Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

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Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

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A peer-reviewed article of this preprint also exists.

Fabio Botelho^*

This version is not peer-reviewed

Submitted:

23 January 2023

Posted:

25 January 2023

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Abstract

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

In this section we establish a dual formulation for a large class of models in non-convex optimization.

The main duality principle is applied to the Ginzburg-Landau system in superconductivity in an absence of a magnetic field.

Such results are based on the works of J.J. Telega and W.R. Bielski [2,3,13,14] and on a D.C. optimization approach developed in Toland [15].

About the other references, details on the Sobolev spaces involved are found in [1]. Related results on convex analysis and duality theory are addressed in [5,6,7,9,12]. Finally, similar models on the superconductivity physics may be found in [4,11].

Remark 1.

It is worth highlighting, we may generically denote

\int_{Ω} [{(- γ \nabla^{2} + K I_{d})}^{- 1} v^{*}] v^{*} d x

simply by

\int_{Ω} \frac{{(v^{*})}^{2}}{- γ \nabla^{2} + K} d x,

where

I_{d}

denotes a concerning identity operator.

Other similar notations may be used along this text as their indicated meaning are sufficiently clear.

Also,

\nabla^{2}

denotes the Laplace operator and for real constants

K_{2} > 0

and

K_{1} > 0

, the notation

K_{2} ≫ K_{1}

means that

K_{2} > 0

is much larger than

K_{1} > 0 .

Finally, 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.

Let

Ω \subset R^{3}

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

\partial Ω .

Firstly we emphasize that, for the Banach space

Y = Y^{*} = L^{2} (Ω)

, we have

{〈 v, v^{*} 〉}_{L^{2}} = \int_{Ω} v v^{*} d x, \forall v, v^{*} \in L^{2} (Ω) .

For the primal formulation we consider the functional

J : U \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}

(3)

Here we assume

α > 0, β > 0, γ > 0

U = W_{0}^{1, 2} (Ω)

f \in L^{2} (Ω)

. Moreover we denote

Y = Y^{*} = L^{2} (Ω) .

Define also

G_{1} : U \to R

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

G_{2} : U \times Y \to R

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

and

F : U \to R

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

where

K ≫ γ .

It is worth highlighting that in such a case

J (u) = G_{1} (u) + G_{2} (u, 0) - F (u) - {〈 u, f 〉}_{L^{2}}, \forall u \in U .

Furthermore, define the following specific polar functionals specified, namely,

G_{1}^{*} : {[Y^{*}]}^{2} \to R

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

(4)

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

\begin{matrix} G_{2}^{*} (v_{2}^{*}, v_{0}^{*}) & = & sup_{(u, v) \in U \times Y} \{{〈 u, v_{2}^{*} 〉}_{L^{2}} + {〈 v, v_{0}^{*} 〉}_{L^{2}} - G_{2} (u, v)\} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*})}^{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}

(5)

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

where

B^{*} = {v_{0}^{*} \in Y^{*} : 2 v_{0}^{*} + K > K / 2 in Ω} .

At this point, we give more details about this calculation.

Observe that

\begin{matrix} G_{2}^{*} (v_{2}^{*}, v_{0}^{*}) & = & sup_{(u, v) \in U \times Y} \{{〈 u, v_{2}^{*} 〉}_{L^{2}} + {〈 v, v_{0}^{*} 〉}_{L^{2}} - G_{2} (u, v)\} \\ = & sup_{(u, v) \in U \times Y} \{{〈 u, v_{2}^{*} 〉}_{L^{2}} + {〈 v, v_{0}^{*} 〉}_{L^{2}} - \frac{α}{2} \int_{Ω} {(u^{2} - β + v)}^{2} d x - \frac{K}{2} \int_{Ω} u^{2} d x\} . \end{matrix}

(6)

Defining

w = u^{2} - β + v,

we have

v = w - u^{2} + β,

so that

\begin{matrix} G_{2}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & sup_{(u, v) \in U \times Y} \{{〈 u, v_{2}^{*} 〉}_{L^{2}} + {〈 v, v_{0}^{*} 〉}_{L^{2}} - \frac{α}{2} \int_{Ω} {(u^{2} - β + v)}^{2} d x - \frac{K}{2} \int_{Ω} u^{2} d x\} \\ = & sup_{(u, w) \in U \times Y} \{{〈 u, v_{2}^{*} 〉}_{L^{2}} + {〈 w - u^{2} + β, v_{0}^{*} 〉}_{L^{2}} - \frac{α}{2} \int_{Ω} {(w)}^{2} d x - \frac{K}{2} \int_{Ω} u^{2} d x\} \\ = & {〈 \tilde{u}, v_{2}^{*} 〉}_{L^{2}} + {〈 \tilde{w} - {\tilde{u}}^{2} + β, v_{0}^{*} 〉}_{L^{2}} - \frac{α}{2} \int_{Ω} {(\tilde{w})}^{2} d x - \frac{K}{2} \int_{Ω} {\tilde{u}}^{2} d x, \end{matrix}

(7)

where

(\tilde{u}, \tilde{w})

are solution of equations (optimality conditions for such a quadratic optimization problem)

v_{0}^{*} - α \tilde{w} = 0,

and

v_{2}^{*} - (2 v_{0}^{*} + K) \tilde{u} = 0,

and therefore

\tilde{w} = \frac{v_{0}^{*}}{α},

and

\tilde{u} = \frac{v_{2}^{*}}{2 v_{0}^{*} + K} .

Replacing such results into (7) we obtain

\begin{matrix} G^{*} (v_{1}^{*}, v_{0}^{*}) & = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*})}^{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}

(8)

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

Finally,

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

is defined by

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

(9)

Define also

A^{*} = {v^{*} = (v_{1}^{*}, v_{2}^{*}, v_{0}^{*}) \in {[Y^{*}]}^{2} \times B^{*} : v_{1}^{*} + v_{2}^{*} - f = 0, in Ω},

J^{*} : {[Y^{*}]}^{4} \to R

J^{*} (v^{*}, z^{*}) = - G_{1}^{*} (v_{1}^{*} + z^{*}) - G_{2}^{*} (v_{2}^{*}, v_{0}^{*}) + F^{*} (z^{*})

and

J_{1}^{*} : {[Y^{*}]}^{4} \times U \to R

J_{1}^{*} (v^{*}, z^{*}, u) = J^{*} (v^{*}, z^{*}) + {〈 u, v_{1}^{*} + v_{2}^{*} - f 〉}_{L^{2}} .

2. The main duality principle, a convex dual formulation and the concerning proximal primal functional

Our main result is summarized by the following theorem.

Theorem 1.

Considering the definitions and statements in the last section, suppose also

({\hat{v}}^{*}, {\hat{z}}^{*}, u_{0}) \in {[Y^{*}]}^{2} \times B^{*} \times Y^{*} \times U

is such that

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

Under such hypotheses, we have

δ J (u_{0}) = 0,

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

and

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

(10)

Proof.

Since

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

from the variation in

v_{1}^{*}

we obtain

- \frac{({\hat{v}}_{1}^{*} + {\hat{z}}^{*})}{- γ \nabla^{2}} + u_{0} = 0 in Ω,

so that

{\hat{v}}_{1}^{*} + {\hat{z}}^{*} = - γ \nabla^{2} u_{0} .

From the variation in

v_{2}^{*}

we obtain

- \frac{{\hat{v}}_{2}^{*}}{2 {\hat{v}}_{0}^{*} + K} + u_{0} = 0, in Ω .

From the variation in

v_{0}^{*}

we also obtain

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

and therefore,

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

From the variation in u we get

{\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} - f = 0, in Ω

and thus

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

Finally, from the variation in

z^{*}

, we obtain

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

so that

- u_{0} + \frac{{\hat{z}}^{*}}{K} = 0,

that is,

{\hat{z}}^{*} = K u_{0} in Ω .

From such results and

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

we get

\begin{matrix} 0 & = & {\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} - f \\ = & - γ \nabla^{2} u_{0} - {\hat{z}}^{*} + 2 (v_{0}^{*}) u_{0} + K u_{0} - f \\ = & - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f, \end{matrix}

(11)

so that

δ J (u_{0}) = 0 .

Also from this and from the Legendre transform proprieties we have

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

G_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) = {〈 u_{0}, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + {〈 0, v_{0}^{*} 〉}_{L^{2}} - G_{2} (u_{0}, 0),

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

and thus we obtain

\begin{matrix} J^{*} ({\hat{v}}^{*}, {\hat{z}}^{*}) & = & - G_{1}^{*} ({\hat{v}}_{1}^{*} + {\hat{z}}^{*}) - G_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F^{*} ({\hat{z}}^{*}) \\ = & - 〈 u_{0}, {\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} 〉 + G_{1} (u_{0}) + G_{2} (u_{0}, 0) - F (u_{0}) \\ = & - {〈 u_{0}, f 〉}_{L^{2}} + G_{1} (u_{0}) + G_{2} (u_{0}, 0) - F (u_{0}) \\ = & J (u_{0}) . \end{matrix}

(12)

Summarizing, we have got

J^{*} ({\hat{v}}^{*}, {\hat{z}}^{*}) = J (u_{0}) .

(13)

On the other hand

\begin{matrix} J^{*} ({\hat{v}}^{*}, {\hat{z}}^{*}) & = & - G_{1}^{*} ({\hat{v}}_{1}^{*} + {\hat{z}}^{*}) - G_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F^{*} ({\hat{z}}^{*}) \\ \leq & - {〈 u, {\hat{v}}_{1}^{*} + {\hat{z}}^{*} 〉}_{L^{2}} - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} - {〈 0, v_{0}^{*} 〉}_{L^{2}} + G_{1} (u) + G_{2} (u, 0) + F^{*} ({\hat{z}}^{*}) \\ = & - {〈 u, f 〉}_{L^{2}} + G_{1} (u) + G_{2} (u, 0) - {〈 u, {\hat{z}}^{*} 〉}_{L^{2}} + F^{*} ({\hat{z}}^{*}) \\ = & - {〈 u, f 〉}_{L^{2}} + G_{1} (u) + G_{2} (u, 0) - F (u) + F (u) - {〈 u, {\hat{z}}^{*} 〉}_{L^{2}} + F^{*} ({\hat{z}}^{*}) \\ = & J (u) + \frac{K}{2} \int_{Ω} u^{2} d x - {〈 u, {\hat{z}}^{*} 〉}_{L^{2}} + F^{*} ({\hat{z}}^{*}) \\ = & J (u) + \frac{K}{2} \int_{Ω} u^{2} d x - K {〈 u, u_{0} 〉}_{L^{2}} + \frac{K}{2} \int_{Ω} u_{0}^{2} d x \\ = & J (u) + \frac{K}{2} \int_{Ω} {| u - u_{0} |}^{2} d x, \forall u \in U . \end{matrix}

(14)

Finally by a simple computation we may obtain the Hessian

\{\frac{\partial^{2} J^{*} (v^{*}, z^{*})}{\partial {(v^{*})}^{2}}\} < 0

{[Y^{*}]}^{2} \times B^{*} \times Y^{*}

, so that we may infer that

J^{*}

is concave in

v^{*}

{[Y^{*}]}^{2} \times B^{*} \times Y^{*}

Therefore, from this, (13) and (14), we have

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

(15)

The proof is complete. □

3. A primal dual variational formulation

In this section we develop a more general primal dual variational formulation suitable for a large class of models in non-convex optimization.

Consider again

U = W_{0}^{1, 2} (Ω)

and let

G : U \to R

and

F : U \to R

be three times Fréchet differentiable functionals. Let

J : U \to R

be defined by

J (u) = G (u) - F (u), \forall u \in U .

Assume

u_{0} \in U

is such that

δ J (u_{0}) = 0

and

δ^{2} J (u_{0}) > 0 .

Denoting

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

, define

J^{*} : U \times Y^{*} \times Y^{*} \to R

J^{*} (u, v^{*}) = \frac{1}{2} ∥ v_{1}^{*} - G^{'} (u) ∥_{2}^{2} + \frac{1}{2} ∥ v_{2}^{*} - F^{'} (u) ∥_{2}^{2} + \frac{1}{2} {∥ v_{1}^{*} - v_{2}^{*} ∥}_{2}^{2}

(16)

Denoting

L_{1}^{*} (u, v^{*}) = v_{1}^{*} - G^{'} (u)

and

L_{2}^{*} (u, v^{*}) = v_{2}^{*} - F^{'} (u)

, define also

C^{*} = \{(u, v^{*}) \in U \times Y^{*} \times Y^{*} : ∥ L_{1}^{*} (u, v_{1}^{*}) ∥_{\infty} \leq \frac{1}{K} and {∥ L_{2}^{*} (u, v_{1}^{*}) ∥}_{\infty} \leq \frac{1}{K}\},

for an appropriate

K > 0

to be specified.

Observe that in

C^{*}

the Hessian of

J^{*}

is given by

{δ^{2} J^{*} (u, v^{*})} = \{\begin{matrix} G^{''} {(u)}^{2} + F^{''} {(u)}^{2} + O (1 / K) & - G^{''} (u) & - F^{''} (u) \\ - G^{''} (u) & 2 & - 1 \\ - F^{''} (u) & - 1 & 2 \end{matrix}\},

(17)

Observe also that

det \{\frac{\partial^{2} J^{*} (u, v^{*})}{\partial v_{1}^{*} \partial v_{2}^{*}}\} = 3,

and

det {δ^{2} J^{*} (u, v^{*})} = {(G^{''} (u) - F^{''} (u))}^{2} + O (1 / K) = {(δ^{2} J (u))}^{2} + O (1 / K) .

Define now

{\hat{v}}_{1}^{*} = G^{'} (u_{0}),

{\hat{v}}_{2}^{*} = F^{'} (u_{0}),

so that

{\hat{v}}_{1}^{*} - {\hat{v}}_{2}^{*} = 0 .

From this we may infer that

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}) \in C^{*}

and

J^{*} (u_{0}, {\hat{v}}^{*}) = 0 = min_{(u, v^{*}) \in C^{*}} J^{*} (u, v^{*}) .

Moreover, for

K > 0

sufficiently big,

J^{*}

is convex in a neighborhood of

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

Therefore, in the last lines, we have proven the following theorem.

Theorem 2.

Under the statements and definitions of the last lines, there exist

r_{0} > 0

and

r_{1} > 0

such that

J (u_{0}) = min_{u \in B_{r_{0}} (u_{0})} J (u)

and

(u_{0}, {\hat{v}}_{1}^{*}, {\hat{v}}_{2}^{*}) \in C^{*}

is such that

J^{*} (u_{0}, {\hat{v}}^{*}) = 0 = min_{(u, v^{*}) \in U \times {[Y^{*}]}^{2}} J^{*} (u, v^{*}) .

Moreover,

J^{*}

is convex in

B_{r_{1}} (u_{0}, {\hat{v}}^{*}) .

4. One more duality principle and a concerning primal dual variational formulation

In this section we establish a new duality principle and a related primal dual formulation.

The results are based on the approach of Toland, [15].

4.1. Introduction

Let

Ω \subset R^{3}

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

\partial Ω .

Let

J : V \to R

be a functional such that

J (u) = G (u) - F (u), \forall u \in V,

where

V = W_{0}^{1, 2} (Ω)

Suppose

G, F

are both three times Fréchet differentiable convex functionals such that

\frac{\partial^{2} G (u)}{\partial u^{2}} > 0

and

\frac{\partial^{2} F (u)}{\partial u^{2}} > 0

\forall u \in V .

Assume also there exists

α_{1} \in R

such that

α_{1} = inf_{u \in V} J (u) .

Moreover, suppose that if

{u_{n}} \subset V

is such that

∥ u_{n} ∥_{V} \to \infty

then

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

At this point we define

J^{* *} : V \to R

J^{* *} (u) = sup_{(v^{*}, α) \in H^{*}} {〈 u, v^{*} 〉 + α},

where

H^{*} = {(v^{*}, α) \in V^{*} \times R : {〈 v, v^{*} 〉}_{V} + α \leq F (v), \forall v \in V} .

Observe that

(0, α_{1}) \in H^{*},

so that

J^{* *} (u) \geq α_{1} = inf_{u \in V} J (u) .

On the other hand, clearly we have

J^{* *} (u) \leq J (u), \forall u \in V,

so that we have got

α_{1} = inf_{u \in V} J (u) = inf_{u \in V} J^{* *} (u) .

Let

u \in V

Since J is strongly continuous, there exist

δ > 0

and

A > 0

such that,

α_{1} \leq J^{* *} (v) \leq J (v) \leq A, \forall v \in B_{δ} (u) .

From this, considering that

J^{* *}

is convex on V, we may infer that

J^{* *}

is continuous at u,

\forall u \in V .

Hence

J^{* *}

is strongly lower semi-continuous on V, and since

J^{* *}

is convex we may infer that

J^{* *}

is weakly lower semi-continuous on V.

Let

{u_{n}} \subset V

be a sequence such that

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

Hence

α_{1} = lim_{n \to \infty} J (u_{n}) = inf_{u \in V} J (u) = inf_{u \in V} J^{* *} (u) .

Suppose there exists a subsequence

{u_{n_{k}}}

{u_{n}}

such that

∥ u_{n_{k}} ∥_{V} \to \infty, as k \to \infty .

From the hypothesis we have

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

which contradicts

α_{1} \in R .

Therefore there exists

K > 0

such that

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

Since V is reflexive, from this and the Katutani Theorem, there exists a subsequence

{u_{n_{k}}}

{u_{n}}

and

u_{0} \in V

such that

u_{n_{k}} ⇀ u_{0}, weakly in V .

Consequently, from this and considering that

J^{* *}

is weakly lower semi-continuous, we have got

α_{1} = \underset{k \to \infty}{lim inf} J^{* *} (u_{n_{k}}) \geq J^{* *} (u_{0}),

so that

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

Define

G^{*}, F^{*} : V^{*} \to R

G^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{V} - G (u)},

and

F^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{V} - F (u)} .

Defining also

J^{*} : V \to R

J^{*} (v^{*}) = F^{*} (v^{*}) - G^{*} (v^{*}),

from the results in [15], we may obtain

inf_{u \in V} J (u) = inf_{v^{*} \in V^{*}} J^{*} (v^{*}),

so that

\begin{matrix} J^{* *} (u_{0}) & = & inf_{u \in V} J^{* *} (u) \\ = & inf_{u \in V} J (u) = inf_{v^{*} \in V^{*}} J^{*} (v^{*}) . \end{matrix}

(18)

Suppose now there exists

\hat{u} \in V

such that

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

From the standard necessary conditions, we have

δ J (\hat{u}) = 0,

so that

\frac{\partial G (\hat{u})}{\partial u} - \frac{\partial F (\hat{u})}{\partial u} = 0 .

Define now

v_{0}^{*} = \frac{\partial F (\hat{u})}{\partial u} .

From these last two equations we obtain

v_{0}^{*} = \frac{\partial G (\hat{u})}{\partial u} .

From such results and the Legendre transform properties, we have

\hat{u} = \frac{\partial F^{*} (v_{0}^{*})}{\partial v^{*}},

\hat{u} = \frac{\partial G^{*} (v_{0}^{*})}{\partial v^{*}},

so that

δ J^{*} (v_{0}^{*}) = \frac{\partial F^{*} (v_{0}^{*})}{\partial v^{*}} - \frac{\partial G^{*} (v_{0}^{*})}{\partial v^{*}} = \hat{u} - \hat{u} = 0,

G^{*} (v_{0}^{*}) = {〈 \hat{u}, v_{0}^{*} 〉}_{V} - G (\hat{u})

and

F^{*} (v_{0}^{*}) = {〈 \hat{u}, v_{0}^{*} 〉}_{V} - F (\hat{u})

so that

\begin{matrix} inf_{u \in V} J (u) & = & J (\hat{u}) \\ = & G (\hat{u}) - F (\hat{u}) \\ = & inf_{v^{*} \in V^{*}} J^{*} (v^{*}) \\ = & F^{*} (v_{0}^{*}) - G^{*} (v_{0}^{*}) \\ = & J^{*} (v_{0}^{*}) . \end{matrix}

(19)

4.2. The main duality principle and a related primal dual variational formulation

Considering these last statements and results, we may prove the following theorem.

Theorem 3.

Let

Ω \subset R^{3}

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

\partial Ω .

Let

J : V \to R

be a functional such that

J (u) = G (u) - F (u), \forall u \in V,

where

V = W_{0}^{1, 2} (Ω)

Suppose

G, F

are both three times Fréchet differentiable functionals such that there exists

K > 0

such that

\frac{\partial^{2} G (u)}{\partial u^{2}} + K > 0

and

\frac{\partial^{2} F (u)}{\partial u^{2}} + K > 0

\forall u \in V .

Assume also there exists

u_{0} \in V

and

α_{1} \in R

such that

α_{1} = inf_{u \in V} J (u) = J (u_{0}) .

Assume

K_{3} > 0

is such that

∥ u_{0} ∥_{\infty} < K_{3} .

Define

\tilde{V} = {{u \in V : ∥ u ∥}_{\infty} \leq K_{3}} .

Assume

K_{1} > 0

is such that if

u \in \tilde{V}

then

max \{∥ F^{'} {(u) ∥}_{\infty}, ∥ G^{'} {(u) ∥}_{\infty}, ∥ F^{''} {(u) ∥}_{\infty}, ∥ F^{'''} {(u) ∥}_{\infty}, ∥ G^{''} {(u) ∥}_{\infty}, {∥ G^{'''} (u) ∥}_{\infty}\} \leq K_{1} .

Suppose also

K ≫ max {K_{1}, K_{3}} .

Define

F_{K}, G_{K} : V \to R

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

and

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

\forall u \in V .

Define also

G_{K}^{*}, F_{K}^{*} : V^{*} \to R

G_{K}^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{V} - G_{K} (u)},

and

F_{K}^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{V} - F_{K} (u)} .

Observe that since

u_{0} \in V

is such that

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

we have

δ J (u_{0}) = 0 .

Let

ε > 0

be a small constant.

Define

v_{0}^{*} = \frac{\partial F_{K} (u_{0})}{\partial u} \in V^{*} .

Under such hypotheses, defining

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

\begin{matrix} J_{1}^{*} (u, v^{*}) & = & F_{K}^{*} (v^{*}) - G_{K}^{*} (v^{*}) \\ + \frac{1}{2 ε} {∥\frac{\partial G_{K}^{*} (v^{*})}{\partial v^{*}} - u∥}_{2}^{2} + \frac{1}{2 ε} {∥\frac{\partial F_{K}^{*} (v^{*})}{\partial v^{*}} - u∥}_{2}^{2} \\ + \frac{1}{2 ε} {∥\frac{\partial G_{K}^{*} (v^{*})}{\partial v^{*}} - \frac{\partial F_{K}^{*} (v^{*})}{\partial v^{*}}∥}_{2}^{2}, \end{matrix}

(20)

we have

\begin{matrix} J (u_{0}) & = & inf_{u \in V} J (u) \\ = & inf_{(u, v^{*}) \in V \times V^{*}} J_{1}^{*} (u, v^{*}) \\ = & J_{1}^{*} (u_{0}, v_{0}^{*}) . \end{matrix}

(21)

Proof.

Observe that from the hypotheses and the results and statements of the last subsection

J (u_{0}) = inf_{u \in V} J (u) = inf_{v^{*} \in Y^{*}} J_{K}^{*} (v^{*}) = J_{K}^{*} (v_{0}^{*}),

where

J_{K}^{*} (v^{*}) = F_{K}^{*} (v^{*}) - G_{K}^{*} (v^{*}), \forall v^{*} \in V^{*} .

Moreover we have

J_{1}^{*} (u, v^{*}) \geq J_{K}^{*} (v^{*}), \forall u \in V, v^{*} \in V^{*} .

Also from hypotheses and the last subsection results,

u_{0} = \frac{\partial F_{K}^{*} (v_{0}^{*})}{\partial v^{*}} = \frac{\partial G_{K}^{*} (v_{0}^{*})}{\partial v^{*}},

so that clearly we have

J_{1}^{*} (u_{0}, v_{0}^{*}) = J_{K}^{*} (v_{0}^{*}) .

From these last results, we may infer that

\begin{matrix} J (u_{0}) & = & inf_{u \in V} J (u) \\ = & inf_{v^{*} \in V^{*}} J_{K}^{*} (v^{*}) \\ = & J_{K}^{*} (v_{0}^{*}) \\ = & inf_{(u, v^{*}) \in V \times V^{*}} J_{1}^{*} (u, v^{*}) \\ = & J_{1}^{*} (u_{0}, v_{0}^{*}) . \end{matrix}

(22)

The proof is complete.

□

Remark 2.

At this point we highlight that

J_{1}^{*}

has a large region of convexity around the optimal point

(u_{0}, v_{0}^{*})

, for

K > 0

sufficiently large and corresponding

ε > 0

sufficiently small.

Indeed, observe that for

v^{*} \in V^{*}

G_{K}^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{V} - G_{K} (u)} = {〈 \hat{u}, v^{*} 〉}_{V} - G_{K} (\hat{u})

where

\hat{u} \in V

is such that

v^{*} = \frac{\partial G_{K} (\hat{u})}{\partial u} = G^{'} (\hat{u}) + K \hat{u} .

Taking the variation in

v^{*}

in this last equation, we obtain

1 = G^{''} (u) \frac{\partial \hat{u}}{\partial v^{*}} + K \frac{\partial \hat{u}}{\partial v^{*}},

so that

\frac{\partial \hat{u}}{\partial v^{*}} = \frac{1}{G^{''} (u) + K} = O (\frac{1}{K}) .

From this we get

\begin{matrix} \frac{\partial^{2} \hat{u}}{\partial {(v^{*})}^{2}} & = & - \frac{1}{{(G^{''} (u) + K)}^{2}} G^{'''} (u) \frac{\partial \hat{u}}{\partial v^{*}} \\ = & - \frac{1}{{(G^{''} (u) + K)}^{3}} G^{'''} (u) \\ = & O (\frac{1}{K^{3}}) . \end{matrix}

(23)

On the other hand, from the implicit function theorem

\frac{\partial G_{K}^{*} (v^{*})}{\partial v^{*}} = u + [v^{*} - G_{K}^{'} (\hat{u})] \frac{\partial \hat{u}}{\partial v^{*}} = u,

so that

\frac{\partial^{2} G_{K}^{*} (v^{*})}{\partial {(v^{*})}^{2}} = \frac{\partial \hat{u}}{\partial v^{*}} = O (\frac{1}{K})

and

\frac{\partial^{3} G_{K}^{*} (v^{*})}{\partial {(v^{*})}^{3}} = \frac{\partial^{2} \hat{u}}{\partial {(v^{*})}^{2}} = O (\frac{1}{K^{3}}) .

Similarly, we may obtain

\frac{\partial^{2} F_{K}^{*} (v^{*})}{\partial {(v^{*})}^{2}} = O (\frac{1}{K})

and

\frac{\partial^{3} F_{K}^{*} (v^{*})}{\partial {(v^{*})}^{3}} = O (\frac{1}{K^{3}}) .

Denoting

A = \frac{\partial^{2} F_{K}^{*} (v_{0}^{*})}{\partial {(v^{*})}^{2}}

and

B = \frac{\partial^{2} G_{K}^{*} (v_{0}^{*})}{\partial {(v^{*})}^{2}},

we have

\frac{\partial^{2} J_{1}^{*} (u_{0}, v_{0}^{*})}{\partial {(v^{*})}^{2}} = A - B + \frac{1}{ε} (2 A^{2} + 2 B^{2} - 2 A B),

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

and

\frac{\partial^{2} J_{1}^{*} (u_{0}, v_{0}^{*})}{\partial (v^{*}) \partial u} = - \frac{1}{ε} (A + B) .

From this we get

\begin{matrix} det (δ^{2} J^{*} (v_{0}^{*}, u_{0})) & = & \frac{\partial^{2} J_{1}^{*} (u_{0}, v_{0}^{*})}{\partial {(v^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (u_{0}, v_{0}^{*})}{\partial u^{2}} - {[\frac{\partial^{2} J_{1}^{*} (u_{0}, v_{0}^{*})}{\partial (v^{*}) \partial u}]}^{2} \\ = & 2 \frac{A - B}{ε} + 2 \frac{{(A - B)}^{2}}{ε^{2}} \\ = & O (\frac{1}{ε^{2}}) \\ ≫ & 0 \end{matrix}

(24)

about the optimal point

(u_{0}, v_{0}^{*}) .

5. A convex dual variational formulation

In this section, again for

Ω \subset R^{3}

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

\partial Ω

γ > 0,

α > 0,

β > 0

and

f \in L^{2} (Ω)

, we denote

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

F_{2} : V \to R

and

G : V \times Y \to R

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

(25)

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

and

G (u, v) = \frac{α}{2} \int_{Ω} {(u^{2} - β + v)}^{2} d x + \frac{K}{2} \int_{Ω} u^{2} d x .

We define also

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

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}},

and

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

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

and

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

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

(26)

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

(27)

and

\begin{matrix} G^{*} (v_{1}^{*}, v_{0}^{*}) & = & sup_{(u, v) \in V \times Y} {{〈 u, v_{1}^{*} 〉}_{L^{2}} - {〈 v, v_{0}^{*} 〉}_{L^{2}} - G (u, v)} \\ = & \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}

(28)

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

where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2 and - γ \nabla^{2} + 2 v_{0}^{*} < - ε I_{d}},

for some small real parameter

ε > 0

and where

I_{d}

denotes a concerning identity operator.

Finally, we also define

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

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

Assuming

K_{2} ≫ K_{1} ≫ K ≫ max {1 / (ε^{2}), 1, γ, α}

by directly computing

δ^{2} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})

we may obtain that for such specified real constants,

J_{1}^{*}

in convex in

v_{2}^{*}

and it is concave in

(v_{1}^{*}, v_{0}^{*})

Y^{*} \times Y^{*} \times B^{*} .

Considering such statements and definitions, we may prove the following theorem.

Theorem 4.

Let

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

be such that

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

and

u_{0} \in V

be such that

u_{0} = \frac{{\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f}{K_{2} - K - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

so that

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in Y^{*}} \{sup_{(v_{1}^{*}, v_{0}^{*}) \in Y^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(29)

Proof.

Observe that

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

so that, since

J_{1}^{*}

is convex in

v_{2}^{*}

and concave in

(v_{1}^{*}, v_{0}^{*})

Y^{*} \times Y^{*} \times B^{*}

, we obtain

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

Now we are going to show that

δ J (u_{0}) = 0 .

From

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

we have

- u_{0} + \frac{{\hat{v}}_{2}^{*}}{K_{2}} = 0,

and thus

{\hat{v}}_{2}^{*} = K_{2} u_{0} .

From

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

we obtain

- u_{0} - \frac{{\hat{v}}_{1}^{*} - f}{2 {\hat{v}}_{0}^{*} + K} = 0,

and thus

{\hat{v}}_{1}^{*} = - 2 {\hat{v}}_{0}^{*} u_{0} - K u_{0} + f .

Finally, denoting

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

from

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

we have

- 2 D u_{0} + u_{0}^{2} - \frac{{\hat{v}}_{0}^{*}}{α} - β = 0,

so that

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

(30)

Observe now that

{\hat{v}}_{1}^{*} + {\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f = (K_{2} - K - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}) u_{0}

so that

\begin{matrix} K_{2} u_{0} - 2 {\hat{v}}_{0} u_{0} - K u_{0} + f \\ = & K_{2} u_{0} - K u_{0} - γ \nabla^{2} u_{0} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) (- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f) . \end{matrix}

(31)

The solution for this last system of equations (30) and (31) is obtained through the relations

{\hat{v}}_{0}^{*} = α (u_{0}^{2} - β)

and

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

so that

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

and

δ \{J (u_{0}) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f)}^{2} d x\} = 0,

and hence, from the concerning convexity in u on V,

J (u_{0}) = min_{u \in V} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} .

Moreover, from the Legendre transform properties

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

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

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

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) & = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - G^{*} ({\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) + G (u_{0}, 0) \\ = & J (u_{0}) . \end{matrix}

(32)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in Y^{*}} \{sup_{(v_{1}^{*}, v_{0}^{*}) \in Y^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{1}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{1}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(33)

The proof is complete.

□

Remark 3.

We could have also defined

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2 and - γ \nabla^{2} + 2 v_{0}^{*} > ε I_{d}},

for some small real parameter

ε > 0

. In this case,

- γ \nabla^{2} + 2 v_{0}^{*}

is positive definite, whereas in the previous case,

- γ \nabla^{2} + 2 v_{0}^{*}

is negative definite.

6. Another convex dual variational formulation

In this section, again for

Ω \subset R^{3}

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

\partial Ω

γ > 0,

α > 0,

β > 0

and

f \in L^{2} (Ω)

, we denote

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

F_{2} : V \to R

and

G : Y \to R

\begin{matrix} F_{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 + \frac{K_{2}}{2} \int_{Ω} u^{2} d x, \end{matrix}

(34)

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

and

G (u^{2}) = \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x .

We define also

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

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}},

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

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

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

and

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

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

and

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

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

(35)

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

(36)

and

\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}

(37)

At this point we define

B_{1}^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2},

B_{2}^{*} = {v_{0}^{*} \in Y^{*} : - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} > 0},

B_{3}^{*} = {v_{0}^{*} \in Y^{*} : - 1 / α + 4 K_{1} [u {(v_{2}^{*}, v_{0}^{*})}^{2}] + 100 / K_{2} \leq 0, \forall v_{2}^{*} \in E_{1}^{*}},

where

u (v_{2}^{*}, v_{0}^{*}) = \frac{φ_{1}}{φ},

φ_{1} = (v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)

and

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

Finally, we also define

E_{1}^{*} = {v_{2}^{*} \in Y^{*} : ∥ v_{2}^{*} ∥_{\infty} \leq (5 / 4) K_{2}} .

E_{2}^{*} = {v_{2}^{*} \in Y^{*} : f v_{2}^{*} > 0, a . e . in Ω},

E^{*} = E_{1}^{*} \cap E_{2}^{*},

B^{*} = B_{1}^{*} \cap B_{3}^{*},

and

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

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

Moreover, assume

K_{2} ≫ K_{1} ≫ K ≫ K_{3} ≫ max {1, γ, α} .

By directly computing

δ^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})

we may obtain that for such specified real constants,

J_{1}^{*}

is concave in

v_{0}^{*}

E^{*} \times B^{*} .

Indeed, recalling that

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

φ_{1} = (v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f),

and

u = \frac{φ_{1}}{φ},

we obtain

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})}{\partial {(v_{2}^{*})}^{2}} = 1 / K_{2} - 1 / φ > 0,

E^{*} \times B_{3}^{*}

and

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

E^{*} \times B^{*}

Considering such statements and definitions, we may prove the following theorem.

Theorem 5.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \in E^{*} \times (B^{*} \cap B_{2}^{*})

be such that

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

and

u_{0} \in V_{1}

be such that

u_{0} = \frac{{\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f}{K_{2} + 2 {\hat{v}}_{0}^{*} - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

so that

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in E^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(38)

Proof.

Observe that

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

so that, since

J_{1}^{*}

concave in

v_{0}^{*}

E^{*} \times B^{*}

v_{0}^{*} \in B_{2}^{*}

and

J_{1}^{*}

is quadratic in

v_{2}^{*}

, we get

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

Consequently, from this and the Min-Max Theorem, we obtain

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

Now we are going to show that

δ J (u_{0}) = 0 .

From

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

we have

- u_{0} + \frac{{\hat{v}}_{2}^{*}}{K_{2}} = 0,

and thus

{\hat{v}}_{2}^{*} = K_{2} u_{0} .

Finally, denoting

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

from

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

we have

- 2 D u_{0} + u_{0}^{2} - \frac{{\hat{v}}_{0}^{*}}{α} - β = 0,

so that

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

(39)

Observe now that

{\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f = (K_{2} - γ \nabla^{2} + 2 {\hat{v}}_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}) u_{0}

so that

\begin{matrix} K_{2} u_{0} - 2 {\hat{v}}_{0} u_{0} - K u_{0} + f \\ = & K_{2} u_{0} - K u_{0} - γ \nabla^{2} u_{0} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) (- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f) . \end{matrix}

(40)

The solution for this last equation is obtained through the relation

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

so that from this and (39), we get

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

Thus,

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

and

δ \{J (u_{0}) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f)}^{2} d x\} = 0,

and hence, from the concerning convexity in u on V,

J (u_{0}) = min_{u \in V} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} .

Moreover, from the Legendre transform properties

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

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

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

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) & = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - G^{*} ({\hat{v}}_{0}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) - {〈 u_{0}^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} + G (u_{0}^{2}) \\ = & J (u_{0}) . \end{matrix}

(41)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} \{J (u) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 {\hat{v}}_{0}^{*} u - f)}^{2} d x\} \\ = & inf_{v_{2}^{*} \in E^{*}} \{sup_{v_{0}^{*} \in B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(42)

The proof is complete.

□

7. A third duality principle and related convex dual variational formulation

In this section, we assume a finite dimensional version for the model in question, in a finite differences or finite elements context, although the concerning spaces and operators have not been relabeled.

Again, for

Ω \subset R^{3}

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

\partial Ω

γ < 0,

α < 0,

β > 0

K_{2} < 0

and

f \in L^{2} (Ω)

, we denote

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

F_{2} : V \to R

and

G : Y \to R

\begin{matrix} F_{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 + \frac{K_{2}}{2} \int_{Ω} u^{2} d x, \end{matrix}

(43)

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

and

G (u^{2}) = \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x .

We define also

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

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}},

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

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

V_{1} = A^{-} \cap V_{1},

and

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

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

and

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

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

(44)

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

(45)

and

\begin{matrix} G^{*} (v_{0}^{*}) & = & inf_{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}

(46)

At this point we define

B_{1}^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 2},

B_{2}^{*} = {v_{0}^{*} \in Y^{*} : - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} > 0},

B_{3}^{*} = {v_{0}^{*} \in Y^{*} : - 1 / α + 4 K_{1} [u {(v_{2}^{*}, v_{0}^{*})}^{2}] + 100 / | K_{2} | > 0, \forall v_{2}^{*} \in E_{1}^{*}},

where

u (v_{2}^{*}, v_{0}^{*}) = \frac{φ_{1}}{φ},

φ_{1} = (v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f)

and

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

By direct computation we may obtain

\frac{\partial^{2} [u {(v_{2}^{*}, v_{0}^{*})}^{2}]}{\partial {(v_{0}^{*})}^{2}} \geq 0, \forall v_{0}^{*} \in B^{*}, \forall v_{2}^{*} \in E_{1}^{*}

so that

B_{3}^{*}

is convex.

Finally, we also define

B_{4}^{*} = {v_{0}^{*} \in Y^{*} : - [- γ \nabla^{2} + 2 v_{0}^{*}] \leq - ε I_{d}},

E_{1}^{*} = {v_{2}^{*} \in Y^{*} : ∥ v_{2}^{*} ∥_{\infty} \leq (5 / 4) | K_{2} |},

E_{2}^{*} = {v_{2}^{*} \in Y^{*} : f v_{2}^{*} > 0, a . e . in Ω},

E^{*} = E_{1}^{*} \cap E_{2}^{*},

And

J_{1}^{*}

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

We assume

K_{2} < 0,

and

| K_{2} | ≫ K_{1} ≫ K ≫ K_{3} ≫ max {| α |, | γ |, β, 1 / ε^{2}} .

Recalling that

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

φ_{1} = (v_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 v_{0}^{*}) f),

and

u = \frac{φ_{1}}{φ},

we obtain

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})}{\partial {(v_{2}^{*})}^{2}} = 1 / K_{2} - 1 / φ > 0,

E^{*} \times B_{3}^{*}

and

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

Also,

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})}{\partial {(v_{2}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} - \frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{0}^{*})}{\partial v_{2}^{*} \partial v_{0}^{*}} \\ = & O (\frac{K_{1}^{2} A_{3} + K_{1} A_{4}}{- α K_{2} φ}), \end{matrix}

(47)

where

A_{3} = 8 α (f^{2} + 4 f (- γ \nabla^{2} + 2 v_{0}^{*}) u + 3 {[(- γ \nabla^{2} + 2 v_{0}^{*}) u]}^{2}

and

A_{4} = {(- γ \nabla^{2} + 2 v_{0}^{*})}^{2} - 12 α [(- γ \nabla^{2} + 2 v_{0}^{*}) u] u .

Observe that at a critical point

A_{3} = 0

and

A_{4} > 0

so that, for the dual formulation, we set the restrictions

A_{3} \geq 0

and

A_{4} \geq ε I_{d} .

Thus, we define

B_{5}^{*} = {v_{0}^{*} \in Y^{*} : A_{3} \geq 0 and A_{4} \geq ε I_{d}},

and

B^{*} = B_{1}^{*} \cap B_{2}^{*} \cap B_{3}^{*} \cap B_{4}^{*} \cap B_{5}^{*} .

Observe also that

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

B^{*} \times E^{*}

, so that

J_{1}^{*}

is convex on

E^{*} \times B^{*}

Considering such statements and definitions, we may prove the following theorem.

Theorem 6.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) \in E^{*} \times B^{*}

be such that

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

and

u_{0} \in V_{1}

be such that

u_{0} = \frac{{\hat{v}}_{2}^{*} - K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f}{K_{2} + 2 {\hat{v}}_{0}^{*} - γ \nabla^{2} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}} .

Under such hypotheses, we have

δ J (u_{0}) = 0,

so that

\begin{matrix} J (u_{0}) & = & sup_{u \in V_{1}} \{J (u) + \frac{K_{2}}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} \\ = & inf_{(v_{2}^{*}, v_{0}^{*}) \in E^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(48)

Proof.

Observe that

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

so that, since

J_{1}^{*}

convex on the convex set

E^{*} \times B^{*}

, we have that

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

Now we are going to show that

δ J (u_{0}) = 0 .

From

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

we have

- u_{0} + \frac{{\hat{v}}_{2}^{*}}{K_{2}} = 0,

and thus

{\hat{v}}_{2}^{*} = K_{2} u_{0} .

Finally, denoting

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

from

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

we have

- 2 D u_{0} + u_{0}^{2} - \frac{{\hat{v}}_{0}^{*}}{α} - β = 0,

so that

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

(49)

Observe now that

{\hat{v}}_{2}^{*} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) f = (K_{2} - γ \nabla^{2} + 2 {\hat{v}}_{0}^{*} + K_{1} {(- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*})}^{2}) u_{0}

so that

\begin{matrix} K_{2} u_{0} - 2 {\hat{v}}_{0} u_{0} - K u_{0} + f \\ = & K_{2} u_{0} - K u_{0} - γ \nabla^{2} u_{0} + K_{1} (- γ \nabla^{2} + 2 {\hat{v}}_{0}^{*}) (- γ \nabla^{2} u_{0} + 2 {\hat{v}}_{0}^{*} u_{0} - f) . \end{matrix}

(50)

The solution for this last equation is obtained through the relation

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

so that from this and (39), we get

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

Thus,

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

and

δ \{J (u_{0}) + \frac{K_{2}}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} = 0,

and hence, from the concerning cocavity in u on V,

J (u_{0}) = min_{u \in V_{1}} \{J (u) + \frac{K_{2}}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} .

Moreover, from the Legendre transform properties

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

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

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

so that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) & = & - F_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) + F_{2}^{*} ({\hat{v}}_{2}^{*}) - G^{*} ({\hat{v}}_{0}^{*}) \\ = & F_{1} (u_{0}, {\hat{v}}_{0}^{*}) - F_{2} (u_{0}) - {〈 u_{0}^{2}, {\hat{v}}_{0}^{*} 〉}_{L^{2}} + G (u_{0}^{2}) \\ = & J (u_{0}) . \end{matrix}

(51)

Finally, observe that

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) & \geq & F_{1} (u, {\hat{v}}_{0}^{*}) - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} ({\hat{v}}_{2}^{*}) - G^{*} ({\hat{v}}_{0}^{*}) \\ \geq & inf_{v_{0}^{*} \in Y^{*}} \{\frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x + \frac{K_{2}}{2} \int_{Ω} u^{2} d x \\ - {〈 u, {\hat{v}}_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} ({\hat{v}}_{2}^{*}) + {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} - \frac{α}{2} \int_{Ω} {(v_{0}^{*})}^{2} d x \\ - β \int_{Ω} v_{0}^{*} d x\} \\ = & \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}} + \frac{K_{2}}{2} \int_{Ω} u^{2} d x \\ - {〈 u, K_{2} u_{0} 〉}_{L^{2}} + \frac{K_{2}}{2} \int_{Ω} u_{0}^{2} d x \\ = & J (u) + \frac{K_{2}}{2} \int_{Ω} {(u - u_{0})}^{2} d x, \end{matrix}

(52)

\forall u \in V_{1}

where we recall that

K_{2} < 0

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & sup_{u \in V_{1}} \{J (u) + \frac{K_{2}}{2} \int_{Ω} {(u - u_{0})}^{2} d x\} \\ = & inf_{(v_{2}^{*}, v_{0}^{*}) \in E^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{0}^{*}) \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(53)

The proof is complete.

□

8. Closely related primal-dual variational formulations

Consider again the 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

α > 0

γ > 0

β > 0

f \in L^{2} (Ω)

and

V = W_{0}^{1, 2} (Ω) .

Observe that

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

(54)

Having obtained

J_{1}^{*} (u, v_{0}^{*})

, we propose the following exactly penalized primal-dual formulation

J_{2}^{*} (u, v_{0}^{*})

, where

J_{2}^{*} (u, v_{0}^{*}) = J_{1}^{*} (u, v_{0}^{*}) + \frac{K_{1}}{2} \int_{Ω} {(- γ \nabla^{2} u + 2 v_{0}^{*} u - f)}^{2} d x,

so that

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

(55)

In particular, if we set

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

and

K_{1} = 1 / (8 K_{3}^{2} α),

we may also define

J_{3} (u) = sup_{v_{0}^{*} \in B^{*}} J_{2}^{*} (u, v_{0}^{*}),

where

B^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K / 8},

for appropriate

K, K_{3} > 0 .

Here we highlight that

J_{2}^{*}

is concave in

v_{0}^{*}

B^{*}

(indeed it is concave on

Y^{*}

) and the parameter

K_{1} > 0

multiplying a positive definite quadratic functional in u improves the convexity conditions of

J_{3}

9. One more duality principle suitable for the primal formulation global optimization

In this section we establish one more duality principle and related convex dual formulation suitable for a global optimization of the primal variational formulation.

Let

Ω \subset R^{3}

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

\partial Ω .

For the primal formulation, we define

V = W_{0}^{1, 2} (Ω)

and 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}

(56)

Here we assume

f \in L^{2} (Ω)

, and define

Y = Y^{*} = L^{2} (Ω)

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

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

and

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

for an appropriate constant

K_{3} > 0

to be specified.

Define also the functionals

F_{1} : V \to R

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

and

G : Y \to R

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

\begin{matrix} F_{2} (u, v_{3}^{*}, v_{0}^{*}) & = & - \frac{γ}{2} \int_{Ω} \nabla u \cdot \nabla u d x - {〈 u^{2}, v_{0}^{*} 〉}_{L^{2}} + \frac{K_{2}}{2} \int_{Ω} {(\nabla^{2} u)}^{2} d x \\ - \frac{K_{1}}{2} \int_{Ω} {(- γ_{1} \nabla^{2} u + 2 v_{3}^{*} u - h_{1})}^{2} d x, \end{matrix}

(57)

and

G (u^{2}) = \frac{α}{2} \int_{Ω} {(u^{2} - β)}^{2} d x,

for appropriate positive constants

K_{1}, K_{2}, K_{3}

to be specified.

Moreover, define

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

and

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

and

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

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

(58)

and

\begin{matrix} F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) & = & sup_{u \in V} {{〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u, v_{3}^{*}, v_{0}^{*})} \\ = & \frac{1}{2} \int_{Ω} \frac{{(v_{2}^{*} + K_{1} (- γ_{1} \nabla^{2} + 2 v_{3}^{*}) h_{1})}^{2}}{K_{2} \nabla^{4} + γ \nabla^{2} - 2 v_{0}^{*} - K_{1} {(- γ_{1} \nabla^{2} + 2 v_{3}^{*})}^{2}} \\ - \frac{K_{1}}{2} \int_{Ω} h_{1}^{2} d x \end{matrix}

for appropriate

γ_{1} > 0

and

H_{1} \in L^{2} (Ω),

and

\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}

(59)

Furthermore, we define

D^{*} = {v_{2}^{*} \in Y^{*} : ∥ v_{2}^{*} ∥_{\infty} \leq (3 / 2) K_{2}},

B^{*} = \{v_{3}^{*} \in Y^{*} : {∥ v_{3}^{*} ∥}_{\infty} \leq K_{4}\},

for an appropriate constant

K_{4} > 0

to be specified.

Define also

C_{1}^{*} = {v_{0}^{*} \in Y^{*} : ∥ v_{0}^{*} ∥_{\infty} \leq K_{4}} .

and

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

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

Moreover, assuming

K_{2} ≫ K_{1} ≫ K_{4} ≫ max {1, K_{3}, α, β, γ, γ_{1} {, ∥ f ∥}_{\infty}, ∥ h_{1} ∥_{\infty}}

By directly computing

δ^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})

denoting

A = - 2 K_{1} h_{1},

B = 4 K_{1} (- γ_{1} \nabla^{2} + 2 v_{3}^{*}),

φ = - K_{2} \nabla^{4} - γ \nabla^{2} + 2 v_{0}^{*} + K_{1} {(- γ_{1} \nabla^{2} + 2 v_{3}^{*})}^{2}),

φ_{1} = v_{2}^{*} - K_{1} (- γ_{1} \nabla^{2} + 2 v_{3}^{*}) h_{1},

u = - \frac{φ_{1}}{φ},

we may obtain, considering that

φ < 0

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} = 4 K_{1} u^{2} - \frac{{(A - u B)}^{2}}{φ} > 0

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

Moreover,

\begin{matrix} \frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{2}^{*})}^{2}} \frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{3}^{*})}^{2}} - {(\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial v_{2}^{*} \partial v_{3}^{*}})}^{2} \\ = & O (\frac{K_{1}^{2} H_{1} + K_{1} H_{2}}{- K_{2} φ}), \end{matrix}

(60)

where

H_{1} = (8 (- h_{1}^{2} + 4 h_{1} (- γ_{1} \nabla^{2} + 2 v_{3}^{*}) u - 3 [{(- γ_{1} \nabla^{2} + 2 v_{3}^{*})}^{2} u] u),

and

H_{2} = [(- γ \nabla^{2} + 2 v_{0}^{*}) u] u .

At a critical point we have

H_{1} = 0

and

H_{2} = f u_{0} > 0, a . e in Ω .

With such results, we may define the restrictions

C_{2}^{*} = {v_{0}^{*} \in Y^{*} : H_{1} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \geq 0, \forall v_{2}^{*} \in D^{*}, v_{3}^{*} \in B^{*}} .

C_{3}^{*} = {v_{0}^{*} \in Y^{*} : H_{2} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \geq 0, \forall v_{2}^{*} \in D^{*}, v_{3}^{*} \in B^{*}} .

Here, we define

C^{*} = C_{1}^{*} \cap C_{2}^{*} \cap C_{3}^{*} .

On the other hand, clearly we have

\frac{\partial^{2} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} < 0

From such results, we may obtain that

J_{1}^{*}

in convex in

(v_{2}^{*}, v_{3}^{*})

and it is concave in

v_{0}^{*}

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

9.1. The main duality principle and a related convex dual formulation

Considering the statements and definitions presented in the previous section, we may prove the following theorem.

Theorem 7.

Let

({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*} {\hat{v}}_{0}^{*}) \in D^{*} \times B^{*} \times C^{*}

be such that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0

and

u_{0} \in V_{1}

be such that

u_{0} = \frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} .

Assume also

u_{0} \neq 0, a . e . in Ω .

Under such hypotheses, we have

δ J (u_{0}) = 0,

- γ_{1} \nabla^{2} u_{0} + 2 {\hat{v}}_{3}^{*} u_{0} - h_{1} = 0,

and

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} J (u) \\ = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(61)

Proof.

Observe that

δ J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = 0

so that, since

J_{1}^{*}

is convex in

(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*} \times C^{*}

and

\frac{\partial^{2} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*})}{\partial {(v_{0}^{*})}^{2}} > 0, \forall v_{0}^{*} \in C_{1}^{*},

we obtain

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, {\hat{v}}_{0}^{*}) = sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*}) .

Consequently, from this and the Saddle Point Theorem, we obtain

J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} .

Now we are going to show that

δ J (u_{0}) = 0 .

From

\frac{\partial J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = 0,

and

\frac{\partial F_{1}^{*} ({\hat{v}}_{2}^{*})}{\partial v_{2}^{*}} = u_{0}

we have

- \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} - u_{0} = 0

and

{\hat{v}}_{2}^{*} = K_{2} \nabla^{4} u_{0} - f .

Observe now that

F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, v_{0}^{*}) = sup_{u \in V} {- {〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u, v_{3}^{*}, v_{0}^{*})} .

Denoting

H (v_{2}^{*}, v_{3}^{*}, v_{)}^{*}, u) = {〈 u, v_{2}^{*} 〉}_{L^{2}} - F_{2} (u, v_{3}^{*}, v_{0}^{*}),

there exists

\hat{u} \in V

such that

\frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} = 0,

and

F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) = H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u}),

so that

\begin{matrix} \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} & = & \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial v_{2}^{*}} \\ + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{2}^{*}} \\ = & \hat{u} . \end{matrix}

(62)

Summarizing, we have got

u_{0} = \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{2}^{*}} = \hat{u} .

From such results and the Legendre tranform proprieties we get

v_{2}^{*} = \frac{\partial F_{1} (u_{0})}{\partial u}

and

v_{2}^{*} = \frac{\partial F_{2} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u} .

On the other hand, from the variation of

J_{1}^{*}

v_{3}^{*}

, we have

\begin{matrix} \frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{3}^{*}} \\ = & - K_{1} (- γ_{1} \nabla^{2} u_{0} + 2 {\hat{v}}_{3}^{*} u_{0} - h_{1}) 2 u_{0} + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{3}^{*}} \\ = & - K_{1} (- γ_{1} \nabla^{2} u_{0} + 2 {\hat{v}}_{3}^{*} u_{0} - h_{1}) 2 u_{0} \\ = & 0 . \end{matrix}

(63)

From such results, since

u_{0} \neq 0, a . e . in Ω,

we get

- γ_{1} \nabla^{2} u_{0} + 2 {\hat{v}}_{3}^{*} u_{0} - h_{1} = 0, a . e . in Ω .

Finally, from the variation of

J_{1}^{*}

v_{0}^{*}

we obtain

\frac{\partial F_{2}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial v_{0}^{*}} - \frac{\partial G^{*} (v_{0}^{*})}{\partial v_{0}^{*}} = 0,

so that

u_{0}^{2} + \frac{\partial H ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}, \hat{u})}{\partial u} \frac{\partial \hat{u}}{\partial v_{0}^{*}} - \frac{v_{0}^{*}}{α} - β = 0 .

Thus,

v_{0}^{*} = α (u_{0}^{2} - β) .

Consequently, from such last results, we have

\begin{matrix} 0 = {\hat{v}}_{2}^{*} - {\hat{v}}_{2}^{*} \\ = & \frac{\partial F_{1} (u_{0})}{\partial u} - \frac{\partial F_{2} (u_{0}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*})}{\partial u} \\ = & K_{2} \nabla^{4} u_{0} - f - K_{2} \nabla^{4} u_{0} - γ \nabla^{2} u_{0} + 2 v_{0}^{*} u_{0} \\ = & - γ \nabla^{2} u_{0} + 2 α (u_{0}^{2} - β) u_{0} - f \\ = & δ J (u_{0}) . \end{matrix}

(64)

Summarizing,

δ J (u_{0}) = 0 .

Furthermore, also from such last results and the Legendre transform properties, we have

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

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

G^{*} ({\hat{v}}_{0}^{*}) = {〈 u_{0}^{2}, v_{0}^{*} 〉}_{L^{2}} - G (u_{0}^{2}),

so that

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

(65)

Finally, observe that

J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \leq F_{1} (u) - {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*}),

\forall u \in V_{1}, v_{2}^{*} \in D^{*}, v_{3}^{*} \in B^{*}, v_{0}^{*} \in C^{*} .

Therefore,

sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) \leq sup_{v_{0}^{*} \in C_{1}^{*}} {- {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{1} (u) + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*})},

so that

\begin{matrix} inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ \leq & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C_{1}^{*}} {- {〈 u, v_{2}^{*} 〉}_{L^{2}} + F_{1} (u) + F_{2}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*}) - G^{*} (v_{0}^{*})}\} \\ = & J (u), \forall u \in V_{1} . \end{matrix}

(66)

Summarizing, we have got

\begin{matrix} J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) & = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ \leq & inf_{u \in V_{1}} J (u) . \end{matrix}

(67)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & inf_{u \in V_{1}} J (u) \\ = & inf_{(v_{2}^{*}, v_{3}^{*}) \in D^{*} \times B^{*}} \{sup_{v_{0}^{*} \in C^{*}} J_{1}^{*} (v_{2}^{*}, v_{3}^{*}, v_{0}^{*})\} \\ = & J_{1}^{*} ({\hat{v}}_{2}^{*}, {\hat{v}}_{3}^{*}, {\hat{v}}_{0}^{*}) . \end{matrix}

(68)

The proof is complete.

□

10. Another duality principle for a related model in phase transition

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

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

where

Q_{F} (u)

refers to a quasi-convex regularization of

F .

We 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}

(69)

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}

(70)

Observe that if

K > 0, K_{1} > 0

is large enough, both

F_{3}

and G are convex.

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.

Set $K \approx 150$ and $K_{1} = K / 20$ and $0 < ε ≪ 1 .$
Choose $(u_{1}, ϕ_{1}) \in V \times V_{0},$ such that $∥ u_{1} ∥_{1, \infty} ≪ K / 4$ and $∥ ϕ_{1} ∥_{1, \infty} ≪ K / 4 .$
Set $n = 1 .$
Calculate $(v_{n}^{*}, {(v_{0}^{*})}_{n})$ solution of the system of equations:

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

and

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

that is

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

and

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

so that

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

and

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

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

and

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

that is

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

and

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

For the case in which

f (x) = 0

, we have obtained numerical results for

K = 1500

and

K_{1} = K / 20

. 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 for

K = 100

and

K_{1} = K / 20

. For such a concerning solution

u_{0}

obtained, please see Figure 2.

Remark 4.

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.

11. A related numerical computation through the generalized method of lines

We start by recalling that the generalized method of lines was originally introduced in the book entitled "Topics on Functional Analysis, Calculus of Variations and Duality" [7], published in 2011.

Indeed, the present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [5,7,8,9]. About the Sobolev spaces involved we would mention [1]. Concerning the applications, related models in physics are addressed in [4,11].

We also emphasize that, in such a method, the domain of the partial differential equation in question is discretized in lines (or more generally, in curves) and the concerning solution is written on these lines as functions of boundary conditions and the domain boundary shape.

In fact, in its previous format, this method consists of an application of a kind of a partial finite differences procedure combined with the Banach fixed point theorem to obtain the relation between two adjacent lines (or curves).

In the present article, we propose an improvement concerning the way we truncate the series solution obtained through an application of the Banach fixed point theorem to find the relation between two adjacent lines. The results obtained are very good even as a typical parameter

ε > 0

is very small.

In the next lines and sections we develop in details such a numerical procedure.

11.1. About a concerning improvement for the generalized method of lines

Let

Ω \subset R^{2}

where

Ω = {(r, θ) \in R^{2} : 1 \leq r \leq 2, 0 \leq θ \leq 2 π} .

Consider the problem of solving the partial differential equation

\{\begin{matrix} - ε (\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial θ^{2}}) + α u^{3} - β u = f, & in Ω, \\ u = u_{0} (θ), & on \partial Ω_{1}, \\ u = u_{f} (θ), & on \partial Ω_{2} . \end{matrix}

(71)

Here

Ω = {(r, θ) \in R^{2} : 1 \leq r \leq 2, 0 \leq θ \leq 2 π},

\partial Ω_{1} = {(1, θ) \in R^{2} : 0 \leq θ \leq 2 π},

\partial Ω_{2} = {(2, θ) \in R^{2} : 0 \leq θ \leq 2 π},

ε > 0, α > 0, β > 0

, and

f \equiv 1, on Ω .

In a partial finite differences scheme, such a system stands for

- ε (\frac{u_{n + 1} - 2 u_{n} + u_{n - 1}}{d^{2}} + \frac{1}{t_{n}} \frac{u_{n} - u_{n - 1}}{d} + \frac{1}{t_{n}^{2}} \frac{\partial^{2} u_{n}}{\partial θ^{2}}) + α u_{n}^{3} - β u_{n} = f_{n},

\forall n \in {1, \dots, N - 1},

with the boundary conditions

u_{0} = 0,

and

u_{N} = 0 .

Here N is the number of lines and

d = 1 / N .

In particular, for

n = 1

we have

- ε (\frac{u_{2} - 2 u_{1} + u_{0}}{d^{2}} + \frac{1}{t_{1}} \frac{(u_{1} - u_{0})}{d} + \frac{1}{t_{1}^{2}} \frac{\partial^{2} u_{1}}{\partial θ^{2}}) + α u_{1}^{3} - β u_{1} = f_{1},

so that

u_{1} = (u_{2} + u_{1} + u_{0} + \frac{1}{t_{1}} (u_{1} - u_{0}) d + \frac{1}{t_{1}^{2}} \frac{\partial^{2} u_{1}}{\partial θ^{2}} d^{2} + (- α u_{1}^{3} + β u_{1} - f_{1}) \frac{d^{2}}{ε}) / 3.0,

We solve this last equation through the Banach fixed point theorem, obtaining

u_{1}

as a function of

u_{2} .

Indeed, we may set

u_{1}^{0} = u_{2}

and

\begin{matrix} u_{1}^{k + 1} & = & (u_{2} + u_{1}^{k} + u_{0} + \frac{1}{t_{1}} (u_{1}^{k} - u_{0}) d + \frac{1}{t_{1}^{2}} \frac{\partial^{2} u_{1}^{k}}{\partial θ^{2}} d^{2} \\ + (- α {(u_{1}^{k})}^{3} + β u_{1}^{k} - f_{1}) \frac{d^{2}}{ε}) / 3.0, \end{matrix}

(72)

\forall k \in N .

Thus, we may obtain

u_{1} = lim_{k \to \infty} u_{1}^{k} \equiv H_{1} (u_{2}, u_{0}) .

Similarly, for

n = 2

, we have

\begin{matrix} u_{2} & = & (u_{3} + u_{2} + H_{1} (u_{2}, u_{0}) + \frac{1}{t_{1}} (u_{2} - H_{1} (u_{2}, u_{0})) d + \frac{1}{t_{1}^{2}} \frac{\partial^{2} u_{2}}{\partial θ^{2}} d^{2} \\ + (- α u_{2}^{3} + β u_{2} - f_{2}) \frac{d^{2}}{ε}) / 3.0, \end{matrix}

(73)

We solve this last equation through the Banach fixed point theorem, obtaining

u_{2}

as a function of

u_{3}

and

u_{0} .

Indeed, we may set

u_{2}^{0} = u_{3}

and

\begin{matrix} u_{2}^{k + 1} & = & (u_{3} + u_{2}^{k} + H_{1} (u_{2}^{k}, u_{0}) + \frac{1}{t_{2}} (u_{2}^{k} - H_{1} (u_{2}^{k}, u_{0})) d + \frac{1}{t_{2}^{2}} \frac{\partial^{2} u_{2}^{k}}{\partial θ^{2}} d^{2} \\ + (- α {(u_{2}^{k})}^{3} + β u_{2}^{k} - f_{2}) \frac{d^{2}}{ε}) / 3.0, \end{matrix}

(74)

\forall k \in N .

Thus, we may obtain

u_{2} = lim_{k \to \infty} u_{2}^{k} \equiv H_{2} (u_{3}, u_{0}) .

Now reasoning inductively, having

u_{n - 1} = H_{n - 1} (u_{n}, u_{0}),

we may get

\begin{matrix} u_{n} & = & (u_{n + 1} + u_{n} + H_{n - 1} (u_{n}, u_{0}) + \frac{1}{t_{n}} (u_{n} - H_{n - 1} (u_{n}, u_{0})) d + \frac{1}{t_{n}^{2}} \frac{\partial^{2} u_{n}}{\partial θ^{2}} d^{2} \\ + (- α u_{n}^{3} + β u_{n} - f_{n}) \frac{d^{2}}{ε}) / 3.0, \end{matrix}

(75)

We solve this last equation through the Banach fixed point theorem, obtaining

u_{n}

as a function of

u_{n + 1}

and

u_{0} .

Indeed, we may set

u_{n}^{0} = u_{n + 1}

and

\begin{matrix} u_{n}^{k + 1} & = & (u_{n + 1} + u_{n}^{k} + H_{n - 1} (u_{n}^{k}, u_{0}) + \frac{1}{t_{n}} (u_{n}^{k} - H_{n - 1} (u_{n}^{k}, u_{0})) d + \frac{1}{t_{n}^{2}} \frac{\partial^{2} u_{n}^{k}}{\partial θ^{2}} d^{2} \\ + (- α {(u_{n}^{k})}^{3} + β u_{n}^{k} - f_{n}) \frac{d^{2}}{ε}) / 3.0, \end{matrix}

(76)

\forall k \in N .

Thus, we may obtain

u_{n} = lim_{k \to \infty} u_{n}^{k} \equiv H_{n} (u_{n + 1}, u_{0}) .

We have obtained

u_{n} = H_{n} (u_{n + 1}, u_{0})

\forall n \in {1, \dots, N - 1} .

In particular,

u_{N} = u_{f} (θ),

so that we may obtain

u_{N - 1} = H_{N - 1} (u_{N}, u_{0}) = H_{N - 1} (0) \equiv F_{N - 1} (u_{N}, u_{0}) = F_{N - 1} (u_{f} (θ), u_{0} (θ)) .

Similarly,

u_{N - 2} = H_{N - 2} (u_{N - 1}, u_{0}) = H_{N - 2} (H_{N - 1} (u_{N}, u_{0})) = F_{N - 2} (u_{N}, u_{0}) = F_{N - 1} (u_{f} (θ), u_{0} (θ)),

an so on, up to obtaining

u_{1} = H_{1} (u_{2}) \equiv F_{1} (u_{N}, u_{0}) = F_{1} (u_{f} (θ), u_{0} (θ)) .

The problem is then approximately solved.

11.2. Software in Mathematica for solving such an equation

We recall that the equation to be solved is a Ginzburg-Landau type one, where

\{\begin{matrix} - ε (\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} u}{\partial θ^{2}}) + α u^{3} - β u = f, & in Ω, \\ u = 0, & on \partial Ω_{1}, \\ u = u_{f} (θ), & on \partial Ω_{2} . \end{matrix}

(77)

Here

Ω = {(r, θ) \in R^{2} : 1 \leq r \leq 2, 0 \leq θ \leq 2 π},

\partial Ω_{1} = {(1, θ) \in R^{2} : 0 \leq θ \leq 2 π},

\partial Ω_{2} = {(2, θ) \in R^{2} : 0 \leq θ \leq 2 π},

ε > 0, α > 0, β > 0

, and

f \equiv 1, on Ω .

In a partial finite differences scheme, such a system stands for

- ε (\frac{u_{n + 1} - 2 u_{n} + u_{n - 1}}{d^{2}} + \frac{1}{t_{n}} \frac{u_{n} - u_{n - 1}}{d} + \frac{1}{t_{n}^{2}} \frac{\partial^{2} u_{n}}{\partial θ^{2}}) + α u_{n}^{3} - β u_{n} = f_{n},

\forall n \in {1, \dots, N - 1},

with the boundary conditions

u_{0} = 0,

and

u_{N} = u_{f} [x] .

Here N is the number of lines and

d = 1 / N .

At this point we present the concerning software for an approximate solution.

Such a software is for

N = 10

(10 lines) and

u_{0} [x] = 0 .

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

$m_{8} = 10$ ; $(N = 10 l i n e s)$
$d = 1 / m 8$ ;
$e_{1} = 0.1$ ; ( $ε = 0.1)$
$A = 1.0$ ;
$B = 1.0$ ;
$F o r [i = 1, i < m 8, i + +, f [i] = 1.0];$ $(f \equiv 1, on Ω)$
$a = 0.0$ ;
$F o r [i = 1, i < m 8, i + +,$

$C l e a r [b, u];$

$t [i] = 1 + i * d;$

$b [x_{-}] = u [i + 1] [x];$
$F o r [k = 1, k < 30, k + +,$ $(we have fixed the number of iterations)$

$z = (u [i + 1] [x] + b [x] + a + \frac{1}{t [i]} (b [x] - a) * d + \frac{1}{t {[i]}^{2}} D [b [x], {x, 2}] * d^{2} + (- A * b {[x]}^{3} + B * u [x] + f [i]) * \frac{d^{2}}{e_{1}}) / 3.0;$

$z = S e r i e s [z, {u [i + 1] [x], 0, 3}, {u {[i + 1]}^{'} [x], 0, 1}, {u {[i + 1]}^{''} [x], 0, 1}, {u {[i + 1]}^{'''} [x], 0, 0}, {u {[i + 1]}^{''''} [x], 0, 0}];$

$z = N o r m a l [z],$

$z = E x p a n d [z];$

$b [x_{-}] = z];$
$a_{1} [i] = z;$
$C l e a r [b];$
$u [i + 1] [x_{-}] = b [x]$ ;
$a = a_{1} [i]$ ];
$b [x_{-}] = u_{f} [x];$
$F o r [i = 1, i < m 8, i + +,$

$A_{1} = a_{1} [m 8 - i];$

$A_{1} = S e r i e s [A_{1}, {u_{f} [x], 0, 3}, {u_{f}^{'} [x], 0, 1}, {u_{f}^{''} [x], 0, 1}, {u_{f}^{'''} [x], 0, 0}, {u_{f}^{''''} [x], 0, 0}];$

$A_{1} = N o r m a l [A_{1}];$

$A_{1} = E x p a n d [A_{1}];$

$u [m 8 - i] [x_{-}] = A_{1};$

$b [x_{-}] = A_{1}];$

$P r i n t [u [m 8 / 2] [x]];$

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

The numerical expressions for the solutions of the concerning

N = 10

lines are given by

\begin{matrix} u [1] [x] & = & 0.47352 + 0.00691 u_{f} [x] - 0.00459 u_{f} {[x]}^{2} + 0.00265 u_{f} {[x]}^{3} + 0.00039 (u_{f}^{''}) [x] \\ - 0.00058 u_{f} [x] (u_{f}^{''}) [x] + 0.00050 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.000181213 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(78)

\begin{matrix} u [2] [x] & = & 0.76763 + 0.01301 u_{f} [x] - 0.00863 u_{f} {[x]}^{2} + 0.00497 u_{f} {[x]}^{3} + 0.00068 (u_{f}^{''}) [x] \\ - 0.00103 u_{f} [x] (u_{f}^{''}) [x] + 0.00088 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00034 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(79)

\begin{matrix} u [3] [x] & = & 0.91329 + 0.02034 u_{f} [x] - 0.01342 u_{f} {[x]}^{2} + 0.00768 u_{f} {[x]}^{3} + 0.00095 (u_{f}^{''}) [x] \\ - 0.00144 u_{f} [x] (u_{f}^{''}) [x] + 0.00122 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00051 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(80)

\begin{matrix} u [4] [x] & = & 0.97125 + 0.03623 u_{f} [x] - 0.02328 u_{f} {[x]}^{2} + 0.01289 u_{f} {[x]}^{3} + 0.00147331 (u_{f}^{''}) [x] \\ - 0.00223 u_{f} [x] (u_{f}^{''}) [x] + 0.00182 u f {[x]}^{2} (u_{f}^{''}) [x] - 0.00074 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(81)

\begin{matrix} u [5] [x] & = & 1.01736 + 0.09242 u_{f} [x] - 0.05110 u_{f} {[x]}^{2} + 0.02387 u_{f} {[x]}^{3} + 0.00211 (u_{f}^{''}) [x] \\ - 0.00378 u_{f} [x] (u_{f}^{''}) [x] + 0.00292 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00132 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(82)

\begin{matrix} u [6] [x] & = & 1.02549 + 0.21039 u_{f} [x] - 0.09374 u_{f} {[x]}^{2} + 0.03422 u_{f} {[x]}^{3} + 0.00147 (u_{f}^{''}) [x] \\ - 0.00634 u_{f} [x] (u_{f}^{''}) [x] + 0.00467 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00200 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(83)

\begin{matrix} u [7] [x] & = & 0.93854 + 0.36459 u_{f} [x] - 0.14232 u_{f} {[x]}^{2} + 0.04058 u_{f} {[x]}^{3} + 0.00259 (u_{f}^{''}) [x] \\ - 0.00747373 u_{f} [x] (u_{f}^{''}) [x] + 0.0047969 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00194 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(84)

\begin{matrix} u [8] [x] & = & 0.74649 + 0.57201 u_{f} [x] - 0.17293 u_{f} {[x]}^{2} + 0.02791 u_{f} {[x]}^{3} + 0.00353 (u_{f}^{''}) [x] \\ - 0.00658 u_{f} [x] (u_{f}^{''}) [x] + 0.00407 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00172 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(85)

\begin{matrix} u [9] [x] & = & 0.43257 + 0.81004 u_{f} [x] - 0.13080 u_{f} {[x]}^{2} + 0.00042 u_{f} {[x]}^{3} + 0.00294 (u_{f}^{''}) [x] \\ - 0.00398 u_{f} [x] (u_{f}^{''}) [x] + 0.00222 u_{f} {[x]}^{2} (u_{f}^{''}) [x] - 0.00066 u_{f} {[x]}^{3} (u_{f}^{''}) [x] \end{matrix}

(86)

11.3. Some plots concerning the numerical results

In this section we present the lines

2, 4, 6, 8

related to results obtained in the last section.

Indeed, we present such mentioned lines, in a first step, for the previous results obtained through the generalized of lines and, in a second step, through a numerical method which is combination of the Newton's one and the generalized method of lines. In a third step, we also present the graphs by considering the expression of the lines as those also obtained through the generalized method of lines, up to the numerical coefficients for each function term, which are obtained by the numerical optimization of the functional J, below specified. We consider the case in which

u_{0} (x) = 0

and

u_{f} (x) = sin (x)

For the procedure mentioned above as the third step, recalling that

N = 10

lines, considering that

u_{f}^{''} (x) = - u_{f} (x)

, we may approximately assume the following general line expressions:

u_{n} (x) = a (1, n) + a (2, n) u_{f} (x) + a (3, n) u_{f} {(x)}^{3} + a (4, n) u_{f} {(x)}^{3}, \forall n \in {1, \dots N - 1} .

Defining

W_{n} = - e_{1} \frac{(u_{n + 1} (x) - 2 u_{n} (x) + u_{n - 1} (x))}{d^{2}} - \frac{e_{1}}{t_{n}} \frac{(u_{n} (x) - u_{n - 1} (x))}{d} - \frac{e_{1}}{t_{n}^{2}} u_{n}^{''} (x) + u_{n} {(x)}^{3} - u_{n} (x) - 1,

and

J ({a (j, n)}) = \sum_{n = 1}^{N - 1} \int_{0}^{2 π} {(W_{n})}^{2} d x

we obtain

{a (j, n)}

by numerically minimizing J.

Hence, we have obtained the following lines for these cases. For such graphs, we have considered 300 nodes in x, with

2 π / 300

as units in

x \in [0, 2 π] .

For the Lines 2, 4, 6, 8, through the generalized method of lines, please see Figure 3, Figure 6, Figure 9 and Figure 12.

For the Lines 2, 4, 6, 8, through a combination of the Newton's and the generalized method of lines, please see Figure 4, Figure 7, Figure 10 and Figure 13.

Finally, for the Line 2, 4, 6, 8 obtained through the minimization of the functional J, please see Figure 5, Figure 8, Figure 11 and Figure 14.

Figure 3. Line 2, solution

u_{2} (x)

through the general method of lines

Figure 3. Line 2, solution

u_{2} (x)

through the general method of lines

Figure 4. Line 2, solution

u_{2} (x)

through the Newton's Method

Figure 4. Line 2, solution

u_{2} (x)

through the Newton's Method

Figure 5. Line 2, solution

u_{2} (x)

through the minimization of functional J

Figure 5. Line 2, solution

u_{2} (x)

through the minimization of functional J

Figure 6. Line 4, solution

u_{4} (x)

through the general method of lines

Figure 6. Line 4, solution

u_{4} (x)

through the general method of lines

Figure 7. Line 4, solution

u_{4} (x)

through the Newton's Method

Figure 7. Line 4, solution

u_{4} (x)

through the Newton's Method

Figure 8. Line 4, solution

u_{4} (x)

through the minimization of functional J

Figure 8. Line 4, solution

u_{4} (x)

through the minimization of functional J

Figure 9. Line 6, solution

u_{6} (x)

through the general method of lines

Figure 9. Line 6, solution

u_{6} (x)

through the general method of lines

Figure 10. Line 6, solution

u_{6} (x)

through the Newton's Method

Figure 10. Line 6, solution

u_{6} (x)

through the Newton's Method

Figure 11. Line 6, solution

u_{6} (x)

through the minimization of functional J

Figure 11. Line 6, solution

u_{6} (x)

through the minimization of functional J

Figure 12. Line 8, solution

u_{8} (x)

through the general method of lines

Figure 12. Line 8, solution

u_{8} (x)

through the general method of lines

Figure 13. Line 8, solution

u_{8} (x)

through the Newton's Method

Figure 13. Line 8, solution

u_{8} (x)

through the Newton's Method

Figure 14. Line 8, solution

u_{8} (x)

through the minimization of functional J

Figure 14. Line 8, solution

u_{8} (x)

through the minimization of functional J

12. Conclusion

In the first part of this article we develop duality principles for non-convex variational optimization. In the final concerning sections we propose dual convex formulations suitable for a large class of models in physics and engineering. In the last article section, we present an advance concerning the computation of a solution for a partial differential equation through the generalized method of lines. In particular, in its previous versions, we used to truncate the series in

d^{2}

however, we have realized the results are much better by taking line solutions in series for

u_{f} [x]

and its derivatives, as it is indicated in the present software.

This is a little difference concerning the previous procedure, but with a great result improvement as the parameter

ε > 0

is small.

Indeed, with a sufficiently large N (number of lines), we may obtain very good qualitative results even as

ε > 0

is very small.

References

R.A. Adams and J.F. Fournier, Sobolev Spaces, 2nd edn. (Elsevier, New York, 2003).
W.R. Bielski, A. Galka, J.J. Telega, The Complementary Energy Principle and Duality for Geometrically Nonlinear Elastic Shells. I. Simple case of moderate rotations around a tangent to the middle surface. Bulletin of the Polish Academy of Sciences, Technical Sciences, Vol. 38, No. 7-9, 1988.
W.R. Bielski and J.J. Telega, A Contribution to Contact Problems for a Class of Solids and Structures, Arch. Mech., 37, 4-5, pp. 303-320, Warszawa 1985.
J.F. Annet, Superconductivity, Superfluids and Condensates, 2nd edn. ( Oxford Master Series in Condensed Matter Physics, Oxford University Press, Reprint, 2010).
F.S. Botelho, Functional Analysis, Calculus of Variations and Numerical Methods in Physics and Engineering, CRC Taylor and Francis, Florida, 2020.
F.S. Botelho, Variational Convex Analysis, Ph.D. thesis, Virginia Tech, Blacksburg, VA -USA, (2009).
F. Botelho, Topics on Functional Analysis, Calculus of Variations and Duality, Academic Publications, Sofia, (2011).
F. Botelho, Existence of solution for the Ginzburg-Landau system, a related optimal control problem and its computation by the generalized method of lines, Applied Mathematics and Computation, 218, 11976-11989, (2012). [CrossRef]
F. Botelho, Functional Analysis and Applied Optimization in Banach Spaces, Springer Switzerland, 2014.
J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, second edition (Philadelphia, 2004).
L.D. Landau and E.M. Lifschits, Course of Theoretical Physics, Vol. 5- Statistical Physics, part 1. (Butterworth-Heinemann, Elsevier, reprint 2008).
R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, (1970).
J.J. Telega, On the complementary energy principle in non-linear elasticity. Part I: Von Karman plates and three dimensional solids, C.R. Acad. Sci. Paris, Serie II, 308, 1193-1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, ibid, pp. 1313-1317 (1989).
A.Galka and J.J.Telega Duality and the complementary energy principle for a class of geometrically non-linear structures. Part I. Five parameter shell model; Part II. Anomalous dual variational priciples for compressed elastic beams, Arch. Mech. 47 (1995) 677-698, 699-724.
J.F. Toland, A duality principle for non-convex optimisation and the calculus of variations, Arch. Rat. Mech. Anal., 71, No. 1 (1979), 41-61. [CrossRef]

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

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Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

Abstract

1. Introduction

2. The main duality principle, a convex dual formulation and the concerning proximal primal functional

3. A primal dual variational formulation

4. One more duality principle and a concerning primal dual variational formulation

4.1. Introduction

4.2. The main duality principle and a related primal dual variational formulation

5. A convex dual variational formulation

6. Another convex dual variational formulation

7. A third duality principle and related convex dual variational formulation

8. Closely related primal-dual variational formulations

9. One more duality principle suitable for the primal formulation global optimization

9.1. The main duality principle and a related convex dual formulation

10. Another duality principle for a related model in phase transition

11. A related numerical computation through the generalized method of lines

11.1. About a concerning improvement for the generalized method of lines

11.2. Software in Mathematica for solving such an equation

11.3. Some plots concerning the numerical results

12. Conclusion

References

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