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

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Abstract
This article develops duality principles and related convex dual formulations suitable for the local and global optimization of non-convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg-Landau type equation.
Keywords: 
Subject: Computer Science and Mathematics  -   Applied Mathematics

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
Ω [ ( γ 2 + K I d ) 1 v * ] v * d x
simply by
Ω ( v * ) 2 γ 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.
Finally, 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 .
At this point we start to describe the primal and dual variational formulations.
Let Ω R 3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by Ω .
For the primal formulation, consider a functional J : V R where
J ( u ) = γ 2 Ω u · u d x + α 2 Ω ( u 2 β ) 2 d x u , f L 2 .
Here γ > 0 , α > 0 , β > 0 and f L 2 ( Ω ) L ( Ω ) .
Moreover, V = W 0 1 , 2 ( Ω ) and we denote Y = Y * = L 2 ( Ω ) .
At this point we define F 1 : V × Y R , F 2 : V R and G : V × Y R by
F 1 ( u , v 0 * ) = γ 2 Ω u · u d x K 2 Ω u 2 d x + K 1 2 Ω ( γ 2 u + 2 v 0 * u f ) 2 d x + K 2 2 Ω u 2 d x ,
F 2 ( u ) = K 2 2 Ω u 2 d x + u , f L 2 ,
and
G ( u , v ) = α 2 Ω ( u 2 β + v ) 2 d x + K 2 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 ) = γ 2 Ω u · u d x + α 2 Ω ( u 2 β ) 2 d x u , f L 2 ,
and F 1 * : [ Y * ] 3 R , F 2 * : Y * R , and G * : [ Y * ] 2 R , by
F 1 * ( v 2 * , v 1 * , v 0 * ) = sup u V { u , v 1 * + v 2 * L 2 F 1 ( u , v 0 * ) } = 1 2 Ω v 1 * + v 2 * + K 1 ( γ 2 + 2 v 0 * ) f 2 ( γ 2 K + K 2 + K 1 ( γ 2 + 2 v 0 * ) 2 ) d x K 1 2 Ω f 2 d x ,
F 2 * ( v 2 * ) = sup u V { u , v 2 * L 2 F 2 ( u ) } = 1 2 K 2 Ω ( v 2 * f ) 2 d x ,
and
G * ( v 1 * , v 0 * ) = sup ( u , v ) V × Y { u , v 1 * L 2 v , v 0 * L 2 G ( u , v ) } = 1 2 Ω ( v 1 * ) 2 2 v 0 * + K d x + 1 2 α Ω ( v 0 * ) 2 d x + β Ω v 0 * d x
if v 0 * B * where
B * = { v 0 * Y * : v 0 * K / 2 } .
Define also
V 2 = { u V : u K 3 } ,
A + = { u V : u f 0 a . e . in Ω } ,
V 1 = V 2 A + ,
B 2 * = { v 0 * Y * : γ 2 K + K 1 ( γ 2 + 2 v 0 * ) 2 > 0 } ,
D 3 * = { ( v 1 * , v 2 * ) Y * × Y * : 1 / α + 4 K 1 [ u ( v 1 * , v 2 * , v 0 * ) 2 ] + 100 / K 2 0 , v 0 * B * } ,
where
u ( v 2 * , v 0 * ) = φ 1 φ ,
φ 1 = ( v 1 * + v 2 * + K 1 ( γ 2 + 2 v 0 * ) f )
and
φ = ( γ 2 K + K 1 ( γ 2 + 2 v 0 * ) 2 + K 2 ) ,
D * = { v 2 * Y * ; v 2 * < K 4 }
E * = { v 1 * Y * : v 1 * K 5 } ,
for some K 3 , K 4 , K 5 > 0 to be specified,
Finally, we also define J 1 * : [ Y * ] 2 × B * 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 * ) .
Assume now K 1 = 1 / [ 4 ( α + ε ) K 3 2 ] ,
K 2 K 1 max { K 3 , K 4 , K 5 , 1 , γ , α , β } .
Observe that, by direct computation, we may obtain
2 J 1 * ( v 2 * , v 1 * , v 0 * ) ( v 0 * ) 2 = 1 α + 4 K 1 u ( v * ) 2 + O ( 1 / K 2 ) < 0 ,
for v 0 * B 3 * .
Considering such statements and definitions, we may prove the following theorem.
Theorem 1.2.
Let ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) ( ( D * × E * ) D 3 * ) × ( B 2 * B * ) be such that
δ J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) = 0
and u 0 V 1 , where
u 0 = v ^ 1 * + v ^ 2 * + K 1 ( γ 2 + 2 v 0 * ) f K 2 K γ 2 + K 1 ( γ 2 + 2 v ^ 0 * ) 2 .
Under such hypotheses, we have
δ J ( u 0 ) = 0 ,
so that
J ( u 0 ) = inf u V 1 J ( u ) + K 1 2 Ω ( γ 2 u + 2 v ^ 0 * u f ) 2 d x = inf v 2 * D * sup ( v 1 * , v 0 * ) E * × B * J 1 * ( v 2 * , v 1 * , v 0 * ) = J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) .
Proof. 
Observe that δ J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) = 0 so that, since ( v ^ 2 * , v ^ 1 * ) D 3 * , v ^ 0 * B 2 * and J 1 * is quadratic in v 2 * , we may infer that
J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) = inf v 2 * Y * J 1 * ( v 2 * , v ^ 1 * , v ^ 0 * ) = sup ( v 1 * , v 0 * ) E * × B * J 1 * ( v ^ 2 * , v 1 * , v 0 * ) .
Therefore, from a standard saddle point theorem, we have that
J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) = inf v 2 * Y * sup ( v 1 * , v 0 * ) E * × B * J 1 * ( v 2 * , v 1 * , v 0 * ) .
Now we are going to show that
δ J ( u 0 ) = 0 .
From
J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) v 2 * = 0 ,
we have
u 0 + v ^ 2 * K 2 = 0 ,
and thus
v ^ 2 * = K 2 u 0 .
From
J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) v 1 * = 0 ,
we obtain
u 0 v ^ 1 * f 2 v ^ 0 * + K = 0 ,
and thus
v ^ 1 * = 2 v ^ 0 * u 0 K u 0 + f .
Finally, denoting
D = γ 2 u 0 + 2 v ^ 0 * u 0 f ,
from
J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) v 0 * = 0 ,
we have
2 D u 0 + u 0 2 v ^ 0 * α β = 0 ,
so that
v ^ 0 * = α ( u 0 2 β 2 D u 0 ) .
Observe now that
v ^ 1 * + v ^ 2 * + K 1 ( γ 2 + 2 v ^ 0 * ) f = ( K 2 K γ 2 + K 1 ( γ 2 + 2 v ^ 0 * ) 2 ) u 0
so that
K 2 u 0 2 v ^ 0 u 0 K u 0 + f = K 2 u 0 K u 0 γ 2 u 0 + K 1 ( γ 2 + 2 v ^ 0 * ) ( γ 2 u 0 + 2 v ^ 0 * u 0 f ) .
The solution for this last system of equations (8) and (9) is obtained through the relations
v ^ 0 * = α ( u 0 2 β )
and
γ 2 u 0 + 2 v ^ 0 * u 0 f = D = 0 ,
so that
δ J ( u 0 ) = γ 2 u 0 + 2 α ( u 0 2 β ) u 0 f = 0
and
δ J ( u 0 ) + K 1 2 Ω ( γ 2 u 0 + 2 v ^ 0 * u 0 f ) 2 d x = 0 .
Moreover, from the Legendre transform properties
F 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) = u 0 , v ^ 2 * + v ^ 1 * L 2 F 1 ( u 0 , v ^ 0 * ) ,
F 2 * ( v ^ 2 * ) = u 0 , v ^ 2 * L 2 F 2 ( u 0 ) ,
G * ( v ^ 1 * , v ^ 0 * ) = u 0 , v ^ 1 * L 2 0 , v ^ 0 * L 2 G ( u 0 , 0 ) ,
so that
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 * ) = F 1 ( u 0 , v ^ 0 * ) F 2 ( u 0 ) + G ( u 0 , 0 ) = J ( u 0 ) .
Observe now that
J ( u 0 ) = J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) γ 2 Ω u · u d x K 2 Ω u 2 d x + K 1 2 Ω ( γ 2 u + v ^ 0 * u f ) 2 d x + u , v ^ 1 * L 2 u , f L 2 1 2 Ω ( v 1 * ) 2 2 v 0 * + K d x 1 2 α Ω ( v 0 * ) 2 d x β Ω v 0 * d x γ 2 Ω u · u d x u , f L 2 + K 1 2 Ω ( γ 2 u + v ^ 0 * u f ) 2 d x + sup ( v 1 * , v 0 * ) D * × B * + u , v ^ 1 * L 2 1 2 Ω ( v 1 * ) 2 2 v 0 * + K d x 1 2 α Ω ( v 0 * ) 2 d x 1 2 α Ω ( v 0 * ) 2 d x β Ω v 0 * d x = J ( u ) + K 1 2 Ω ( γ 2 u + 2 v ^ 0 * u f ) 2 d x ,
u V 1 .
Hence, we have got
J ( u 0 ) = inf u V 1 J ( u ) + K 1 2 Ω ( γ 2 u + 2 v ^ 0 * u f ) 2 d x .
Joining the pieces, we have got
J ( u 0 ) = inf u V J ( u ) + K 1 2 Ω ( γ 2 u + 2 v ^ 0 * u f ) 2 d x = inf v 2 * Y * sup ( v 1 * , v 0 * ) E * × ( B * B r ( v ^ 0 * ) ) J 1 * ( v 2 * , v 1 * , v 0 * ) = J 1 * ( v ^ 2 * , v ^ 1 * , v ^ 0 * ) .
The proof is complete.

2. Another duality principle suitable for a local optimization of the primal formulation

In this section we develop a second duality principle which the dual formulation is concave.
We start by describing the primal formulation.
Let Ω R 3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by Ω .
For the primal formulation, consider a functional J : V R where
J ( u ) = γ 2 Ω u · u d x + α 2 Ω ( u 2 β ) 2 d x u , f L 2 .
Here γ > 0 , α > 0 , β > 0 and f L 2 ( Ω ) L ( Ω ) .
Moreover, V = W 0 1 , 2 ( Ω ) and we denote Y = Y * = L 2 ( Ω ) .
Define the functionals F 1 : V × Y R , F 2 : V R by
F 1 ( u , v 0 * ) = γ 2 Ω u · u d x + u 2 , v 0 * L 2 K 1 2 Ω ( γ 2 u + 2 v 0 * u f ) 2 d x + K 2 2 Ω ( 2 u ) 2 d x u , f L 2 1 2 α Ω ( v 0 * ) 2 d x β Ω v 0 * d x ,
and
F 2 ( u ) = K 2 2 Ω ( 2 u ) 2 d x .
We define also F 1 * : [ Y * ] 2 R and F 2 * : Y * R , by
F 1 * ( v 2 * , v 0 * ) = sup u V { u , v 2 * L 2 F 1 ( u , v 0 * ) } = 1 2 Ω ( v 2 * + f K 1 ( γ 2 + 2 v 0 * ) f ) 2 K 2 4 γ 2 + 2 v 0 * K 1 ( γ 2 + 2 v 0 * ) 2 d x + K 1 2 Ω f 2 d x
and,
F 2 * ( v 2 * ) = sup u V { u , v 2 * L 2 F 2 ( u ) } = 1 2 K 2 Ω ( v 2 * ) 2 4 d x .
Here we denote
B * = { v 0 * Y * : 2 v 0 * < K / 2 } ,
for an appropriate real constant K > 0 .
Furthermore, we define
D * = { v 2 * Y * : v 2 * 5 K 2 / 4 }
and J 1 * : D * × B * R , by
J 1 * ( v 2 * , v 0 * ) = F 1 * ( v 2 * , v 0 * ) + F 2 * ( v 2 * ) .
Assuming 0 < α 1 (through a re-scaling, if necessary) and
K 2 K 1 K max { f , α , β , γ , 1 }
by directly computing δ 2 J 1 * ( v 2 * , v 0 * ) we may easily obtain that for such specified real constants, J 1 * in concave in ( v 2 * , v 0 * ) on D * × B * .

2.1. The main duality principle and a concerning convex dual formulation

Considering the statements and definitions presented in the previous section, we may prove the following theorem.
Theorem 2.1.
Let ( v ^ 2 * , v ^ 0 * ) D * × B * be such that
δ J 1 * ( v ^ 2 * , v ^ 0 * ) = 0
and u 0 V be such that
u 0 = F 2 * ( v ^ 2 * ) v 2 * .
Under such hypotheses, we have
δ J ( u 0 ) = 0 ,
and
J ( u 0 ) = inf u V J ( u ) + 1 2 K 2 Ω v ^ 2 * 2 K 2 ( 2 u ) 2 d x = sup ( v 2 * , v 0 * ) D * × B * J 1 * ( v 2 * , v 0 * ) = J 1 * ( v ^ 2 * , v ^ 0 * ) .
Proof. 
Observe that δ J 1 * ( v ^ 2 * , v ^ 0 * ) = 0 so that, since J 1 * is concave in ( v 2 * , v 0 * ) on D * × B * , we obtain
J 1 * ( v ^ 2 * , v ^ 0 * ) = sup ( v 2 * , v 0 * ) D * × B * J 1 * ( v 2 * , v 0 * ) .
Now we are going to show that
δ J ( u 0 ) = 0 .
From
J 1 * ( v ^ 2 * , v ^ 0 * ) v 2 * = 0 ,
and
F 2 * ( v ^ 2 * ) v 2 * = u 0
we have
F 1 * ( v ^ 2 * , v 0 * ) v 2 * + u 0 = 0
and
v ^ 2 * K 2 4 u 0 = 0 .
Observe now that denoting
H ( v 2 * , v 0 * , u ) = u , v 2 * L 2 F 1 ( u , v 0 * ) ,
there exists u ^ V such that
H ( v ^ 2 * , v ^ 0 * , u ^ ) u = 0 ,
and
F 1 * ( v ^ 2 * , v ^ 0 * ) = H ( v ^ 2 * , v ^ 0 * , u ^ ) ,
so that
F 1 * ( v ^ 2 * , v ^ 0 * ) v 2 * = H ( v ^ 2 * , v ^ 0 * , u ^ ) v 2 * + H ( v ^ 2 * , v ^ 0 * , u ^ ) u u ^ v 2 * = u ^ .
Summarizing, we have got
u 0 = F 1 * ( v ^ 2 * , v ^ 0 * ) v 2 * = u ^ .
Also, denoting
A ( u 0 , v ^ 0 * ) = γ 2 u 0 + 2 v ^ 0 * u 0 f ,
from
H ( v 2 ^ * , v ^ 0 * , u 0 ) u = 0 ,
we have
( γ 2 u 0 + 2 v ^ 0 * u 0 f K 1 ( γ 2 + 2 v ^ 0 * ) A ( u 0 , v ^ 0 * ) v ^ 2 * + K 2 4 u 0 ) = 0 ,
so that
A ( u 0 , v ^ 0 * ) K 1 ( γ 2 + 2 v ^ 0 * ) A ( u 0 , v ^ 0 * ) = 0 .
From such results, we may infer that
A ( u 0 , v ^ 0 * ) = γ 2 u 0 + 2 v ^ 0 * f = 0 , in Ω .
Moreover, from
J 1 * ( v ^ 2 * , v ^ 0 * ) v 0 * = 0 ,
we have
K 1 A ( u 0 , v ^ 0 * ) 2 u 0 v ^ 0 * α + u 0 2 β = 0 ,
so that
v 0 * = α ( u 0 2 β ) .
From such last results we get
γ 2 u 0 + 2 α ( u 0 2 β ) u 0 f = 0 ,
and thus
δ J ( u 0 ) = 0 .
Furthermore, also from such last results and the Legendre transform properties, we have
F 1 * ( v ^ 2 * , v ^ 0 * ) = u 0 , v ^ 2 * L 2 F 1 ( u 0 , v ^ 0 * ) ,
F 2 * ( v ^ 2 * ) = u 0 , v ^ 2 * L 2 F 2 ( u 0 ) ,
so that
J 1 * ( v ^ 2 * , v ^ 0 * ) = F 1 * ( v ^ 2 * , v ^ 0 * ) + F 2 * ( v ^ 2 * ) = F 1 ( u 0 , v ^ 0 * ) F 2 ( u 0 ) = J ( u 0 ) .
Finally, observe that
J 1 * ( v 2 * , v 0 * ) u , v 2 * L 2 + F 1 ( u , v 0 * ) + F 2 * ( v 2 * ) ,
u V , v 2 * D * , v 0 * B * .
Thus, we may obtain
J 1 * ( v ^ 2 * , v ^ 0 * ) u , v ^ 2 * L 2 + γ 2 Ω u · u d x + u 2 , v ^ 0 * L 2 K 1 2 Ω ( γ 2 u + 2 v ^ 0 * u f ) 2 d x + F 2 ( u ) + F 2 * ( v ^ 2 * ) u , f L 2 1 2 α Ω ( v ^ 0 * ) 2 d x β Ω v ^ 0 * d x u , v ^ 2 * L 2 + γ 2 Ω u · u d x + u 2 , v ^ 0 * L 2 + F 2 ( u ) + F 2 * ( v ^ 2 * ) u , f L 2 1 2 α Ω ( v ^ 0 * ) 2 d x β Ω v ^ 0 * d x sup v 0 * Y * u , v ^ 2 * L 2 + γ 2 Ω u · u d x + u 2 , v ^ 0 * L 2 + F 2 ( u ) + F 2 * ( v ^ 2 * ) u , f L 2 1 2 α Ω ( v 0 * ) 2 d x β Ω v 0 * d x = J ( u ) + F 2 ( u ) u , v ^ 2 * L 2 + F 2 * ( v ^ 2 * ) , u V .
Summarizing, we have got
J 1 * ( v ^ 2 * , v ^ 0 * ) J ( u ) + F 2 ( u ) u , v ^ 2 * L 2 + F 2 * ( v ^ 2 * ) , u V .
Joining the pieces, from a concerning convexity in u, we have got
J ( u 0 ) = inf u V J ( u ) + 1 2 K 2 Ω v ^ 2 * 2 K 2 ( 2 u ) 2 d x = sup ( v 2 * , v 0 * ) D * × B * J 1 * ( v 2 * , v 0 * ) = J 1 * ( v ^ 2 * , v ^ 0 * ) .
The proof is complete.

3. A third duality principle also suitable for the primal formulation local optimization

In this section we establish one more duality principle and related convex dual formulation suitable for a local optimization of the primal variational formulation.
Let Ω R 3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by Ω .
For the primal formulation, we define V = W 0 1 , 2 ( Ω ) and consider a functional J : V R where
J ( u ) = γ 2 Ω u · u d x + α 2 Ω ( u 2 β ) 2 d x u , f L 2 .
Here we assume f L 2 ( Ω ) , and define Y = Y * = L 2 ( Ω ) ,
V 2 = { u V : u K 4 } ,
A + = { u V : u f > 0 , a . e . in Ω } ,
and
V 1 = A + V 2 ,
for an appropriate constant K 4 > 0 to be specified.
Define also the functionals F 1 : V × [ Y ] 2 R and G : Y R by
F 1 ( u , v 3 * , v 0 * ) = γ 2 Ω u · u d x + u 2 , v 0 * L 2 u , f L 2 + K 1 2 Ω ( v 3 * u K 3 ) 2 d x ,
and
G ( u 2 ) = α 2 Ω ( u 2 β ) 2 d x ,
for appropriate positive constants K 1 , K 3 , K 4 to be specified.
Moreover, define F 1 * : [ Y * ] 2 R and G * : Y * R , by
F 1 * ( v 3 * , v 0 * ) = sup u V { F 2 ( u , v 3 * , v 0 * ) } = 1 2 Ω ( f + K 1 K 3 v 3 * ) 2 γ 2 + 2 v 0 * + K 1 ( v 3 * ) 2 d x K 1 2 Ω K 3 2 d x
and
G * ( v 0 * ) = sup v Y { v , v 0 * L 2 G ( v ) } = 1 2 α Ω ( v 0 * ) 2 d x + β Ω v 0 * d x .
Furthermore, we define
B * = { v 3 * Y * : u 1 ( v 3 * ) V 1 } ,
where
u 1 ( v 3 * ) = K 3 v 3 * .
Define also
C 1 * = { v 0 * Y * : v 0 * K 2 }
for an appropriate real constant K 2 > 0 to be specified, and J 1 * : B * × C 1 * R by
J 1 * ( v 3 * , v 0 * ) = F 2 * ( v 3 * , v 0 * ) G * ( v 0 * ) .
Moreover, we assume K 1 K 2 max { 1 , K 3 , K 4 , α , β , γ , f } .
By directly computing δ 2 J 1 * ( v 3 * , v 0 * ) denoting
A = K 1 K 3 ,
B = 2 K 1 v 3 * ,
φ = γ 2 + 2 v 0 * + K 1 ( v 3 * ) 2 ,
φ 1 = f + K 1 K 3 v 3 * ,
u = φ 1 φ ,
we may also obtain,
2 J 1 * ( v 3 * , v 0 * ) ( v 3 * ) 2 = ( A u B ) 2 φ + K 1 u 2 = K 1 ( K 1 K 3 2 ( 3 u 2 4 u u 1 + u 1 2 ) u 2 u 1 ( γ 2 + 2 v 0 * ) u 1 ) K 1 K 3 2 + u 1 ( γ 2 + 2 v 0 * ) u 1 = K 1 2 H 1 + K 1 H 2 K 1 K 3 2 + u 1 ( γ 2 2 v 0 * ) u 1 )
on B * × C 1 * .
where
u 1 = u 1 ( v 3 * ) = K 3 v 3 * ,
H 1 = K 3 2 ( 3 u 2 4 u u 1 + u 1 2 ) ,
and
H 2 = u 2 [ ( γ 2 + 2 v 0 * ) u 1 ] u 1 .
At a critical point we have H 1 = 0 and
H 2 = u 0 2 f u 0 > 0 , a . e in Ω .
With such results, for a sufficiently small ε 1 > 0 , we may define the restrictions
C v 0 * = { v 3 * B * : | H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2 in Ω }
and
( C 3 ) v 0 * = { v 3 * B * : H 2 ( v 3 * , v 0 * ) 0 in Ω } .
At this point, we prove that ( C ) v 0 * is a convex set.
Firstly, fixing v 0 * C 1 * observe that for K 7 sufficiently large
| H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2 + K 7 ( v 3 * ) 2
is convex in v 3 * on B * .
Observe also that
| H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2
is equivalent to
| H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2 + K 7 ( v 3 * ) 2 K 7 ( v 3 * ) 2 ,
which is equivalent to
| H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2 + K 7 ( v 3 * ) 2 K 7 | v 3 * | .
Moreover, this last inequality is equivalent to
H 5 ( v 3 * , v 0 * ) = | H 1 ( v 3 * , v 0 * ) | ε 1 ( v 3 * ) 2 + K 7 ( v 3 * ) 2 K 7 | v 3 * | 0 .
Since for v 3 * B * we have v 3 * f 0 , a . e . in Ω , we may obtain that K 7 | v 3 * | is a convex function on B * , so that H 5 is convex in v 3 * on B *
From such results, we may easily infer that C v 0 * is a convex set, v 0 * C 1 * .
Similarly, we may prove that ( C 3 ) v 0 * is a convex set, v 0 * C 1 * .
On the other hand, clearly we have
2 J 1 * ( v 3 * , v 0 * ) ( v 0 * ) 2 < 0 in B * × C 1 * .

3.1. A concerning 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 3.1.
Let ( v ^ 3 * , v ^ 0 * ) ( C 3 ) v ^ 0 * × C 1 * be such that
δ J 1 * ( v ^ 3 * , v ^ 0 * ) = 0
and u 0 V 1 be such that
u 0 = φ 1 φ = f + K 1 K 3 v ^ 3 * γ 2 + 2 v ^ 0 * + K 1 ( v ^ 3 * ) 2 .
Assume also
u 0 0 , a . e . in Ω .
Under such hypotheses, denoting B 1 * = C v ^ 0 * ( C 3 ) v ^ 0 * , we have
δ J ( u 0 ) = 0 ,
v ^ 3 * u 0 K 3 = 0 , a . e . in Ω ,
and
J ( u 0 ) = inf u V 1 J ( u ) + K 1 2 Ω ( v ^ 3 * u K 3 ) 2 d x = inf v 3 * B 1 * sup v 0 * C 1 * J 1 * ( v 3 * , v 0 * ) = J 1 * ( v ^ 3 * , v ^ 0 * ) .
Proof. 
Observe that δ J 1 * ( v ^ 3 * , v ^ 0 * ) = 0 so that, since v ^ 0 * C * and v ^ 3 * C v ^ 0 * , from the results in the previous lines, for a sufficiently small ε 1 > 0 , we have that
J 1 * ( v ^ 3 * , v ^ 0 * ) = inf v 3 * B 1 * J 1 * ( v 3 * , v ^ 0 * ) = sup v 0 * C 1 * J 1 * ( v ^ 3 * , v 0 * ) .
Consequently, from this and the Saddle Point Theorem, we have
J 1 * ( v ^ 3 * , v ^ 0 * ) = inf v 3 * B 1 * sup v 0 * C 1 * J 1 * ( v 3 * , v 0 * ) .
Now we are going to show that
δ J ( u 0 ) = 0 .
Firstly, observe that
F 2 * ( v 3 * , v 0 * ) = sup u V { F 2 ( u , v 3 * , v 0 * ) } .
Denoting
H ( v 3 * , v 0 * , u ) = F 2 ( u , v 3 * , v 0 * ) ,
there exists u ^ V such that
H ( v ^ 3 * , v ^ 0 * , u ^ ) u = 0 ,
and
F 1 * ( v ^ 3 * , v ^ 0 * ) = H ( v ^ 3 * , v ^ 0 * , u ^ ) ,
so that
u 0 2 = F 1 * ( v ^ 3 * , v ^ 0 * ) v 0 * = H ( v ^ 3 * , v ^ 0 * , u ^ ) v 0 * + H ( v ^ 3 * , v ^ 0 * , u ^ ) u u ^ v 0 * = u ^ 2 .
Summarizing, we this last equation is satisfied through the relation
u 0 = u ^ .
Hence from the variation of J 1 * in v 0 * , we obtain
u 0 2 v 0 * α β = 0 ,
so that
v 0 * = α ( u 0 2 β ) .
On the other hand, from the variation of J 1 * in v 3 * , we have
F 1 * ( v ^ 3 * , v ^ 0 * ) v 3 * = K 1 ( v ^ 3 * u 0 K 3 ) u 0 + H ( v ^ 3 * , v ^ 0 * , u ^ ) u u ^ v 3 * = K 1 ( v ^ 3 * u 0 K 3 ) u 0 = 0 .
From such results, since
u 0 0 , a . e . in Ω ,
we get
v ^ 3 * u 0 K 3 = 0 , a . e . in Ω .
Consequently, from such last results and from
u 0 = f + K 1 K 3 v ^ 3 * γ 2 + 2 v ^ 0 * + K 1 ( v ^ 3 * ) 2 ,
we obtain
γ 2 u 0 + 2 v 0 * u 0 + K 1 ( v ^ 3 * ) 2 u 0 f K 1 K 3 v ^ 3 * = γ 2 u 0 + 2 α ( u 0 2 β ) u 0 f = δ J ( u 0 ) = 0 .
Summarizing,
δ J ( u 0 ) = 0 .
Furthermore, also from such last results and the Legendre transform properties, we have
F 1 * ( v ^ 3 * , v ^ 0 * ) = F 1 ( u 0 , v ^ 3 * , v ^ 0 * ) ,
G * ( v ^ 0 * ) = u 0 2 , v 0 * L 2 G ( u 0 2 ) ,
so that
J 1 * ( v ^ 0 * ) = F 1 * ( v ^ 3 * , v ^ 0 * ) G * ( v ^ 0 * ) = J ( u 0 ) .
Finally, observe that
J 1 * ( v 3 * , v 0 * ) F 1 ( u , v 3 * , v 0 * ) G * ( v 0 * ) ,
u V 1 , v 3 * B * , v 0 * C 1 * .
Therefore,
J 1 * ( v ^ 3 * , v ^ 0 * ) sup v 0 * C 1 * { F 2 ( u , v ^ 3 * , v 0 * ) G * ( v 0 * ) } = J ( u ) + K 1 2 Ω ( v ^ 3 * u K 3 ) 2 d x ,
u V 1 .
Summarizing, we have obtained
J ( u 0 ) = inf u V 1 J ( u ) + K 1 2 Ω ( v ^ 3 * u K 3 ) 2 d x = inf v 3 * B 1 * sup v 0 * C 1 * J 1 * ( v 3 * , v 0 * ) = J 1 * ( v ^ 3 * , v ^ 0 * ) .
The proof is complete.

4. Conclusion

In this article we have developed convex 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.

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