > n_{m a x}$ , then stop. Otherwise, $n : = n + 1,$ and go to item 2.

Here we highlight that if

{\hat{u}}_{0} = {lim}_{n \to \infty} u_{n}

with corresponding limits

{\hat{v}}_{1}^{*}

and

{\hat{z}}^{*} = K {\hat{u}}_{0}

, the solution of equation indicated in the item 2, given by

\frac{𝜕 J^{*} ({\hat{u}}_{0}, {\hat{v}}_{1}^{*}, {\hat{z}}^{*})}{𝜕 v_{1}^{*}} = 0,

will stand for

- ε {\hat{u}}_{0}^{''} + A {\hat{u}}_{0}^{3} - B {\hat{u}}_{0} - f = 0, in Ω .

We have obtained numerical results for

ε = 0.1

A = B = 1

and

f \equiv 1, in Ω .

For such an optimal solution

{\hat{u}}_{0}

obtained please see Figure 44.

Here we present the software in MAT-LAB through which we have obtained such numerical results.

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

clear all

global m8 d yo K uo u z A B e1 v1

m8=100;

d=1/m8;

K=10;

A=1;

B=1;

e1=0.1;

z(:,1)=0.1*ones(m8,1);

yo(:,1)=ones(m8,1);

uo(:,1)=1.2*ones(m8,1);

for i=1:m8

xo(i,1)=1.2;

end

b12=1.0;

k=1;

while $(b 12 > 10^{- 4})$ && $(k < 100)$

k

k=k+1;

b14=1.0;

k1=1;

while $(b 14 > 10^{- 4})$ && $(k 1 < 35)$

k1

k1=k1+1;

X=fminunc(’funJune2024C10’,xo);

b14=max(abs(X-xo));

xo=X;

u(m8/2,1)

end;

b12=max(abs(u-uo));

uo=u;

z=K*u;

end;

for i=1:m8

x(i,1)=i*d;

end;

plot(x,u);

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

With the auxiliary function "funJune2024C10", where

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

function S=funJune2024C10(x)

global m8 d yo K uo u z A B e1 v1

for i=1:m8

v1(i,1)=x(i,1);

end;

v1(m8,1)=0;

d2v1(1,1)=(-2*v1(1,1)+v1(2,1))/ $d^{2}$ ;

for i=2:m8-1

d2v1(i,1)=(v1(i+1,1)-2*v1(i,1)+v1(i-1,1))/ $d^{2}$ ;

end;

for i=1:m8-1

u(i,1)= $(e 1 * d 2 v 1 (i, 1) + z (i, 1) - 3 * A * u o {(i, 1)}^{2} * v 1 (i, 1) + B * v 1 (i, 1)) / K;$

end;

u(m8,1)=0;

S=0;

for i=1:m8-1

S=S+ $(- e 1 * d 2 v 1 (i, 1) * u (i, 1)) + v 1 (i, 1) * 3 * A * u o {(i, 1)}^{2} * u (i, 1)$

$- B * v 1 (i, 1) * u (i, 1) + K * u {(i, 1)}^{2} / 2 - y o (i, 1) * v 1 (i, 1) - 2 * A * u o {(i, 1)}^{3} * v 1 (i, 1);$

S=S $- z (i, 1) * u (i, 1) - v 1 {(i, 1)}^{2} / 2$ ;

end;

S=-S;

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

63. A Dual Variational Formulation for a General Non-Convex Primal One

Let

Ω \subset R^{3}

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

𝜕 Ω .

Consider a 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}

(457)

where

γ > 0

α > 0,

β > 0

and

f \in L^{2} (Ω) .

Here

u \in V = W_{0}^{1, 2} (Ω)

and we denote

Y = Y^{*} = L^{2} (Ω) .

At this point, we define the functionals

F_{1} : V \to R

F_{2} : V \times Y \to R

and

F_{3} : V \to R,

where

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

F_{2} (u, v) = - \frac{α}{2} \int_{Ω} {(u^{2} - β + v)}^{2} d x + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, f 〉}_{L^{2}};

and

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

for some constant

K > 0

Moreover, we define the following polar functionals

\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}

(458)

\begin{matrix} F_{2}^{*} (v_{1}^{*}, v_{0}^{*}) & = & sup_{u \in V} \{inf_{v \in Y} {{〈 u, - v_{1}^{*} + z^{*} 〉}_{L^{2}} + {〈 v, v_{0}^{*} 〉}_{L^{2}} - F_{2} (u, v)}\} \\ = & - \frac{1}{2} \int_{Ω} \frac{{(- v_{1}^{*} + z^{*} + f)}^{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}

(459)

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

, where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ 2 v_{0}^{*} ∥_{\infty} \leq \sqrt[4]{K}}

and,

\begin{matrix} F_{3}^{*} (z^{*}) & = & sup_{v \in Y} {{〈 v, z^{*} 〉}_{L^{2}} - F_{3} (v)} \\ = & \frac{1}{2 K} \int_{Ω} {(z^{*})}^{2} d x . \end{matrix}

(460)

Finally, denoting

D^{*} = \{v_{1}^{*} \in Y^{*} : {∥ v_{1}^{*} ∥}_{\infty} \leq \frac{5}{4} K\},

we also define

J^{*} : D^{*} \times B^{*} \times Y^{*} \to R

\begin{matrix} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & = & F_{1}^{*} (v_{1}^{*} - F_{2}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) + F_{3}^{*} (z^{*}) \\ - K_{1} {∥\frac{- v_{1}^{*} + z + f}{2 v_{0}^{*} - K} + \frac{v_{1}^{*}}{- γ \nabla^{2}}∥}_{0, 2}^{2} - K_{1} {∥\frac{- v_{1}^{*} + z + f}{2 v_{0}^{*} - K} + \frac{z^{*}}{K}∥}_{0, 2}^{2} . \end{matrix}

(461)

Observe that

\begin{matrix} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & \geq & F_{1}^{*} (v_{1}^{*}) - F_{2}^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) + {〈 u, z^{*} 〉}_{L^{2}} - \frac{K}{2} \int_{Ω} u^{2} d x \\ - K_{1} {∥\frac{- v_{1}^{*} + z + f}{2 v_{0}^{*} - K} + \frac{v_{1}^{*}}{- γ \nabla^{2}}∥}_{0, 2}^{2} - K_{1} {∥\frac{- v_{1}^{*} + z + f}{2 v_{0}^{*} - K} + \frac{z^{*}}{K}∥}_{0, 2}^{2}, \end{matrix}

(462)

so that

\begin{matrix} sup_{(v_{1}^{*}, z^{*}) \in Y^{*} \times Y^{*}} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & \geq & F_{1}^{*} (v_{1}^{*}) - F_{2}^{*} (v_{1}^{*}), v_{0}^{*}, z^{*}) + {〈 u, z^{*} 〉}_{L^{2}} - \frac{K}{2} \int_{Ω} u^{2} d x \end{matrix}

(463)

Let

{\tilde{v}}_{1}^{*} \in Y^{*}

be such that

\frac{- {\tilde{v}}_{1}^{*} + z^{*} + f}{2 v_{0}^{*} - K} + \frac{{\tilde{v}}_{1}^{*}}{- γ \nabla^{2}} = 0 .

Let us now denote

\tilde{u} = \tilde{u} (z^{*}, v_{0}^{*}) = \frac{- {\tilde{v}}_{1}^{*} + z + f}{2 v_{0}^{*} - K} = - \frac{{\tilde{v}}_{1}^{*}}{- γ \nabla^{2}} .

Consequently

\begin{matrix} sup_{(v_{1}^{*}, z^{*}) \in Y^{*} \times Y^{*}} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & \geq & - \frac{γ}{2} \int_{Ω} \nabla \tilde{u} \cdot \nabla \tilde{u} d x \\ + {〈 {\tilde{u}}^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{K}{2} \int_{Ω} {\tilde{u}}^{2} d x \\ - {〈 \tilde{u}, z^{*} 〉}_{L^{2}} + {〈 \tilde{u}, f 〉}_{L^{2}} \\ + {〈 u, z^{*} 〉}_{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, \forall z^{*} \in Y^{*} . \end{matrix}

(464)

Considering that

z^{*}

is arbitrary, in particular for

\tilde{u} = \tilde{u} (z^{*}, v_{0}^{*}) = u,

we obtain

\begin{matrix} sup_{(v_{1}^{*}, z^{*}) \in Y^{*} \times Y^{*}} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & \geq & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x \\ + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, z^{*} 〉}_{L^{2}} \\ + {〈 u, z^{*} 〉}_{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 \\ \geq & inf_{v_{0}^{*} \in Y^{*}} \{- \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + {〈 u, f 〉}_{L^{2}} \\ + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{1}{2 α} \int_{Ω} {(v_{0}^{*})}^{2} d x + β \int_{Ω} v_{0}^{*} d x\} . \end{matrix}

(465)

From such results, we may infer that

\begin{matrix} sup_{(v_{1}^{*}, z^{*}) \in Y^{*} \times Y^{*}} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*}) & \geq & - \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}} \\ = & - J (u), \forall u \in V, v_{0}^{*} \in Y^{*} . \end{matrix}

(466)

Therefore,

{sta}_{v_{0}^{*} \in Y^{*}} \{sup_{z^{*} \in Y^{*}} \{sup_{(v_{1}^{*} \in Y^{*}} J^{*} (v_{1}^{*}, v_{0}^{*}, z^{*})\}\} \geq sup_{u \in V} J (u) .

Now let

({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) \in D^{*} \times B^{*} \times Y^{*}

be such that

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

Let

u_{0} \in V

be such that

u_{0} = \frac{{\hat{z}}^{*}}{K} .

Similarly as in the previous sections, we may prove that

δ J (u_{0}) = 0,

and

\begin{matrix} - J (u_{0}) & = & sup_{u \in V} {- J (u)} \\ = & J^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}, {\hat{z}}^{*}) \\ = & sup_{(z^{*}, v_{1}^{*}) \in Y^{*} \times Y^{*}} J^{*} (v_{1}^{*}, {\hat{v}}_{0}^{*}, z^{*}) . \end{matrix}

(467)

The objective of this section is complete.

64. 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.

The author declares no conflict of interest concerning this article.

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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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.
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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

Figure 29. Optimal solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 29. Optimal solution

u_{0} (x)

for the case

f (x) = sin (π x) / 2

Figure 30. Optimal solution

u_{0} (x)

for the case

f (x) = cos (π x) / 2

Figure 30. Optimal solution

u_{0} (x)

for the case

f (x) = cos (π x) / 2

Figure 31. Optimal solution

u_{0} (x)

for the case

f (x) = 0

Figure 31. Optimal solution

u_{0} (x)

for the case

f (x) = 0

Figure 32. Optimal solution

u_{0} (x)

through the concerning dual formulation.

Figure 32. Optimal solution

u_{0} (x)

through the concerning dual formulation.

Figure 33. Optimal solution

u_{1} (x)

through the concerning primal formulation.

Figure 33. Optimal solution

u_{1} (x)

through the concerning primal formulation.

Figure 34. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 34. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 35. Optimal solution

u (x)

for the case

f (x) = 0

Figure 35. Optimal solution

u (x)

for the case

f (x) = 0

Figure 36. Optimal solution

u (x)

for the case A.

Figure 36. Optimal solution

u (x)

for the case A.

Figure 37. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 37. Optimal solution

u (x)

for the case

f (x) = sin (π x) / 2

Figure 38. Optimal solution

u (x)

for the case

f (x) = 0

Figure 38. Optimal solution

u (x)

for the case

f (x) = 0

Figure 39. Optimal solution

L_{1} (x)

for the case A.

Figure 39. Optimal solution

L_{1} (x)

for the case A.

Figure 40. Optimal shape

λ_{1} (x, y)

for the beam of case A.

Figure 40. Optimal shape

λ_{1} (x, y)

for the beam of case A.

Figure 41. Optimal shape

λ_{1} (x, y)

for the beam of case B.

Figure 41. Optimal shape

λ_{1} (x, y)

for the beam of case B.

Figure 42. Solution

u (x, y)

for

ε = 0.01

Figure 42. Solution

u (x, y)

for

ε = 0.01

Figure 43. Solution

u (x)

for

ε = 0.01

Figure 43. Solution

u (x)

for

ε = 0.01

Figure 44. Solution

{\hat{u}}_{0} (x)

through the dual functional for

ε = 0.1

Figure 44. Solution

{\hat{u}}_{0} (x)

through the dual functional for

ε = 0.1

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