On Duality Principles and Related Convex Dual Formulations Suitable for Local and Global Non-Convex Variational Optimization

1. Introduction

In this article we establish a duality principle and a related convex dual formulation suitable for the local optimization of the primal 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 the 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 1.1.

It is worth highlighting, we may generically denote

$${\int }_{\Omega }\left[{(-\gamma {\nabla }^2+K{I}_d)}^{-1}{v}^{\ast }\right]v^{\ast }dx$$

simply by

$${\int }_{\Omega }\frac{{\left(v^{\ast }\right)}^2}{-\gamma {\nabla }^2+K}dx,$$

where $I_d$ denotes a concerning identity operator.

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

Finally, ${\nabla }^2$ denotes the Laplace operator and for real constants $K_2>0$ and $K_1>0$, the notation $K_2\gg K_1$ means that $K_2>0$ is much larger than $K_1>0.$

Now we present some basic definitions and statements.

Definition 1.2.

Let V be a Banach space. We define the topological dual space of V, denoted by $V^{\prime }$, as the set of all continuous and linear functionals defined on V.

We assume $V^{\prime }$ may be represented through another Banach space denoted by $V^{\ast }$ and a bilinear form

$${\langle ·,·\rangle }_V:V\times V^{\ast }\to \mathbb{R}.$$

More specifically, for each $f\in V^{\prime }$, we suppose there exists a unique $u^{\ast }\in V^{\ast }$ such that

$$f\left(u\right)={\langle u,u^{\ast }\rangle }_V,\forall u\in V.$$

Moreover, we define the norm of f, denoted by

$${\parallel f\parallel }_{V^{\ast }}$$

by

$${\parallel f\parallel }_{V^{\ast }}=sup\left\{\right|{\langle u,u^{\ast }\rangle }_V:u\in {V\mathrm{and}\parallel u\parallel }_V\le 1\}\equiv \parallel u^{\ast }{\parallel }_{V^{\ast }}.$$

For an open, bounded and connected set $\Omega \subset {\mathbb{R}}^N$ and $Y=Y^{\ast }=L^2(\Omega )$ we recall that

$${\langle u,u^{\ast }\rangle }_{L^2}={\int }_{\Omega }u{u}^{\ast }dx.$$

More specifically, for each continuous and linear functional $f:Y\to \mathbb{R}$ there exists a unique $u^{\ast }\in Y^{\ast }=L^2(\Omega )$ such that

$$f\left(u\right)={\int }_{\Omega }u{u}^{\ast }dx,\forall u\in Y=L^2(\Omega ).$$

Definition 1.3

(Polar functional). Let V be a Banach space and let $F:V\to \mathbb{R}$ be a functional.

We define the polar functional of F, denoted by $F^{\ast }:V^{\ast }\to \mathbb{R}$, by

$$F^{\ast }\left(u^{\ast }\right)=\underset{u\in V}{sup}\{{\langle u,u^{\ast }\rangle }_{L^2}-F\left(u\right)\},\forall u^{\ast }\in V^{\ast }.$$

Another important definition refers to the Legendre transform one and respective relevant propriety, which are summarized in the next theorem.

Theorem 1.4

(Legendre transform theorem). Let V be a Banach space and let $F:V\to \mathbb{R}$ be a twice continuously Fréchet differentiable functional.

Let $u^{\ast }\in V^{\ast }$. Assume there exists a unique $\widehat{u}\in V$ such that

$$u^{\ast }=\frac{\partial F\left(\widehat{u}\right)}{\partial u}.$$

Suppose also

$$det\left\{\frac{{\partial }^{2}F\left(u\right)}{\partial u^2}\right\}\ne 0,$$

in a neighborhood of $\widehat{u}.$

Under such hypotheses, defining the Legendre tranform of F at $u^{\ast }$ by $F_L^{\ast }\left(u^{\ast }\right)$ where

$$F_L^{\ast }\left(u^{\ast }\right)={\langle \widehat{u},u^{\ast }\rangle }_V-F\left(\widehat{u}\right)$$

we have that

$$\widehat{u}=\frac{\partial F_L^{\ast }\left(u^{\ast }\right)}{\partial u^{\ast }}.$$

Remark 1.5.

Concerning such a last definition, observe that if F is convex on V, then the extremal condition

$$u^{\ast }=\frac{\partial F\left(\widehat{u}\right)}{\partial u},$$

corresponds to globally maximize

$$H\left(u\right)={\langle u,u^{\ast }\rangle }_V-F\left(u\right)$$

on V, so that, in such a case,

$$F^{\ast }\left(u^{\ast }\right)=H\left(\widehat{u}\right)={\langle \widehat{u},u^{\ast }\rangle }_V-F\left(\widehat{u}\right)=F_L^{\ast }\left(u^{\ast }\right).$$

Summarizing, if F is convex, under the hypotheses of the last theorem, the polar functional $F^{\ast }\left(u^{\ast }\right)$ coincides with the Legendre transform of F on $V^{\ast }$ already denoted by $F_L^{\ast }$, that is,

$$F^{\ast }\left(u\right)=F_L^{\ast }\left(u^{\ast }\right),\forall u^{\ast }\in V^{\ast }.$$

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

Let $\Omega \subset {\mathbb{R}}^3$ be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

Consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx\hfill \\ & & +\frac{\alpha }{2}{\int }_{\Omega }{(u^2-\beta )}^{2}dx-{\langle u,f\rangle }_{L^2}.\hfill \end{array}$$

(1)

Here $\gamma >0,$$\alpha >0$, $\beta >0$ and $f\in L^2(\Omega )\cap L^{\infty }(\Omega ).$

Moreover, $V=W_0^{1,2}(\Omega )$ and we denote $Y=Y^{\ast }=L^2(\Omega ).$

Define the functionals $F_1:V\times Y\to \mathbb{R}$, $F_2:V\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill F_1(u,v_0^{\ast })& =& \frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx+{\langle u^2,v_0^{\ast }\rangle }_{L^2}\hfill \\ & & -\frac{K_1}{2}{\int }_{\Omega }{(-\gamma {\nabla }^{2}u+2v_0^{\ast }u-f)}^{2}dx+\frac{K_2}{2}{\int }_{\Omega }{\left({\nabla }^{2}u\right)}^{2}dx\hfill \\ & & -{\langle u,f\rangle }_{L^2}-\frac{1}{2\alpha }{\int }_{\Omega }{\left(v_0^{\ast }\right)}^{2}dx\hfill \\ & & -\beta {\int }_{\Omega }{v}_0^{\ast }dx,\hfill \end{array}$$

(2)

and

$$F_2\left(u\right)=\frac{K_2}{2}{\int }_{\Omega }{\left({\nabla }^{2}u\right)}^{2}dx.$$

At this point we assume a finite dimensional version for this concerning model. For example, we may define a new domain for the primal functional considering the projection of V on the space spanned by the first N (in general N=10, is enough) eigen-vectors of the Laplace operator, corresponding to the first N eigen-values. On this new not relabeled finite dimensional space V, we recall that the Laplace operator is bounded, so that we assume the following restriction,

$$-\gamma {\nabla }^2+2v_0^{\ast }\le \mathbf{0}.$$

Hence, we denote

$$B^{\ast }=\{v_0^{\ast }\in Y^{\ast }:\parallel 2v_0^{\ast }{\parallel }_{\infty }<K/2\mathrm{and}-\gamma {\nabla }^2+2v_0^{\ast }\le \mathbf{0}\},$$

for an appropriate real constant $K>0$.

We define also $F_1^{\ast }:Y^{\ast }\times B^{\ast }\to \mathbb{R}$ and $F_2^{\ast }:Y^{\ast }\to \mathbb{R},$ by

$$\begin{array}{ccc}& & F_1^{\ast }(v_2^{\ast },v_0^{\ast })\hfill \\ & =& \underset{u\in V}{sup}\{{\langle u,v_2^{\ast }\rangle }_{L^2}-F_1(u,v_0^{\ast })\}\hfill \\ & =& \frac{1}{2}{\int }_{\Omega }\frac{{(v_2^{\ast }+f-K_1(-\gamma {\nabla }^2+2v_0^{\ast })f)}^2}{K_2{\nabla }^4-\gamma {\nabla }^2+2v_0^{\ast }-K_1{(-\gamma {\nabla }^2+2v_0^{\ast })}^2}dx\hfill \\ & & +\frac{1}{2\alpha }{\int }_{\Omega }{\left(v_0^{\ast }\right)}^{2}dx+\beta {\int }_{\Omega }{v}_0^{\ast }dx,+\frac{K_1}{2}{\int }_{\Omega }{f}^{2}dx\hfill \end{array}$$

(3)

and,

$$\begin{array}{ccc}\hfill F_2^{\ast }\left(v_2^{\ast }\right)& =& \underset{u\in V}{sup}\{{\langle u,v_2^{\ast }\rangle }_{L^2}-F_2\left(u\right)\}\hfill \\ & =& \frac{1}{2K_2}{\int }_{\Omega }\frac{{\left(v_2^{\ast }\right)}^2}{{\nabla }^4}dx.\hfill \end{array}$$

(4)

Furthermore, we define

$$D^{\ast }=\{v_2^{\ast }\in Y^{\ast }:\parallel v_2^{\ast }{\parallel }_{\infty }\le 5K_2/4\}$$

and $J_1^{\ast }:D^{\ast }\times B^{\ast }\to \mathbb{R},$ by

$$J_1^{\ast }(v_2^{\ast },v_0^{\ast })=-F_1^{\ast }(v_2^{\ast },v_0^{\ast })+F_2^{\ast }\left(v_2^{\ast }\right).$$

Assuming $0<\alpha \ll 1$ (through a re-scaling, if necessary) and

$$K_2\gg K_1\gg K\gg {max\{\parallel f\parallel }_{\infty },\alpha ,\beta ,\gamma ,1\}$$

by directly computing ${\delta }^2{J}_1^{\ast }(v_2^{\ast },v_0^{\ast })$ we may easily obtain that for such specified real constants, $J_1^{\ast }$ in concave in $(v_2^{\ast },v_0^{\ast })$ on $D^{\ast }\times B^{\ast }.$

2. The main duality principle and a concerning convex (in fact concave) dual formulation

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

Theorem 2.1.

Let $({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })\in D^{\ast }\times B^{\ast }$ be such that

$$\delta J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })=\mathbf{0}$$

and $u_0\in V$ be such that

$$u_0=\frac{\partial F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)}{\partial v_2^{\ast }}.$$

Under such hypotheses, we have

$$\delta J\left(u_0\right)=\mathbf{0},$$

and

$$\begin{array}{ccc}\hfill J\left(u_0\right)& =& \underset{u\in V}{inf}\left\{J\left(u\right)+\frac{1}{2K_2}{\int }_{\Omega }{\left(\frac{{\widehat{v}}_2^{\ast }}{-{\nabla }^2}-K_2(-{\nabla }^{2}u)\right)}^{2}dx\right\}\hfill \\ & =& \underset{(v_2^{\ast },v_0^{\ast })\in D^{\ast }\times B^{\ast }}{sup}{J}_1^{\ast }(v_2^{\ast },v_0^{\ast })\hfill \\ & =& J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast }).\hfill \end{array}$$

(5)

Proof. 

Observe that $\delta J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })=\mathbf{0}$ so that, since $J_1^{\ast }$ is concave in $(v_2^{\ast },v_0^{\ast })$ on $D^{\ast }\times B^{\ast }$, we obtain

$$J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })=\underset{(v_2^{\ast },v_0^{\ast })\in D^{\ast }\times B^{\ast }}{sup}{J}_1^{\ast }(v_2^{\ast },v_0^{\ast }).$$

Now we are going to show that

$$\delta J\left(u_0\right)=\mathbf{0}.$$

From

$$\frac{\partial J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })}{\partial v_2^{\ast }}=\mathbf{0},$$

and

$$\frac{\partial F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)}{\partial v_2^{\ast }}=u_0$$

we have

$$-\frac{\partial F_1^{\ast }({\widehat{v}}_2^{\ast },v_0^{\ast })}{\partial v_2^{\ast }}+u_0=\mathbf{0}$$

and

$${\widehat{v}}_2^{\ast }-K_2{\nabla }^4{u}_0=\mathbf{0}.$$

Observe now that denoting

$$H(v_2^{\ast },v_0^{\ast },u)={\langle u,v_2^{\ast }\rangle }_{L^2}-F_1(u,v_0^{\ast }),$$

there exists $\widehat{u}\in V$ such that

$$\frac{\partial H({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast },\widehat{u})}{\partial u}=\mathbf{0},$$

and

$$F_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })=H({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast },\widehat{u}),$$

so that

$$\begin{array}{ccc}\hfill \frac{\partial F_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })}{\partial v_2^{\ast }}& =& \frac{\partial H({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast },\widehat{u})}{\partial v_2^{\ast }}\hfill \\ & & +\frac{\partial H({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast },\widehat{u})}{\partial u}\frac{\partial \widehat{u}}{\partial v_2^{\ast }}\hfill \\ & =& \widehat{u}.\hfill \end{array}$$

(6)

Summarizing, we have got

$$u_0=\frac{\partial F_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })}{\partial v_2^{\ast }}=\widehat{u}.$$

Also, denoting

$$A(u_0,{\widehat{v}}_0^{\ast })=-\gamma {\nabla }^2{u}_0+2{\widehat{v}}_0^{\ast }{u}_0-f,$$

from

$$\frac{\partial H({\widehat{v_2}}^{\ast },{\widehat{v}}_0^{\ast },u_0)}{\partial u}=\mathbf{0},$$

we have

$$-(-\gamma {\nabla }^2{u}_0+2{\widehat{v}}_0^{\ast }{u}_0-f-K_1(-\gamma {\nabla }^2+2{\widehat{v}}_0^{\ast })A(u_0,{\widehat{v}}_0^{\ast })-{\widehat{v}}_2^{\ast }+K_2{\nabla }^4{u}_0)=\mathbf{0},$$

so that

$$A(u_0,{\widehat{v}}_0^{\ast })-K_1(-\gamma {\nabla }^2+2{\widehat{v}}_0^{\ast })A(u_0,{\widehat{v}}_0^{\ast })=\mathbf{0}.$$

(7)

From such results, we may infer that

$$A(u_0,{\widehat{v}}_0^{\ast })=-\gamma {\nabla }^2{u}_0+2{\widehat{v}}_0^{\ast }-f=0,\mathrm{in}\Omega .$$

Moreover, from

$$\frac{\partial J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })}{\partial v_0^{\ast }}=\mathbf{0},$$

we have

$$-K_{1}A(u_0,{\widehat{v}}_0^{\ast })2u_0-\frac{{\widehat{v}}_0^{\ast }}{\alpha }+u_0^2-\beta =\mathbf{0},$$

so that

$$v_0^{\ast }=\alpha (u_0^2-\beta ).$$

From such last results we get

$$-\gamma {\nabla }^2{u}_0+2\alpha (u_0^2-\beta )u_0-f=\mathbf{0},$$

and thus

$$\delta J\left(u_0\right)=\mathbf{0}.$$

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

$$F_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })={\langle u_0,{\widehat{v}}_2^{\ast }\rangle }_{L^2}-F_1(u_0,{\widehat{v}}_0^{\ast }),$$

$$F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)={\langle u_0,{\widehat{v}}_2^{\ast }\rangle }_{L^2}-F_2\left(u_0\right),$$

so that

$$\begin{array}{ccc}\hfill & & J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })\hfill \\ & =& -F_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)\hfill \\ & =& F_1(u_0,{\widehat{v}}_0^{\ast })-F_2\left(u_0\right)\hfill \\ & =& J\left(u_0\right).\hfill \end{array}$$

(8)

Finally, observe that

$$J_1^{\ast }(v_2^{\ast },v_0^{\ast })\le -{\langle u,v_2^{\ast }\rangle }_{L^2}+F_1(u,v_0^{\ast })+F_2^{\ast }\left(v_2^{\ast }\right),$$

$\forall u\in V,v_2^{\ast }\in D^{\ast },v_0^{\ast }\in B^{\ast }.$

Thus, we may obtain

$$\begin{array}{ccc}\hfill & & J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })\hfill \\ & \le & -{\langle u,{\widehat{v}}_2^{\ast }\rangle }_{L^2}+\frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx+{\langle u^2,{\widehat{v}}_0^{\ast }\rangle }_{L^2}\hfill \\ & & -\frac{K_1}{2}{\int }_{\Omega }{(-\gamma {\nabla }^{2}u+2{\widehat{v}}_0^{\ast }u-f)}^{2}dx+F_2\left(u\right)+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)-{\langle u,f\rangle }_{L^2}\hfill \\ & & -\frac{1}{2\alpha }{\int }_{\Omega }{\left({\widehat{v}}_0^{\ast }\right)}^{2}dx-\beta {\int }_{\Omega }{\widehat{v}}_0^{\ast }dx\hfill \\ & \le & -{\langle u,{\widehat{v}}_2^{\ast }\rangle }_{L^2}+\frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx+{\langle u^2,{\widehat{v}}_0^{\ast }\rangle }_{L^2}\hfill \\ & & +F_2\left(u\right)+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)-{\langle u,f\rangle }_{L^2}\hfill \\ & & -\frac{1}{2\alpha }{\int }_{\Omega }{\left({\widehat{v}}_0^{\ast }\right)}^{2}dx-\beta {\int }_{\Omega }{\widehat{v}}_0^{\ast }dx\hfill \\ & \le & \underset{v_0^{\ast }\in Y^{\ast }}{sup}\left\{-{\langle u,{\widehat{v}}_2^{\ast }\rangle }_{L^2}+\frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx+{\langle u^2,v_0^{\ast }\rangle }_{L^2}\right.\hfill \\ & & +F_2\left(u\right)+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right)-{\langle u,f\rangle }_{L^2}\hfill \\ & & \left.-\frac{1}{2\alpha }{\int }_{\Omega }{\left(v_0^{\ast }\right)}^{2}dx-\beta {\int }_{\Omega }{v}_0^{\ast }dx\right\}\hfill \\ & =& J\left(u\right)+F_2\left(u\right)-{\langle u,{\widehat{v}}_2^{\ast }\rangle }_{L^2}+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right),\forall u\in V.\hfill \end{array}$$

(9)

Summarizing, we have got

$$\begin{array}{c}\hfill J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast })\le J\left(u\right)+F_2\left(u\right)-{\langle u,{\widehat{v}}_2^{\ast }\rangle }_{L^2}+F_2^{\ast }\left({\widehat{v}}_2^{\ast }\right),\forall u\in V.\end{array}$$

(10)

Joining the pieces, from a concerning convexity in u, we have got

$$\begin{array}{ccc}\hfill J\left(u_0\right)& =& \underset{u\in V}{inf}\left\{J\left(u\right)+\frac{1}{2K_2}{\int }_{\Omega }{\left(\frac{{\widehat{v}}_2^{\ast }}{-{\nabla }^2}-K_2(-{\nabla }^{2}u)\right)}^{2}dx\right\}\hfill \\ & =& \underset{(v_2^{\ast },v_0^{\ast })\in D^{\ast }\times B^{\ast }}{sup}{J}_1^{\ast }(v_2^{\ast },v_0^{\ast })\hfill \\ & =& J_1^{\ast }({\widehat{v}}_2^{\ast },{\widehat{v}}_0^{\ast }).\hfill \end{array}$$

(11)

The proof is complete.

□

3. A convex primal dual formulation for a local optimization of the primal one

In this section we develop a convex primal dual formulation corresponding to a non-convex primal formulation.

We start by describing the primal formulation.

Let $\Omega \subset {\mathbb{R}}^3$ be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

For the primal formulation, consider a functional $J:V\to \mathbb{R}$ where

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx\hfill \\ & & +\frac{\alpha }{2}{\int }_{\Omega }{(u^2-\beta )}^{2}dx-{\langle u,f\rangle }_{L^2}.\hfill \end{array}$$

(12)

Here $\gamma >0,$$\alpha >0$, $\beta >0$ and $f\in L^2(\Omega )\cap L^{\infty }(\Omega ).$

Moreover, $V=W_0^{1,2}(\Omega )$ and we denote $Y=Y^{\ast }=L^2(\Omega ).$

Define the functional $J_1^{\ast }:V\times {\left[Y^{\ast }\right]}^2\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill J_1^{\ast }(u,v_3^{\ast },v_0^{\ast })& =& \frac{\gamma }{2}{\int }_{\Omega }\nabla u·\nabla udx+{\langle u^2,v_0^{\ast }\rangle }_{L^2}\hfill \\ & & +\frac{K_1}{2}{\int }_{\Omega }{(-\gamma {\nabla }^{2}u+2v_3^{\ast }u-f)}^{2}dx+\frac{K_1}{2}{\int }_{\Omega }{(v_3^{\ast }-\alpha (u^2-\beta ))}^{2}dx\hfill \\ & & -{\langle u,f\rangle }_{L^2}-\frac{1}{2\alpha }{\int }_{\Omega }{\left(v_0^{\ast }\right)}^{2}dx\hfill \\ & & -\beta {\int }_{\Omega }{v}_0^{\ast }dx.\hfill \end{array}$$

(13)

We define also

$$B^{\ast }=\{v_0^{\ast }\in Y^{\ast }:\parallel 2v_0^{\ast }{\parallel }_{\infty }<K/2\},$$

for an appropriate real constant $K>0$.

Furthermore, we define

$$D^{\ast }=\{v_3^{\ast }\in Y^{\ast }:\parallel v_3^{\ast }{\parallel }_{\infty }\le K_2\}$$

$$A^{+}=\{u\in V:uf\ge 0,\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \},$$

$$V_2={\{u\in V:\parallel u\parallel }_{\infty }\le K_3\}$$

for an appropriate real constant $K_3>0$ and

$$V_1=A^{+}\cap V_2.$$

Now observe that denoting ${\phi }_1=v_3^{\ast }-\alpha (u^2-\beta )$, we have

$$\begin{array}{ccc}\hfill \frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial u^2}& =& K_1{(-\gamma {\nabla }^2+2v_3^{\ast })}^2+4K_1{\alpha }^2{u}^2\hfill \\ & & -2K_1\alpha {\phi }_1-\gamma {\nabla }^2+2v_0^{\ast }\hfill \end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill \frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial {\left(v_3^{\ast }\right)}^2}& =& K_1+4K_1{u}^2.\hfill \end{array}$$

(15)

Denoting $\phi =-\gamma {\nabla }^{2}u+2v_0^{\ast }u-f$ we have also that

$$\begin{array}{ccc}\hfill \frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial u\partial v_3^{\ast }}& =& K_1(-2\gamma {\nabla }^{2}u+2v_3^{\ast }u)+2K_1\phi -2K_1\alpha u.\hfill \end{array}$$

(16)

In such a case, we obtain

$$\begin{array}{ccc}\hfill det\left\{\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial u\partial v_3^{\ast }}\right\}& =& \frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial {\left(v_3^{\ast }\right)}^2}\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial u^2}-{\left(\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial v_3^{\ast }\partial u}\right)}^2\hfill \\ & =& K_1^2{(-\gamma {\nabla }^2+2v_3^{\ast }+4\alpha u^2)}^2\hfill \\ & & +(-\gamma {\nabla }^2+2v_0^{\ast })\mathcal{O}\left(K_1\right)\hfill \\ & & -4K_1^2{\phi }^2-4K_1^2\phi \left[(-\gamma {\nabla }^{2}u+2v_0^{\ast }u)-2\alpha u\right]\hfill \\ & & -2K_1^2\alpha {\phi }_1(1+4u^2).\hfill \end{array}$$

(17)

Observe that at a critical point

$$\phi =\mathbf{0}$$

and

$${\phi }_1=0.$$

From such results we may infer that

$$det\left\{\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })}{\partial u\partial v_3^{\ast }}\right\}>\mathbf{0}$$

around any critical point.

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

Theorem 3.1.

Let $(u_0,{\widehat{v}}_3^{\ast },{\widehat{v}}_0^{\ast })\in V_1\times D^{\ast }\times B^{\ast }$ be such that

$$\delta J_1^{\ast }(u_0,{\widehat{v}}_3^{\ast },{\widehat{v}}_0^{\ast })=\mathbf{0}.$$

Under such hypotheses, we have

$$\delta J\left(u_0\right)=-\gamma {\nabla }^2{u}_0+2\alpha (u_0^2-\beta )u_0-f=\mathbf{0}$$

and there exists $r>0$ such that

$$\begin{array}{ccc}\hfill J\left(u_0\right)& =& \underset{v_0^{\ast }\in B^{\ast }}{sup}\left\{\underset{(u,v_3^{\ast })\in B_r(u_0,{\widehat{v}}_3^{\ast })}{inf}{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })\right\}\hfill \\ & =& J_1^{\ast }(u_0,{\widehat{v}}_3^{\ast },{\widehat{v}}_0^{\ast }).\hfill \end{array}$$

(18)

Proof. 

The proof that

$$\delta J\left(u_0\right)=\mathbf{0}$$

and

$$J\left(u_0\right)=J_1^{\ast }(u_0,{\widehat{v}}_3^{\ast },{\widehat{v}}_0^{\ast })$$

may be done similarly as in the previous sections.

Observe that, as previously obtained, there exists $r>0$ such that

$$det\left\{\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },{\widehat{v}}_0^{\ast })}{\partial u\partial v_3^{\ast }}\right\}>\mathbf{0},\forall (u,v_3^{\ast })\in B_r(u_0,{\widehat{v}}_3^{\ast })$$

and

$$\frac{{\partial }^2{J}_1^{\ast }(u_0,{\widehat{v}}_3^{\ast },v_0^{\ast })}{\partial {\left(v_0^{\ast }\right)}^2}<\mathbf{0},\forall v_0^{\ast }\in B^{\ast }.$$

Since for a sufficiently large $K_1>0$ we have

$$\frac{{\partial }^2{J}_1^{\ast }(u,v_3^{\ast },{\widehat{v}}_0^{\ast })}{\partial u^2}>\mathbf{0},\mathrm{in}{B}_r(u_0,{\widehat{v}}_3^{\ast }),$$

from these last results and the standard Saddle point theorem, we have

$$J\left(u_0\right)=J_1^{\ast }(u_0,{\widehat{v}}_3^{\ast },{\widehat{v}}_0^{\ast })=\underset{v_0^{\ast }\in B^{\ast }}{sup}\left\{\underset{(u,v_3^{\ast })\in B_r(u_0,{\widehat{v}}_3^{\ast })}{inf}{J}_1^{\ast }(u,v_3^{\ast },v_0^{\ast })\right\}.$$

The proof is complete. □

4. 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 $\Omega \subset {\mathbb{R}}^n$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega =\Gamma .$

For $1<p<+\infty$, consider a functional $J:V\to \mathbb{R}$ where

$$J\left(u\right)=G(\nabla u)+F\left(u\right)-{\langle u,f\rangle }_{L^2},$$

where

$$V=\left\{u\in W^{1,p}(\Omega ;{\mathbb{R}}^N):u=u_0\mathrm{in}\partial \Omega \right\}$$

and $f\in L^2(\Omega ;{\mathbb{R}}^N).$

We assume $G:Y\to \mathbb{R}$ and $F:V\to \mathbb{R}$ are Fréchet differentiable and F is also convex.

Also

$$G(\nabla u)={\int }_{\Omega }g(\nabla u)dx,$$

where $g:{\mathbb{R}}^{N\times n}$ it is supposed to be Fréchet differentiable. Here we have denoted $Y=L^p(\Omega ;{\mathbb{R}}^{N\times n})$.

We define also $J_1:V\times Y_1\to \mathbb{R}$ by

$$J_1(u,\varphi )=G_1(\nabla u+{\nabla }_y\varphi )+F\left(u\right)-{\langle u,f\rangle }_{L^2},$$

where

$$Y_1=W^{1,p}(\Omega \times \Omega ;{\mathbb{R}}^N)$$

and

$$G_1(\nabla u+{\nabla }_y\varphi )=\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }g(\nabla u\left(x\right)+{\nabla }_y\varphi (x,y))dxdy.$$

Moreover, we define the relaxed functional $J_2:V\to \mathbb{R}$ by

$$J_2\left(u\right)=\underset{\varphi \in V_0}{inf}{J}_1(u,\varphi ),$$

where

$$V_0=\{\varphi \in Y_1:\varphi (x,y)=0,\mathrm{in}\Omega \times \partial \Omega \}.$$

Now observe that

$$\begin{array}{ccc}\hfill J_1(u,\varphi )& =& G_1(\nabla u+{\nabla }_y\varphi )+F\left(u\right)-{\langle u,f\rangle }_{L^2}\hfill \\ & =& -\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·(\nabla u+{\nabla }_y\varphi (x,y))dydx+G_1(\nabla u+{\nabla }_y\varphi )\hfill \\ & & +\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·(\nabla u+{\nabla }_y\varphi (x,y))dydx+F\left(u\right)-{\langle u,f\rangle }_{L^2}\hfill \\ & \ge & \underset{v\in Y_2}{inf}\left\{-\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·v(x,y)dydx+G_1\left(v\right)\right\}\hfill \\ & & +\underset{(v,\varphi )\in V\times V_0}{inf}\left\{\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·(\nabla u+{\nabla }_y\varphi (x,y))dydx+F\left(u\right)-{\langle u,f\rangle }_{L^2}\right\}\hfill \\ & =& -G_1^{\ast }\left(v^{\ast }\right)-F^{\ast }\left({\mathrm{div}}_x\left(\frac{1}{|\Omega |}{\int }_{\Omega }{v}^{\ast }(x,y)dy\right)+f\right)\hfill \\ & & +\frac{1}{|\Omega |}{\int }_{\partial \Omega }\left({\int }_{\Omega }{v}^{\ast }(x,y)dy\right)·u_{0}d\Gamma ,\hfill \end{array}$$

(19)

$\forall (u,\varphi )\in V\times V_0,v^{\ast }\in A^{\ast }$, where

$$A^{\ast }=\{v^{\ast }\in Y_2^{\ast }:{\mathrm{div}}_y{v}^{\ast }(x,y)=0,\mathrm{in}\Omega \}.$$

Here we have denoted

$$G_1^{\ast }\left(v^{\ast }\right)=\underset{v\in Y_2}{sup}\left\{\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·v(x,y)dydx-G_1\left(v\right)\right\},$$

where $Y_2=L^p(\Omega \times \Omega ;{\mathbb{R}}^{N\times n}),$$Y_2^{\ast }=L^q(\Omega \times \Omega ;{\mathbb{R}}^{N\times n}),$ and where

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

Furthermore,

$$\begin{array}{ccc}& & F^{\ast }\left({\mathrm{div}}_x\left(\frac{1}{|\Omega |}{\int }_{\Omega }{v}^{\ast }(x,y)dy\right)+f\right)-\frac{1}{|\Omega |}{\int }_{\partial \Omega }\left({\int }_{\Omega }{v}^{\ast }(x,y)dy\right)·u_{0}d\Gamma \hfill \\ & =& \underset{(v,\varphi )\in V\times V_0}{sup}\left\{-\frac{1}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }{v}^{\ast }(x,y)·(\nabla u+{\nabla }_y\varphi (x,y))dydx-F\left(u\right)+{\langle u,f\rangle }_{L^2}\right\},\hfill \end{array}$$

(20)

Therefore, denoting $J_3^{\ast }:Y_2^{\ast }\to \mathbb{R}$ by

$$J_3^{\ast }\left(v^{\ast }\right)=-G_1^{\ast }\left(v^{\ast }\right)-F^{\ast }\left({\mathrm{div}}_x\left({\int }_{\Omega }{v}^{\ast }(x,y)dy\right)+f\right)+\frac{1}{|\Omega |}{\int }_{\partial \Omega }\Omega \left({\int }_{\Omega }{v}^{\ast }(x,y)dy\right)·u_{0}d\Gamma ,$$

we have got

$$\underset{u\in V}{inf}{J}_2\left(u\right)\ge \underset{v^{\ast }\in A^{\ast }}{sup}{J}_3^{\ast }\left(v^{\ast }\right).$$

Finally, we highlight such a dual functional $J_3^{\ast }$ is convex (in fact concave).

5. Conclusion

In this article we have developed convex dual and primal dual variational formulations suitable for the local optimization of non-convex primal formulations.

It is worth highlighting, the results may be applied to a large class of models in physics and engineering.

We also emphasize the duality principles here presented are applied to a Ginzburg-Landau type equation. In a future research, we intend to extend such results for some models of plates and shells and other models in the elasticity theory.