A relaxation process for bifunctionals of displacement-Young measure state variables: A model of multi-material with micro-structured strong interface

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A relaxation process for bifunctionals of

displacement-Young measure state variables: A model of multi-material with micro-structured strong interface

Anne-Laure Bessoud, Françoise Krasucki, Gérard Michaille

To cite this version:

Anne-Laure Bessoud, Françoise Krasucki, Gérard Michaille. A relaxation process for bifunctionals

of displacement-Young measure state variables: A model of multi-material with micro-structured

strong interface. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2010, 27 (2),

pp.447-469. �10.1016/j.anihpc.2010.01.007�. �hal-00790256�

A relaxation process for bifunctionals of displacement-Young measure state variables: a

model of multi-material with micro-structured strong interface

Anne Laure Bessoud ^∗ , Fran¸coise Krasucki ^† and G´ erard Michaille ^‡ July 17, 2008

Abstract

The gradient displacement field of a micro-structured strong inter- face of a three-dimensional multi-material is regarded as a gradient-Young measure so that the stored strain energy of the material is defined as a bifunctional of displacement/Young measure state variables. We propose a new model by computing a suitable variational limit of this bifunctional when the thickness and the stiffness of the strong material are of order ε and

¹ε

respectively. The stored strain energy functional associated with the model in pure displacements living in a Sobolev space is obtained as the marginal map of the limit bifunctional. We also obtain a new asymptotic formulation in terms of Young measure state variable when considering the other marginal map.

AMS subject classifications: 49J45, 74N15, 35B40

Keywords: Micro structures, Young measures, Variational convergences, Γ- convergence.

∗

LMGC, UMR-CNRS 5508 and ACSIOM, UMR-CNRS 5149, Universit´ e Montpellier II, Case courier 048, Place Eug` ene Bataillon, 34095 Montpellier Cedex 5, [email protected] montp2.fr

†

ACSIOM, UMR-CNRS 5149, Universit´ e Montpellier II, Case courier 051, Place Eug` ene Bataillon, 34095 Montpellier Cedex 5, [email protected]

‡

ACSIOM and AVA, UMR-CNRS 5149, Universit´ e Montpellier II et Universit´ e de Nˆımes,

Case courier 051, Place Eug` ene Bataillon, 34095 Montpellier Cedex 5, [email protected]

montp2.fr

1 Introduction

In [1] and [10] a variational model of multi-material with a very rigid interface is obtained by identifying the classical Γ-limit of the stored strain energy func- tional when the magnitude order ε of the interface thickness goes to zero and the stiffness of the material occupying the interface grows as ¹ _ε . In this paper we assume that the thin structure is occupied by a material which undergoes reversible solid/solid phase transformation, while the strain of the soft material occupying the complementary set of the layer can be high. As the main me- chanical features are high strain of the soft material and oscillations of gradient displacement in the layer of hight stiffness, we deal with the asymptotic analysis of the problem by means of a new variational convergence where competing ob- jects are pairs (u, ν) of displacements/gradient Young measures state variables.

The advantage of using the second argument lies in the fact that ν encodes the gradient oscillations of u restricted to the layer. We obtain a new formu- lation (¯ u, ν ¯ ) ∈ argmin F (u, ν) − L(u) of the problem by identifying the limit (u, ν) 7→ F (u, ν) of the stored strain energy functional u 7→ F _ε (u) rewritten as a bifunctional (u, µ) 7→ F ε (u, µ) (we write L(u) for the exterior loading).

Let Ω be the reference configuration occupied by the material and S×]0, ε[

the thin inclusion. The limit energy functional of F ε obtained in [10] is of the form u 7→ F (u) := R

Ω Qf (∇u) dx + R

S Qg 0 ( ˆ ∇γ S (u)) dˆ x for all Sobolev- functions u with smooth trace γ S (u) on the two-dimensional interface S, where Qf and Qg 0 denote the quasiconvexifications of f and g 0 . As a straightforward consequence of our formulation we find the stored strain energy F as to be the marginal map u 7→ inf ν F(u, ν) of the energy functional F when the Young measure ν = ν x ˆ ⊗dˆ x is then regarded as an internal state variable. By comparing the two variational formulations ¯ u ∈ argmin (F −L) and (¯ u, ν ¯ ) ∈ argmin (F −L), we obtain an integral representation with respect to the probability measure ¯ ν ˆ x

on the set M ^3×2 of 3 × 2-matrices, of the significant macroscopic quantities

∇γ ˆ S (¯ u) and Qg 0 ( ˆ ∇γ S (¯ u)). In some sense we may think the variable ¯ ν as the microscopic description of ˆ ∇γ S (¯ u) and Qg 0 ( ˆ ∇γ S (¯ u)).

Another way for obtaining a variational formulation of the problem is to consider the marginal map G of F − L when the displacement field u is now regarded as an internal variable. We show that the energy functional ν 7→

G(ν) := inf u (F(u, ν) − L(u)) is a variational limit of µ 7→ inf u (F ε (u, µ) − L(u)) so that ¯ ν ∈ argmin G is a new formulation of the problem in terms of gradient Young measures parametrized on the interface S. By comparing it with the formulation (¯ u, ¯ ν) ∈ argmin (F − L), we show that ¯ u is a solution of the nonlinear Dirichlet problem min( R

Ω\S Qf (∇u) dx − L(u)) subjected to the boundary condition ¯ u(ˆ x) = ˆ ∇

⁻¹

( R

M^3×2

ˆ λd¯ ν x ˆ ) on the interface S. Consequently, one may think the surface energy R

S Qg 0 ( ˆ ∇γ S (u)) dˆ x obtained in [10] as a relaxation of the boundary condition above (it should be noted the analogy with the relaxation of boundary conditions in BV -spaces).

This paper illustrates, in the modeling of multi-materials, the following gen-

eral strategy: in order to capture various convergence phenomena on minimizing

sequences regarded as Sobolev variables of a problem (P ε ), one defines a suit- able measure state variable µ connected to the Sobolev state variable u (Young measure, concentration measure....), an energy bifunctional (u, µ) 7→ F ε (u, µ) modeling (P ε ), a suitable variational convergence process, and identify its limit F. We recover the limit energy in terms of Sobolev variables as the marginal functional of F when µ is regarded as an internal variable. This idea as al- ready been used in the framework of relaxation theory for a reduction dimen- sion problem in [12], and for control problems in [22]. We also obtain a new asymptotic formulation in terms of measure state variable by considering the marginal functional of F when u is an internal variable. Here, in the context of asymptotic analysis, we have to first establish a variational convergence of the two marginals.

The paper is organized as follows. Section 2 provides the abstract setting of the problem. We introduce a suitable variational convergence for bifunctionals and establish the variational convergence of their marginal maps. In Section 3, after a brief exposition of the mechanical setting, according to Section 2, we set up notation and terminology for the problem and prove the variational convergence of the bifunctional F ε to the bifunctional F (Theorem 2). The two last sections are devoted to the asymptotic analysis of the marginal maps and their consequences (Theorem 3, Corollaries 1 and 2). For making the paper as self contained as possible, we repeat the material from Young measures without proofs in Appendix 6.

2 A variational convergence of sequences of func- tionals defined on topological product spaces

2.1 The abstract setting

Given three first countable topological spaces, X, Y , ˆ Y , a map A from Y to ˆ Y , and extended real-valued functionals F n : X × Y −→ R ∪ {+∞}, F : X × Y ˆ −→

R ∪ {+∞}, our purpose is to define a variational convergence of the sequence (F n ) n∈

N

toward the functional F so that, under a suitable convergence process associated with A and some compactness hypotheses, the following implication holds true when n −→ +∞:

F n −→ F ⇒







inf X×Y F n −→ min _X× Y ˆ F ;

arg min X×Y F n ∋ (x n , y n ) −→ (x, y) ˆ ∈ arg min _X× Y ˆ F at least for a subsequence.

We begin by introducing a new weak notion of convergence between elements of Y and ˆ Y , and next, between elements of X × Y and X × Y ˆ .

Definition 1. Let (y n ) n∈

N

be a sequence in Y and y ˆ in Y ˆ . We say that y n

A-converges to ˆ y and we write y n

⇀

A

y ˆ

iff there exists y in Y such that y _n → y and y ˆ = A(y).

Let ((x n , y _n )) n∈

N

be a sequence in X × Y and (x, y) ˆ in X × Y ˆ . We say that (x n , y _n ) converges to (x, y) ˆ and we write

(x n , y _n ) ^I×A ⇀ (x, y) ˆ iff x n converges to x, and y n A-converges to y. ˆ

We introduce now the variational convergence associated with the previous convergence.

Definition 2. We say that F n Γ _X,Y, Y ˆ -converges to F and we write F n

Γ

_X,Y,Yˆ

−→ F

iff for all (x, y) ˆ in X × Y ˆ , both following assertions hold:

(i) ∀(x n , y n ) ∈ X × Y s.t. (x n , y n ) ^I ⇀

^×A

(x, y), ˆ F (x, y) ˆ ≤ lim inf

n→+∞ F n (x n , y n ), (ii) ∃(x n , y _n ) ∈ X × Y s.t. (x n , y _n ) ^I×A ⇀ (x, y), ˆ F(x, y) ˆ ≥ lim sup

n→+∞ F n (x n , y _n ).

Note that this convergence is closely related to the Γ-convergence. When X = {0} or, which is equivalent, when F n and F do not depend on x, we denote it briefly by Γ _Y, Y ˆ . When Y = {0} and ˆ Y = {ˆ0} i.e. when F n and F do not depend on y and ˆ y, we will write it simply Γ X and our definition agrees with the classical Γ-convergence. Note also that when Y = ˆ Y and A is the identity map, Γ X,Y,Y is the Γ X×Y -convergence. The proposition below expresses the variational nature of the Γ _X,Y, Y ˆ -convergence.

Proposition 1. Let us assume that (F n ) n∈

N

Γ _X,Y, Y ˆ -converges to F and let ((x n , y n )) n∈

N

be a sequence of X × Y satisfying

F n (x n , y n ) ≤ inf

(x,y)∈X×Y F n (x, y) + 1 n .

Assume furthermore that {(x n , y n ) : n ∈ N } is relatively compact for the conver- gence ^I×A ⇀ defined above. Then any cluster point (¯ x, y) ¯ˆ ∈ X × Y ˆ is a minimizer of F and

n→+∞ lim inf{F n (x, y) : (x, y) ∈ X × Y } = F(¯ x, y) ¯ˆ

Proof. The proof is similar to that of Theorem 12.1.1 in [6] and left to the reader.

2.2 The variational convergence of marginal functionals

Let us consider the following marginal functionals F _n , F : X −→ R ∪ {+∞}, G _n : Y −→ R ∪ {+∞}, and G : ˆ Y −→ R ∪ {+∞} defined by :

F n (x) = inf

y∈Y F n (x, y) , F(x) = inf

ˆ y∈ Y ˆ

F(x, y) ˆ G n (y) = inf

x∈X F n (x, y) , G(ˆ y) = inf

x∈X F(x, y) ˆ

The variational convergence of the functionals F n yields the variational conver- gence of their marginal maps, precisely:

Theorem 1. Let us assume that (F n ) n∈

N

Γ _X,Y, Y ˆ -converges to F. Assume fur- thermore that the following inf-compactness property holds: for every sequence ((x n , y n )) n∈

N

satisfying sup _n∈

N

F n (x n , y n ) < +∞, there exists a subsequence (x σ(n) , y _σ(n) ) in X × Y and (x, y) ˆ in X × Y ˆ such that (x σ(n) , y _σ(n) ) ^I×A ⇀ (x, y). ˆ Then

(i) F n Γ

X

−→ F (ii) G n

Γ

_Y,Yˆ

−→ G

Proof. Proof of assertion (i). On account of Theorem 12.1.1 in [6], we are going to establish that for any subsequence of F n , one can extract a subsequence which Γ X -converges to F . Let (x n ) n∈

N

be a sequence converging to x in X and consider a sequence (y n ) n∈

N

in Y such that

F n (x n , y n ) − 1 n ≤ inf

y∈Y F n (x n , y) := F n (x n ). (1) We can assume that sup _n∈

N

F n (x n ) < +∞ (otherwise there is nothing to prove) so that sup _n∈

N

F n (x n , y n ) < +∞. Thus, according to the inf-compactness as- sumption, there exist a subsequence (x σ(n) , y σ(n) ) and ˆ y in ˆ Y such that x σ(n)

converges to x, and y σ(n) A-converges to ˆ y. Furthermore, since (F n ) n∈

N

Γ _X,Y, Y ˆ - converges to F, one has

F (x, y) ˆ ≤ lim inf

n→+∞ F σ(n) (x σ(n) , y σ(n) ).

Since from (1) one has F _σ(n) (x σ(n) ) ≥ F σ(n) (x σ(n) , y _σ(n) ) − _σ(n) ¹ , we deduce F(x) ≤ F(x, y) ˆ ≤ lim inf

n→+∞ F σ(n) (x σ(n) ).

From now on, to shorten notation, we write n instead of σ(n). Let (ˆ y p ) p∈

N

be a sequence in ˆ Y satisfying

F (x) = inf

ˆ y∈ Y ˆ

F(x, y) = lim ˆ

p→+∞ F(x, y ˆ p ). (2)

Combining (i) and (ii) of Definition 2, for every fixed p there exists a sequence ((x ^p _n , y p,n )) n∈

N

in X × Y such that

( x ^p _n → x y p,n

⇀

A

y ˆ p

and satisfying

F(x, y ˆ _p ) = lim

n→+∞ F n (x ^p _n , y _p,n ). (3)

From (2) and (3), we obtain F(x) = lim

p→+∞ lim

n→+∞ F n (x ^p _n , y p,n ).

Then, using a standard diagonalization argument, there exists a map n 7→ p(n) satisfying p(n) → +∞ whenever n → +∞ for which one has

F(x) = lim

n→+∞ F n (x ^p(n) _n , y _p(n),n ) ≥ lim sup

n→+∞

F _n (x ^p(n) _n ).

The sequence defined by x _n = x ^p(n) n then satisfies assertion (ii) of Definition 2 and the proof of (i) is complete. The proof of assertion (ii) is very similar and left to the reader.

2.3 A concrete example

In this section, we present a concrete example entering within the general frame- work described above. It is the main subject of the paper which will be treated in details in the next section. We will deal with a second example in a forthcoming paper where we will take into account the concentration gradient phenomenon.

For the analysis of concentration effects we refer the reader to [18] and [25].

We denote the sets of 3 × 3 and 3 × 2 matrices with real numbers entries by M ^3×3 and M ^3×2 respectively. Considering the space M ^3×3 as the product M ^3×2 × R ³ , we will denote by ˆ λ the first coordinate in M ^3×2 of any element λ of M ^3×3 . We write P

_M^3×2

to denote the projection mapping M ^3×3 to M ^3×2 . In what follows, we use notation of Theorem 4 in Appendix 6.

Let Ω, B be two open bounded subsets of R ³ , and S ⊂ R ² such that B = S ×(0, 1), we define the sets of Young measures Y 3×3 (B) and Y 3×2 (S) as follow:

µ ∈ Y 3×3 (B) ⇔ µ ∈ M ⁺ (B × M ^3×3 ) and P B #µ = L, ν ∈ Y 3×2 (S) ⇔ ν ∈ M ⁺ (S × M ^3×2 ) and P _S #ν = ˆ L

where P B #µ (resp. P S #ν) denotes the image of the measure µ (resp. ν) by the projection P _B : B × M ^3×3 −→ B (resp. P _S : S × M ^3×2 −→ S) and L (resp.

L) the Lebesgue measure on ˆ B (resp. S). For every probability measure P on M ^3×3 or M ^3×2 , we write bar(P) for its barycenter, i.e. bar(P) = R

λ dP(λ).

The map A is defined by:

A : Y 3×3 (B) −→ Y 3×2 (S) µ = µ x ⊗ dx 7→ ν = Z 1

0

ˆ µ x,s ˆ ds

⊗ dˆ x where ˆ µ _x = P

_M^3×2

#µ x , x ∈ Ω, and R 1

0 µ ˆ ˆ x,s ds is the probability measure param- eterized by ˆ x ∈ S, which acts on all ϕ ∈ C 0 (M ^3×2 ) as follows:

h Z 1

0

ˆ

µ ˆ x,s ds , ϕi :=

Z 1 0

Z

M^3×2

ϕ(ˆ λ)dˆ µ ˆ x,s ds.

Given a sequence (µ ε ) ε>0 in the space Y 3×3 (B) equipped with the narrow convergence (see Appendix 6) and ν in Y 3×2 (S), according to Definition 1 one has

µ ε

⇀ ν

A

⇔ ∃µ ∈ Y 3×3 (B) s.t.

( µ ε nar ⇀ µ ν = A(µ)

and, for (v ε , µ ε ) in L ^p (Ω, R ³ ) × Y 3×3 (B), and (v, ν) in L ^p (Ω, R ³ ) × Y 3×2 (S):

(v ε , µ _ε ) ^I×A ⇀ (v, ν) ⇔

( v _ε → v in L ^p (Ω, R ³ ) µ ε

⇀ ν.

A

Given two locally Lipschitz functions f, g : M ^3×3 → R ⁺ satisfying a growth condition of order p, we consider the integral functional F ε defined by

F ε : L ^p (Ω, R ³ ) × Y 3×3 (B) −→ R ∪ {+∞}

F ε (u, µ) :=



 



 

 Z

Ω

ε

f (∇u) dx + Z

B×M

^3×3

g(ˆ λ| 1

ε λ 3 ) dµ + I _ε (u, µ)

if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ), +∞ otherwise,

where

I _ε (u, µ) :=

( 0 if µ = δ

_∇v(x)

⊗ dx, v = 1

B

r _ε u +∞ otherwise.

and r ε u is defined by r ε u(ˆ x, x 3 ) = u(ˆ x, εx 3 ).

Let ˆ ∇Y 3×2 (S) denote the subset of Y 3×2 (S) made up of all Young measures generated by gradients of W ^1,p (S, R ³ )-Sobolev functions, γ S the trace operator from W ^1,p (Ω \ S, R ³ ) into L ^p (S, R ³ ) and define the function g 0 : M ^3×2 → R for all ˆ λ in M ^3×2 by g 0 (ˆ λ) := inf _ξ∈

R3

g(ˆ λ|ξ). Setting X = L ^p (Ω, R ³ ), Y = Y 3×3 (B) and ˆ Y := Y 3×2 (S), in the next section we prove that the sequence (F ε ) ε>0

Γ _X,Y, Y ˆ -converges to the functional F, defined by

F : L ^p (Ω, R ³ ) × Y 3×2 (S) −→ R ∪ {+∞}

F(u, ν) :=



 



 

 Z

Ω

Qf (∇u) dx + Z

S×

M^3×2

g 0 (ˆ λ) dν + I(u, ν) if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ) γ S u ∈ W ^1,p (S, R ³ ), +∞ otherwise,

where

I(u, ν) :=

( 0 if ν ∈ ∇Y ˆ 3×2 (S), bar(ν x ˆ ) = ˆ ∇γ S (u)(ˆ x) a.e in S.

+∞ otherwise.

Theorem 1 applied to the sequence of the two marginal maps sheds new light

on the mechanical problem.

3 A model in terms of displacement/Young mea- sures: analysis of microstructures of the strong material

In the three dimensional Euclidean space E ³ referred to the orthonormal frame (0; e 1 , e 2 , e 3 ), we consider a domain Ω with a C ¹ boundary Γ. Let Ω

^±

= Ω ∩ [±x 3 > 0], the interior S = {∂Ω ⁺ ∩ ∂Ω

⁻

}

^◦

of the common part of the boundaries of Ω

^±

is assumed to have a positive H ² -measure and, to shorten the proofs, included in the plane [x 3 = 0]. The set Ω is the physical reference configuration of the assembly of two materials. More precisely, given a small dimensionless parameter ε and a global characteristic length h (for example the diameter of Ω), the set B ε = {x + εz : 0 < z < h, x ∈ S} is the reference configuration of a strong material (whose stiffness is of order ¹ _ε ) while Ω ε = Ω \ B ε is the reference configuration of a material with stiffness of order 1. (see figure 1).

The structure is clamped on a part Γ 0 of Γ with a positive H ² -measure, the complementary part Γ ψ of Γ 0 is submitted to surface loads ψ and we assume that H ¹ (Γ 0 ∩ S) ¯ > 0. Obviously one can there consider other type of boundary conditions (e.g. a combination of some components of the stress vector and of the displacement). Moreover the structure is submitted to applied body forces Φ.

S Ω⁺ǫ

ǫh Γ0

Γϕ

Ω⁻ǫ

Bǫ B h= 1

S

Ω⁺ Γ₀

Γϕ

Ω⁻

Figure 1: Bonded assembly - Left: the physical configuration-Right: the rescaled

layer - Below: the limit configuration.

Let p ≥ 1, we say that a Borel function W : M ^3×3 → R satisfies a (C p ) condition if

∃α, β, C ∈ R ⁺ s. t. ∀(λ, λ

^′

) ∈ M ^3×3 , α|λ| ^p ≤ W (λ) ≤ β(1 + |λ| ^p )

|W (λ) − W (λ

^′

)| ≤ C|λ − λ

^′

|(1 + |λ| ^p−1 + |λ

^′

| ^p−1 ) (4) We say that a quasiconvex function φ : M ^3×3 → R (resp φ : M ^3×2 → R ) satisfies a growth condition of order p if there exists γ ≥ 0 such that

∀λ ∈ M ^3×3 , |φ(λ)| ≤ γ(1 + |λ| ^p ) (resp ∀ λ ˆ ∈ M ^3×2 , |φ(ˆ λ)| ≤ γ(1 + | λ| ˆ ^p )).

The soft and the strong materials are modeled as hyperelastic and the bulk energy densities f , g of the two materials occupying Ω ε and B _ε satisfy a (C p ) condition with p > 1. To shorten notation, we assume that (C p ) is satisfied with the same constants α, β and C. We make the assumption that the strain of the soft material can be high and that the thin structure B ε is occupied by a material which undergoes reversible solid-solid phase transformation as for instance crystalline solids. In this context, the densities f and g are not convex and g entails a multi-well structure. It is worth pointing out that the assumed growth condition violates the mechanical principle which asserts that it needs infinite amount of energy to squeeze a small piece of material down to a point.

We also do not take into account preservation of orientation and injectivity conditions on the deformation fields so that the model presented in this section is a first attempt to account large purely elastic deformation. We hope to deal with this much more complex situation in a future work (for some results where these constraints are taken into account, we refer the reader to [20], [3, 4]).

We assume that the global characteristic length h is equal to 1 and the stored strain energy associated with a displacement field u is given by the functional

F _ε (u) :=

Z

Ω

ε

f (∇u) dx + 1 ε

Z

B

ε

g(∇u) dx.

The equilibrium configuration of the structure is given by the displacement field u _ε , solution-more generally ε-approximate solution-of the problem

inf{F ε (u) − L(u) : u ∈ W _Γ ^1,p

₀

(Ω, R ³ )}

where L(u) = Z

Ω

Φ dx + Z

Γ

ψ

ψ dH ² . We want to analyze the behavior of u _ε when ε tends to zero and to identify the variational problem whose limit is a solution. But since the material in the layer B _ε possesses a fine micro- structure, the gradient minimizing sequence (∇u ε ) ε>0 develops oscillations we would like to integrate into the variational problem. This is why we write the strain energy ¹ _ε R

B

ε

g(∇u) dx in terms of Young measures so that the limit

problem also accounts for a two-dimensional microstructure (for existence of

microstructures see [9] and for microstructures in thin films, we refer the reader

to [11], [17], [21] and references therein).

Since the behavior of the displacement is radically different in Ω ε and B _ε , in a first stage, it is convenient to write the energy functional F _ε in terms of two arguments, one u, the displacement on Ω ε , the other v, the displacement on B _ε occupied by the strong material. On the other hand, in order to work in a fixed space for the variable v, the change of scale (ˆ x, x 3 ) = (ˆ x, εy 3 ) transforming (ˆ x, x 3 ) ∈ B ε into (ˆ x, y 3 ) ∈ B := S × (0, 1) leads to consider the following functional

G ε : L ^p (Ω, R ³ ) × L ^p (B, R ³ ) −→ R ∪ {+∞}

G ε (u, v) :=





 Z

Ω

ε

f (∇u) dx + Z

B

g( ˆ ∇v| 1 ε

∂v

∂x 3

) dx + I ε (u, v) if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ), +∞ otherwise,

where, for all (u, v) ∈ W _Γ ^1,p

₀

(Ω, R ³ ) × W ^1,p (B, R ³ ) I ε (u, v) :=

( 0 if 1

_B

r ε u = v +∞ otherwise.

and r ε u(ˆ x, y 3 ) := u(ˆ x, εy 3 ). Now we write G ε in terms of pairs of displace- ments/Young measures by defining the functional F ε as follows

F ε : L ^p (Ω, R ³ ) × Y 3×3 (B) −→ R ∪ {+∞}

F ε (u, µ) :=





 Z

Ω

ε

f (∇u) dx + Z

B×M

^3×3

g(ˆ λ| 1

ε λ 3 ) dµ + I _ε (u, µ) if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ), +∞ otherwise,

where

I _ε (u, µ) :=

( 0 if µ = δ

_∇v(x)

⊗ dx, v = 1

B

r _ε u +∞ otherwise.

Clearly F ε is one way of writing G ε and from the strict variational point of view, it is equivalent to identify the variational limit of F _ε and that of F ε in the spirit of the previous section. Indeed we have

inf

u∈W

^1,p

Γ0

(Ω,

R³

)

F ε (u) − L(u)

= inf

(u,v)∈L

^p

(Ω,

R³

)×L

^p

(B,

R³

)

G ε (u, v) − L(u)

= inf

(u,µ)∈L

^p

(Ω,

R³

)×Y

3×3

(B)

F ε (u, µ) − L(u) . Nevertheless, we want to point out that the last formulation has the advantage to encode the gradient oscillations of ε-minimizers in the layer B _ε thanks to the Young measure state variable.

In order to apply Proposition 1 we begin by establishing the following com- pactness lemma

Lemma 1 (Compactness). Let ((u ε , v _ε , µ _ε )) ε>0 be a sequence in L ^p (Ω, R ³ ) × L ^p (B, R ³ ) × Y 3×3 (B) satisfying sup _ε>0 G ε (u ε , v _ε ) = sup _ε>0 F ε (u ε , µ _ε ) < +∞.

Then there exist (u, v) ∈ W _Γ ^1,p

₀

(Ω, R ³ ) × W ^1,p (B, R ³ ), ν ∈ ∇Y ˆ 3×2 (S ) and a

subsequence not relabeled such that

i) u _ε → u weakly in W _Γ ^1,p

₀

(Ω, R ³ ) and strongly in L ^p (Ω, R ³ ), v _ε → v weakly in W ^1,p (B, R ³ ) and strongly in L ^p (B, R ³ );

ii) ^∂v _∂y

^ε₃

→ 0 strongly in L ^p (B, R ³ ), _∂y ^∂v

₃

= 0 and v ∈ W ^1,p (S, R ³ );

iii) γ S (u) = v on S where γ S denotes the trace operator from W ^1,p (Ω \ S, R ³ ) into L ^p (S);

iv) (u ε , µ ε ) ^I×A ⇀ (u, ν), with ν = A(µ) where µ = (ˆ µ x ⊗ δ 0

_R3

) ⊗ dx and µ ˆ :=

ˆ

µ x ⊗ dx is the Young measure generated by ( ˆ ∇v ε ) ε>0 . Moreover u and ν are connected as follows: bar(ν x ˆ ) = ˆ ∇γ S (u)(ˆ x) for a.e. x ˆ ∈ S.

Proof. Assertions i) and ii) are straightforward consequences of the coerciveness condition in (4), Poincar´e’s inequality and the isometry between W ^1,p (S, R ³ ) and {v ∈ W ^1,p (B, R ³ ) : _∂y ^∂v

3

= 0}. We are going to establish assertion iii). For a. e. x ∈ B we have

v ε (ˆ x, y 3 ) = u ε (ˆ x, εy 3 ) = u ε (ˆ x, 0) + Z εy

3

0

∂u ε

∂y 3

(ˆ x, s) ds

where u ε (ˆ x, 0) must be taken in the trace sense. Let Q ρ (ˆ x 0 ) = ˆ Q ρ (ˆ x 0 ) × (0, ρ) be the cylinder of R ³ where ˆ Q ρ (ˆ x 0 ) is the ball of R ² centered at ˆ x 0 ∈ S with ρ > 0 small enough so that Q ρ (ˆ x 0 ) ⊂ B. Integrating the previous equality over Q ρ (ˆ x 0 ) yields

− Z

Q

ρ

(ˆ x

0

)

v ε dx = − Z

Q

ρ

(ˆ x

0

)

u ε (ˆ x, 0) dx + − Z

Q

ρ

(ˆ x

0

)

Z εy

3

0

∂u ε

∂y 3

(ˆ x, s) ds dx.

Letting ε → 0, according to the continuity of the trace operator and to i), ii), we obtain

− Z

Q ˆ

ρ

(ˆ x

0

)

v dˆ x = − Z

Q ˆ

ρ

(ˆ x

0

)

γ S (u) dˆ x + lim sup

ε→0

I ρ,ε (5)

where I ρ,ε := − Z

Q

ρ

(ˆ x

0

)

Z εy

3

0

∂u ε

∂y 3

(ˆ x, s) ds dx. Let us estimate I ρ,ε . An easy calculation using H¨ older’s inequality gives

|I ρ,ε | ≤ − Z

Q

ρ

(ˆ x

0

)

(εy 3 ) ¹⁻

^1p

Z εy

3

0

| ∂u ε

∂y 3 (ˆ x, s)| ^p ds

¹_p

dˆ x dy 3

≤ (ερ) ¹⁻

^1p

− Z

Q ˆ

ρ

(ˆ x

0

)

Z ερ 0

| ∂u ε

∂y 3

(ˆ x, s)| ^p ds

¹_p

dˆ x

≤ (ερ) ¹⁻

^1p

Z

Q ˆ

ρ

(ˆ x

0

)

Z ερ 0

| ∂u ε

∂y 3

(ˆ x, s)| ^p ds dˆ x

¹_p

≤ (ερ) ¹⁻

^1p

Z

B

ε

|∇u ε | ^p dx

_p¹

.

But since sup _ε>0 G ε (u ε , v _ε ) = sup _ε>0 F _ε (u ε ) < +∞, from the coerciveness prop- erty satisfied by g one has

Z

B

ε

|∇u ε | ^p dx ≤ ε

α so that the previous estimate yields |I ρ,ε | ≤ C(ρ, α)ε where C(ρ, α) is a positive constant depending only on ρ and α. From this estimate, (5) becomes

− Z

Q ˆ

ρ

(ˆ x

0

)

v dˆ x = − Z

Q ˆ

ρ

(ˆ x

0

)

γ S (u) dˆ x

for all ρ > 0. Letting ρ → 0 finally gives γ _S (u)(x 0 ) = v(x 0 ) for a.e. x 0 in S.

It remains to establish assertion iv). Since F ε (u ε , µ ε ) < +∞ we have µ ε = δ

_∇vε

_(x) ⊗ dx and sup _ε>0 R

B |∇v ε | ^p dx < +∞ so that the Young measures µ ε

and δ

∇v

ˆ

ε

(x) ⊗ dx are tight. According to the Prokhorov compactness theorem (Theorem 5 of Appendix 6), there exist a subsequence that we do not relabel, and µ ∈ ∇Y 3×3 (B), ˆ µ ∈ Y 3×2 (B) such that

µ ε = δ

∇vε

(x) ⊗ dx

^nar

⇀ µ, δ

∇v

ˆ

ε

(x) ⊗ dx

^nar

⇀ µ. ˆ (6) On the other hand, from assertion ii)

δ

∂vε

∂x3

(x) ⊗ dx

^nar

⇀ δ 0

_R3

⊗ dx. (7)

Combining (6) and (7) one easily deduce that µ x = ˆ µ x ⊗ δ 0

_R3

, and setting ν = ν x ˆ ⊗ dˆ x where ν x ˆ := R 1

0 µ ˆ ˆ x,s ds, we finally obtain that µ _ε ⇀ ν.

^A

We have to prove that ν belongs to ˆ ∇Y 3×2 (S). According to the Kinderlehrer- Pedregal characterization theorem (Theorem 6 of Appendix 6), it is equivalent to establish the three following assertions:

(KP) 1 there exists w ∈ W ^1,p (S, R ³ ) such that bar(ν x ˆ ) = ˆ ∇w(ˆ x) for a.e. x ∈ S;

(KP) 2

Z

M^3×2

| λ| ˆ ^p dν x ˆ < +∞ for a.e. x ∈ S;

(KP) 3 For all quasiconvex function φ satisfying a growth condition of order p, φ(bar(ν x ˆ )) ≤

Z

M^3×2

φ(ˆ λ) dν x ˆ for a.e. ˆ x ∈ S.

Proof of (KP) 1 : From assertion i) and classical properties on Young mea- sures, we have ˆ ∇v(x) = R

M^3×2

λ d ˆ µ ˆ x for a.e. x ∈ B and since _∂x ^∂v

3

= 0,

∇v(ˆ ˆ x) = Z 1

0

Z

M^3×2

λ d ˆ µ ˆ x,s ˆ ds = Z

M^3×2

λ dν ˆ x ˆ

for a.e. ˆ x in S so that v is the suitable Sobolev-function w.

Proof of (KP) 2 : From the definition of ν x ˆ and the lower semicontinuity

property for Young measures (Proposition 5 of Appendix 6) we have Z

S

Z

M^3×2

| λ| ˆ ^p dν x ˆ dˆ x = Z

B

Z

M^3×2

| λ| ˆ ^p dˆ µ x dx

= Z

B

Z

M^3×3

| λ| ˆ ^p (dˆ µ x ⊗ δ 0

_R3

) dx

= Z

B×

M^3×3

| ˆ λ| ^p dµ ≤ Z

B×

M^3×3

|λ| ^p dµ

≤ lim inf

ε→0

Z

B

|∇v ε | ^p dx ≤ sup

ε>0 F ε (u ε , µ ε ) < +∞

which proves that R

M^3×2

| λ| ˆ ^p dν x ˆ is finit for a.e. ˆ x in S.

Proof of (KP) 3 : Let φ : M ^3×2 → R be a quasiconvex function satisfying a growth condition of order p and define the function ˜ φ : M ^3×3 → R by ˜ φ(λ) = φ(ˆ λ). It is easy to check that ˜ φ is quasiconvex and clearly satisfies the same growth condition. Since µ ∈ ∇Y 3×3 (B) we have for a.e. x in B

φ( ˆ ∇v(ˆ x)) = ˜ φ(∇v(x)) ≤ Z

M^3×3

φ(λ) ˜ dµ _x

= Z

M^3×3

φ(ˆ λ) dˆ µ _x ⊗ δ 0

R3

= Z

M^3×2

φ(ˆ λ) dˆ µ _x so that for a.e. x in S

φ( ˆ ∇v(ˆ x)) = Z 1

0

φ( ˆ ∇v(ˆ x)) ds ≤ Z 1

0

Z

M^3×2

φ(ˆ λ) dˆ µ ˆ x,s ds = Z

M^3×2

φ(ˆ λ) dν ˆ x . Remark 1. In the proof above we established A(∇Y 3×3 (B)) ⊂ ∇Y ˆ 3×2 (S). In fact it is easy to check that A(∇Y 3×3 (B)) = ˆ ∇Y 3×2 (S).

Consider the functional defined in Section 2.3:

F : L ^p (Ω, R ³ ) × Y 3×2 (S) −→ R ∪ {+∞}

F(u, ν) :=



 



 

 Z

Ω

Qf (∇u) dx + Z

S×M

^3×2

g 0 (ˆ λ) dν + I(u, ν) if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ) γ S u ∈ W ^1,p (S, R ³ ), +∞ otherwise.

We establish the Γ _X,Y, Y ˆ -convergence of the sequence (F ε ) ε>0 to the functional F by means of the two next propositions.

Proposition 2 (Lower bound). Let (u, ν) be any pair in L ^p (Ω, R ³ ) × Y 3×2 (S).

Then, for all sequence ((u ε , µ ε )) ε>0 in L ^p (Ω, R ³ )×Y 3×3 (B) converging to (u, ν), we have

F(u, ν) ≤ lim inf

ε→0 F ε (u ε , µ _ε ).

Proof. One can assume lim inf ε→0 F ε (u ε , µ _ε ) < +∞ otherwise there is nothing to prove. Consequently, from Lemma 1 we have

u ∈ W _Γ ^1,p

₀

(Ω, R ³ );

ν ∈ ∇Y ˆ 3×2 (S); bar(ν x ˆ ) = ˆ ∇γ S (u)(ˆ x) for a.e. ˆ x ∈ S.

This proves that I(u, ν) = 0 and it suffices to establish the two following esti-

mates: Z

S×M

^3×2

g 0 (ˆ λ) dν ≤ lim inf

ε→0

Z

B

g( ˆ ∇v ε | 1 ε

∂v ε

∂x 3 ) dx; (8) Z

Ω

Qf(∇u) dx ≤ lim inf

ε→0

Z

Ω

ε

f (∇u ε ) dx. (9) Proof of (8): From the lower semicontinuity property for Young measures (see Proposition 5 of Appendix) and assertion iv) of Lemma 1, it follows that lim inf

ε→0

Z

B×

M^3×3

g(ˆ λ| 1

ε λ 3 ) dµ ε ≥ lim inf

ε→0

Z

B×

M^3×3

g 0 (ˆ λ) dµ ε

≥ Z

B×

M^3×3

g 0 (ˆ λ) dµ

= Z

B×

M^3×3

g 0 (ˆ λ) d(ˆ µ x ⊗ δ ₀

³

R

) ⊗ dx

= Z

S×M

^3×2

g 0 (ˆ λ) dν ˆ x ⊗ dˆ x = Z

S×M

^3×2

g 0 (ˆ λ) dν.

Proof of (9): For fixed η > ε we have Z

Ω

ε

f (∇u ε ) dx ≥ Z

Ω

η

f (∇u ε ) dx ≥ Z

Ω

η

Qf (∇u ε ) dx and, since w 7→

Z

Ω

η

Qf (∇w) dx is lower semicontinuous for the weak conver- gence in W ^1,p (Ω η , R ³ ), we deduce

lim inf

ε→0

Z

Ω

ε

f (∇u ε ) dx ≥ Z

Ω

η

Qf (∇u) dx.

We end the proof by letting η → 0.

For establishing the upper bound in the definition of our variational conver- gence, we need to prove the following relaxation result

Lemma 2. Let (u, ν) in W _Γ ^1,p

₀

(Ω, R ³ ) × ∇Y ˆ 3×2 (S) with ∇γ ˆ S (u)(ˆ x) = bar(ν x ˆ ) for a.e. x ˆ in S and a sequence ((u n , v n )) n∈

N

in W _Γ ^1,p

₀

(Ω, R ³ ) × W ^1,p (S, R ³ ) such that u n weakly converges to u in W _Γ ^1,p

₀

(Ω, R ³ ), δ

∇v

ˆ

n

(ˆ x) ⊗ dx ⇀ ν

^nar

in Y 3×2 (S) and

n→+∞ lim Z

Ω

f (∇u n ) dx = Z

Ω

Qf (∇u) dx

n→+∞ lim Z

S

g 0 ( ˆ ∇v n ) dˆ x = Z

S×

M^3×2

g 0 (ˆ λ) dν.

(10)

Then there exists a sequence (˜ u _n ) n∈

N

satisfying all the conditions fulfilled by (u n ) n∈

N

and furthermore which satisfies γ _S (˜ u _n ) = v _n .

Proof. Such a sequence ((u n , v _n )) n∈

N

exists, consult for instance [6], Theorem 11.2.1 and Theorem 11.4.2. Note that one may assume (|∇u n | ^p ) n∈

N

uniformly integrable. Indeed, consider the sequence (˜ u _n ) n∈

N

whose gradients generate the same Young measure µ and such that (|∇˜ u n | ^p ) n∈

N

is uniformly integrable (see Proposition 7 in Appendix). By using Propositions 5, 6 in Appendix and standard lower semicontinuity results in Sobolev spaces, we have

Z

Ω

Qf (∇u) dx = lim

n→+∞

Z

Ω

f (∇u n ) dx

≥ Z

Ω×

M^3×3

f (λ) dµ

= lim

n→+∞

Z

Ω

f (∇ u ˜ n ) dx

≥ Z

Ω

Qf (∇u) dx, so that

n→+∞ lim Z

Ω

f (∇˜ u _n ) dx = lim

n→+∞

Z

Ω

f (∇u n ) dx = Z

Ω

Qf (∇u) dx

which proves the thesis. In what follows, we still denote by (u n ) n∈

N

the se- quence (˜ u n ) n∈

N

. We are going to modify the function u n near S so that the constraint γ S (u n ) = v n holds. The function v n will be indifferently considered as a W ^1,p (S, R ³ )-function or a W ^1,p (Ω, R ³ )-function with ^∂v _∂x

ⁿ

3

= 0.

By coerciveness of g 0 , R

S | ∇v ˆ n | ^p dˆ x is bounded, thus v _n strongly converges in L ^p (S, R ³ ) to a function v which classically satisfies ˆ ∇v(ˆ x) = bar(ν ˆ x ) for a.e. ˆ x in S. Then γ _S (u) = v on S. Let η > 0 intended to go to 0 and set Σ η := S × (− ^η ₂ , ^η ₂ ). We are going to modify u _n on Σ η in order that the trace on S of the new function be equal to v n , and in such a way to decrease lim n→+∞ R

Ω f (∇u n ) dx. Let ϕ ∈ C _c

^∞

( R ) satisfying

ϕ η = 1 on Ω \ Σ 2η , ϕ η = 0 on Σ η , 0 ≤ ϕ η ≤ 1,

|∇ϕ η | ≤ 1 η and define

u n,η = ϕ η (u n − v n ) + v n . (11) Clearly u n,η belongs to W _Γ ^1,p

₀

(Ω, R ³ ) and γ s (u n,η ) = v n . Moreover

Z

Ω

f (∇u n,η ) dx = Z

Σ

η

f (∇u n,η ) dx + Z

Σ

2η\Ση

f (∇u n,η ) dx + Z

Ω\Σ

2η

f (∇u n,η ) dx

≤ Z

Σ

η

f (∇v n ) dx + Z

Σ

2η\Ση

f (∇u n,η ) dx + Z

Ω

f (∇u n ) dx.

Thus, from the growth condition in (4), Z

Ω

f (∇u n,η ) dx ≤ C η + 1

η ^p Z

Σ

2η

|u n − v _n | ^p dx + Z

Σ

2η

(|∇u n | ^p ) dx +

Z

Ω

f (∇u n ) dx

where, from now on, C denotes various positive constants depending only on β, p and Ω. Letting n → +∞, from (10) we obtain

lim sup

n→+∞

Z

Ω

f (∇u n,η ) dx ≤ C η + 1

η ^p Z

Σ

2η

|u

−

v ^p

_|

dx + sup

n∈

N

Z

Σ

2η

(|∇u n | ^p ) dx +

Z

Ω

Qf (∇u) dx.

But since γ _S (u) = v on S, the following Poincar´e inequality holds Z

Σ

2η

|u − v| ^p dx ≤ η ^p Z

Σ

2η

| ∂u

∂x 3

| ^p dx

so that, letting η → 0, from the uniform integrability of (|∇u n |) n∈

N

lim sup

η→+∞ lim sup

n→∞

Z

Ω

f (∇u n,η ) dx ≤ Z

Ω

Qf (∇u) dx.

We conclude by a standard diagonalization argument: there exists n 7→ η(n) such that

lim sup

n→+∞

Z

Ω

f (∇u n,η(n) ) dx ≤ Z

Ω

Qf (∇u) dx.

It is easily checked that sequence (˜ u _n ) n∈

N

defined by ˜ u _n = u _n,η(n)) converges to u strongly in L ^p (Ω, R ³ ) and weakly in W _Γ ^1,p

₀

(Ω, R ³ ), which completes the proof.

Proposition 3 (Upper bound). For all (u, ν) ∈ L ^p (Ω, R ³ ) × Y 3×2 (S) there exists a sequence ((u ε , µ ε )) ε>0 in L ^p (Ω, R ³ ) × Y 3×3 (B) converging to (u, ν) and satisfying

F(u, ν) ≥ lim sup

ε→0

F ε (u ε , µ ε ).

Proof. One can assume F(u, ν) < +∞ so that (u, ν) ∈ W _Γ ^1,p

₀

(Ω, R ³ )× ∇Y ˆ 3×2 (S).

Classically, there exists v n ∈ W ^1,p (S, R ³ ) such that (| ∇v ˆ n | ^p ) n∈

N

is uniformly integrable and δ

∇v

ˆ

n

(ˆ x) ⊗ dˆ x ⇀ ν

^nar

in Y 3×2 (S) (Proposition 7 of Appendix 6).

Note that we also have

(δ

∇v

ˆ

n

(ˆ x) ⊗ δ 0

_R3

) ⊗ dx

^nar

⇀ (ν x ˆ ⊗ δ 0

_R3

) ⊗ dx (12) in Y 3×3 (B) when n → +∞. Since g 0 satisfies a growth condition of order p, we have (Proposition 6 of Appendix 6)

n→+∞ lim Z

S

g 0 ( ˆ ∇v n ) dˆ x = Z

S×

M^3×2

g 0 (ˆ λ) dν.

But, according to a classical interchange argument between infimum and inte- grals (see [2])

Z

S

g 0 ( ˆ ∇v n ) dˆ x = inf

ξ∈D(S,

R³

)

Z

S

g( ˆ ∇v n |ξ) dˆ x.

Let ξ n in D(S, R ³ ) satisfying

Z

S

g( ˆ ∇v n |ξ n ) dˆ x − Z

S

g 0 ( ˆ ∇v n ) dˆ x ≤ 1

n , then,

n→+∞ lim Z

S

g( ˆ ∇v n |ξ n ) dˆ x = Z

S×

M^3×2

g 0 (ˆ λ) dν. (13) Consider the function v _n,ε in W ^1,p (B, R ³ ) defined by

v n,ε (x) = v n (ˆ x) + εx 3 ξ n (ˆ x).

For fixed n, we first claim that

µ n,ε := δ

∇vn,ε

(x) ⊗ dx

^nar

⇀ (δ

∇v

ˆ

n

(ˆ x) ⊗ δ 0

_R3

) ⊗ dx (14) in Y 3×3 (B) when ε → 0. Indeed, setting ˜ v n (x) := v n (ˆ x), since ∇v n,ε − ∇˜ v n → 0 strongly in L ^p (B, M ^3×3 ), ∇v n,ε and ∇˜ v n generates the same Young measure (δ

∇v

ˆ

n

(ˆ x) ⊗ δ 0

_R3

) ⊗ dx in Y 3×3 (B).

On the other hand, according to the classical relaxation theory in Sobolev spaces, there exists u n ∈ W _Γ ^1,p

₀

(Ω, R ³ ) strongly converging to u in L ^p (Ω, R ³ ) such that

n→+∞ lim Z

Ω

f (∇u n ) dx = Z

Ω

Qf (∇u) dx (15)

and we can modify u n near S in such a way that γ S (u n ) = v n (see Lemma 2 of Appendix). Let η > ε and consider the function u n,ε,η defined for all x in Ω by

u _n,ε,η (ˆ x, x 3 ) := θ(x 3 )v n,ε (ˆ x, x 3

ε ) + (1 − θ(x 3 ))u n (ˆ x, x 3 ) where θ is a C ¹ -function satisfying 0 ≤ θ ≤ 1, | _∂x ^∂θ

₃

| ≤ _η−ε ¹ and

θ =

( 1 in B ε

0 in Ω \ B _η .

To shorten notation, we do not indicate the dependance on η and ε for θ. Note that v _n,ε,η ∈ W _Γ ^1,p

₀

(Ω, R ³ ) and r _ε u _n,ε,η = v _n,ε on B. From the local Lipschitz property in (4) satisfied by g one can easily establish

ε→0 lim Z

B

g( ˆ ∇v n,ε | 1 ε

∂v n,ε

∂x 3

) dx = Z

S

g( ˆ ∇v n |ξ n ) dˆ x (16)

Let us write Z

Ω

ε

f (∇u n,ε,η ) dx = Z

Ω\B

η

f (∇u n ) dx + Z

B

η\Bε

f (∇u n,ε,η ) dx. (17)

We claim that lim

η→0 lim

ε→0

Z

B

η\Bε

f (∇u n,ε,η ) dx. = 0. Indeed since

∇u ˆ n,ε,η = θ( ˆ ∇v n + x 3 ∇ξ ˆ n ) + (1 − θ) ˆ ∇u n

∂u

n,ε,η

∂x

3

= ^∂θ _x

₃

(v n − u _n ) + x 3 ξ _n ^∂θ _x

3

+ θξ _n + (1 − θ) ^∂u _x

₃ⁿ

, the following estimate holds:

Z

B

η\Bε

f (∇u n,ε,η dx

≤ C Z

B

η\Bε

h(ξ n , ∇ξ ˆ n , ∇u ˆ n , ∇v ˆ n

∂u n

∂x 3

) dx +

Z

B

η\Bε

∂θ x 3

p

|v n − u n | ^p dx + Z

B

η\Bε

|x 3 ξ n

∂θ x 3

| ^p dx

(18) where h is Lebesgue integrable and does not depend on ε and η. Clearly the first term in (18) tends to 0 when ε then η go to 0. We estimate the two last terms. Since γ _S (u n ) = v _n on S, Poincar´e’s inequality yields

Z

B

η\Bε

∂θ x 3

p

|v n − u n | ^p dx ≤ 1 (η − ε) ^p

Z

B

η

|v n − u n | ^p dx

≤ η ^p (η − ε) ^p

Z

B

η

|∇(v n − u _n )| ^p dx which tends to 0 when ε then η go to 0. On the other hand

Z

B

η\Bε

|x 3 ξ n

∂θ x 3

| ^p dx ≤ η ^p (η − ε) ^p

Z

B

η

|ξ n | ^p dx

which tends to 0 for the same reason. Therefore, combining (16), (17) and (18) we obtain

η→0 lim lim

ε→0 F ε (u n,ε,η , µ n,ε ) = Z

Ω

f (∇u n ) dx + Z

S

g( ˆ ∇v n |ξ n ) dˆ x.

Then, using a standard diagonalization argument, there exists a map ε 7→ η(ε) satisfying η(ε) → 0 whenever ε → 0 and, setting u _n,ε := u _n,ε,η(ε) ,

ε→0 lim F ε (u n,ε , µ n,ε ) = Z

Ω

f (∇u n ) dx + Z

S

g( ˆ ∇v n |ξ n ) dˆ x (19) for all fixed n ∈ N . Moreover, going back to the expression of u n,ε we have

Z

Ω

|u n,ε − u _n | ^p dx ≤ Z

B

η(ε)

|v n + x 3 ξ _n − u _n | ^p dx,

thus u _n,ε strongly converges to u _n in L ^p (Ω, R ³ ) when ε → 0.

Collecting (12), (13), (14), (15) and (19), we deduce the following conver- gence scheme where the first arrow indicates a convergence with respect to ε and the second to n:

µ _n,ε

^nar

⇀ (δ

∇v

ˆ

n

(ˆ x) ⊗ δ 0

_R3

) ⊗ dx

^nar

⇀ (ν ˆ x ⊗ δ 0

_R3

) ⊗ dx;

u n,ε → u n → u strongly in L ^p (Ω, R ³ );

F ε (u n,ε , µ n,ε ) → Z

Ω

f (∇u n ) dx + Z

S

g( ˆ ∇v n |ξ n ) dˆ x

→ Z

Ω

Qf (∇u) dx + Z

S

g 0 (ˆ λ) dν.

The conclusion of Proposition 3 then follows by using a standard diagonalization argument ¹ in the product space L ^p (Ω, R ³ ) × Y 3×3 (B) × R and noticing that A((ν x ˆ ⊗ δ 0

_R3

) ⊗ dx) = ν.

On account of Definition 2, Proposition 1, Lemma 1 and Propositions 2, 3 above, we can state the main theorem of this section.

Theorem 2. The sequence of functionals (F ε − L) ε>0 Γ _X,Y, Y ˆ -converges to the functional F − L. In addition, if (¯ u ε , µ ¯ ε ) ∈ L ^p (Ω, R ³ ) × Y 3×3 (B) is an ε- minimizer of F ε − L, i.e. satisfies

F ε (¯ u _ε , µ ¯ _ε ) − L(¯ u _ε ) ≤ ε + inf

(u,µ)∈L

^p

(Ω,

R3

)×Y

3×3

(B)

F ε (u, µ) − L(u) , then there exists a subsequence of ((¯ u ε , µ ¯ ε )) ε>0 converging to (¯ u, ν ¯ ) which is a minimizer of inf _(u,ν)∈L

^p

_(Ω,

R3

)×Y

3×2

(S)

F(u, ν)−L(u)

. Moreover γ S (¯ u) belongs to W ^1,p (S, R ³ ) and, for a.e. ˆ x in S bar(¯ ν x ˆ ) = ˆ ∇γ S (¯ u)(ˆ x).

Proof. The proof is a straightforward consequence of Lemma 1, Propositions 2 and Proposition 3. Indeed, L is easily seen to be a continuous perturbation of F ε , so that F ε

Γ

_X,Y,Yˆ

−→ F = ⇒ F ε − L ^Γ

^X,Y,

−→ F −

^Yˆ

L.

4 The Young measure considered as an internal variable: the formulation in terms of displace- ments

In this section, we derive the classical model obtained in [1] in the context of the linear elasticity or in [10] in a more general setting, from the model obtained in Section 3. We show that the stored strain energy functional associated with the classical model is the marginal map of the functional limit F obtained in the previous section when we consider the Young measure ν , which represents

1

Such an argument is valid because the set Y

3×3

(B) endowed with the narrow topology

is first countable (see [13] Proposition 2.3.1).

the fine microstructure of the layer, as an internal variable. In some sense the formulation in terms of displacement can be regarded as the macroscopic version of the model suggested in Section 3 (see Corollary 1). With the notations of Sections 2, 3 we define the two functionals

H ε , H : L ^p (Ω, R ³ ) → R ∪ {+∞}

by

H ε (u) :=





 Z

Ω

ε

f (∇u) dx + Z

B

g( ˆ ∇r ε u| 1 ε

∂r ε u

∂x 3

) dx if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ), +∞ otherwise

and

H (u) :=



 



 

 Z

Ω

Qf (∇u) dx + Z

S

Qg 0 ( ˆ ∇γ S (u)) dˆ x if u ∈ W _Γ ^1,p

₀

(Ω, R ³ ), γ S (u) ∈ W ^1,p (S, R ³ ) +∞ otherwise.

Clearly, H _ε is the marginal map associated with the functional F ε , i.e., for all u ∈ L ^p (Ω, R ³ ) we have

H _ε (u) = inf

µ∈Y

3×3

(B) F ε (u, µ).

On the other hand H is the stored strain energy functional obtained in [10]

in non linear elasticity, or in [1] in the linear elasticity framework where we have to replace ∇u with the linearized strain tensor e(u) = ¹ ₂ (∇u + ^t ∇u) and the quasiconvexifications of f and g 0 with their convexifications. In the next proposition, we establish that H is the marginal map associated with the limit functional F.

Proposition 4. The functional H is the marginal map associated with the functional F when the Young measure ν is considered as an internal variable.

More precisely, for all u ∈ L ^p (Ω, R ³ ) we have H (u) = inf

ν∈Y

3×2

(S) F (u, ν).

Proof. We begin by introducing various sets. For every ˆ A in M ^3×2 we define the set adm( ˆ A) of probability measures on M ^3×2 by:

P ∈ adm( ˆ A) ⇐⇒



 

 

 

 

 

  A ˆ =

Z

M^3×2

ˆ λ dP;

Z

M^3×2

| ˆ λ| ^p dP < +∞;

φ( ˆ A) ≤ Z

M^3×2

φ(ˆ λ) dP

for all quasiconvex function φ satisfying a growth condition of order p. On the other hand, for each fixed u ∈ W _Γ ^1,p

₀

(Ω, R ³ ) such that γ _S (u) ∈ W ^1,p (S, R ³ ) we define the subset Adm(u) of Y 3×2 (S) by:

ν ∈ Adm(u) ⇐⇒

ν ∈ ∇Y ˆ 3×2 (S);

bar(ν x ˆ ) = ˆ ∇γ S (u)(ˆ x) for a.e. ˆ x ∈ S.

The proof is based on the following localization Lemma.

Lemma 3. With the notations above we have (i) inf n Z

M^3×2

g 0 (ˆ λ) dP : P ∈ adm( ˆ A) o

= Qg 0 ( ˆ A) for all A ˆ ∈ M ^3×2 ; (ii) inf

ν∈ Adm (u)

Z

S×M

^3×2

g 0 (ˆ λ) dν = Z

S

inf

P∈

adm ( ˆ

∇γS

(u)(ˆ x))

Z

M^3×2

g 0 (ˆ λ) dP dˆ x.

Proof of Lemma 3. Proof of (i). For every P ∈ adm( ˆ A) one has Z

M^3×2

g 0 (ˆ λ) dP ≥ Z

M^3×2

Qg 0 (ˆ λ) dP ≥ Qg 0 ( ˆ A) so that inf n Z

M^3×2

g 0 (ˆ λ) dP : P ∈ adm( ˆ A) o

≥ Qg 0 ( ˆ A).

For the converse inequality, set ˆ Y =]0, 1[ ² , fix ψ ∈ W ₀ ^1,p ( ˆ Y , R ³ ) and define the probability measure P _ψ by

P _ψ :=

Z

Y ˆ

δ A+∇ψ(ˆ ˆ y) dˆ y

which acts on every continuous function ϕ : M ^3×2 → R satisfying a growth condition of order p as follows:

Z

M^3×2

ϕ(ˆ λ) dP ψ :=

Z

Y ˆ

ϕ( ˆ A + ∇ψ(ˆ y)) dˆ y.

Clearly P _ψ ∈ adm( ˆ A), consequently

E( ˆ A) := {P ψ : ψ ∈ W ₀ ^1,p ( ˆ Y , R ³ )} ⊂ adm( ˆ A).

It follows that inf n Z

M^3×2

g 0 (ˆ λ) dP : P ∈ adm( ˆ A) o

≤ inf n Z

M^3×2

g 0 (ˆ λ) dP : P ∈ E( ˆ A) o

= inf

φ∈W

₀^1,p

( ˆ Y ,

R³

)

Z

Y ˆ

g 0 ( ˆ A + φ(ˆ y)) dˆ y

= Qg 0 ( ˆ A).

In the last equality we have used the quasiconvex envelop formula for real valued

functions satisfying a growth condition of order p (see [15]).

Proof of (ii). Since ν ∈ Adm(u) yields ν x ˆ ∈ adm( ˆ ∇γ S (u)(ˆ x)), clearly we have

inf

ν∈ Adm (u)

Z

S

Z

M^3×2

g 0 (ˆ λ) dν x ˆ dˆ x ≥ Z

S

inf

P∈

adm ( ˆ

∇γS

(u)(ˆ x))

Z

M^3×2

g 0 (ˆ λ) dP dˆ x.

Conversely, for all η > 0, and ˆ x ∈ S, let P ^η _{ˆ x} in adm( ˆ ∇γ S (u)(ˆ x)) satisfying inf

P∈

adm ( ˆ

∇γS

(u)(ˆ x))

Z

M^3×2

g 0 (ˆ λ) dP

≥ Z

M^3×2

g 0 (ˆ λ) dP ^η _{x ˆ} − η. (20) We can assume that the map ˆ x 7→ P ^η _{x ˆ} is measurable (see [14]). Set ν := P ^η _{x ˆ} ⊗dˆ x.

As P ^η _{x ˆ} ∈ adm( ˆ ∇γ S (u)(ˆ x)), the Young measure ν belongs to Adm(u) so that (20) yields, since η is arbitrary,

Z

S

inf

P∈

adm ( ˆ

∇γS

(u)(ˆ x))

Z

M^3×2

g 0 (ˆ λ) dP

dˆ x ≥ inf

ν∈ Adm (u)

Z

S

Z

M^3×2

g 0 (ˆ λ) dν ˆ x dˆ x which completes the proof of Lemma 3.

Proof of Proposition 4 continued. According to Lemma 3 we obtain

ν∈Y inf

3×2

(S) F(u, ν) = Z

Ω

Qf (∇u) dx + inf

ν∈ Adm ^(u) Z

S×M

^3×2

g 0 (ˆ λ) dν

= Z

Ω

Qf (∇u)dx + Z

S

inf

P∈

adm ^{( ˆ}

^∇γS

(u)(ˆ x))

Z

M^3×2

g 0 (ˆ λ) dP dˆ x

= Z

Ω

Qf (∇u) dx + Z

S

Qg 0 ( ˆ ∇γ S (u)) dˆ x which proves the proposition.

Applying Proposition 4, Lemma 1 and Theorem 1 we recover the classical nonlinear model of multimaterial with strong interface obtained in [1] and [10].

Precisely

Theorem 3. Let us equip the space X = L ^p (Ω, R ³ ) with the strong convergence.

Then the sequence of functionals (H ε ) ε>0 Γ X -converges to the functional H . In addition, if u ¯ ε ∈ L ^p (Ω, R ³ ) is an ε-minimizer of H ε − L, i.e. which satisfies

H ε (¯ u ε ) − L(¯ u ε ) ≤ ε + inf

u∈L

^p

(Ω,

R3

)

H ε − L(u) ,

then there exists a subsequence of (¯ u _ε ) ε>0 strongly converging in L ^p (Ω, R ³ ) and weakly in W _Γ ^1,p

₀

(Ω, R ³ ) to some u ¯ which is a minimizer of the classical nonlinear problem

u∈L inf

^p

(Ω,

R³

)

H(u) − L(u)

.

Let (¯ u, ¯ ν) be a minimizer of inf (u,ν)∈L

^p

(Ω,

R³

)×Y

3×2

(S)

F(u, ν) − L(u) . The next corollary states that ¯ u is a minimizer of the classical nonlinear problem and that at a.e. ˆ x in S, one may think ¯ ν as the microscopic description of the macroscopic quantities ˆ ∇γ S (¯ u)(ˆ x) and Qg 0 ( ˆ ∇γ S (¯ u)(ˆ x)).

Corollary 1. Let (¯ u, ν ¯ ) be a minimizer of inf (u,ν)∈L

^p

(Ω,

R³

)×Y

3×2

(S)

F(u, ν) −

L(u)

, then u ¯ is a minimizer of inf u∈L

^p

(Ω,

R3

)

H(u)− L(u)

. Moreover for a.e.

ˆ

x in S one has

∇γ ˆ S (¯ u)(ˆ x) = Z

M^3×2

ˆ λ d¯ ν x ˆ ; Qg 0 ( ˆ ∇γ S (¯ u)(ˆ x)) =

Z

M^3×2

g 0 (ˆ λ) d¯ ν ˆ x .

Proof. We begin by proving the two equalities. The proof of the first one is straightforward since ¯ ν ∈ Adm(¯ u). For the same reason, for a.e. ˆ x in S we have

Qg 0 ( ˆ ∇γ S (¯ u)(ˆ x)) ≤ Z

M^3×2

Qg 0 (ˆ λ) d¯ ν x ˆ

≤ Z

M^3×2

g 0 (ˆ λ) d¯ ν x ˆ .

The converse inequality is more involved and requires a localization argu- ment. Let ˆ x 0 be a fixed point of S, Q _ρ (ˆ x 0 ) = ˆ Q _ρ (ˆ x 0 )×(−ρ, ρ) the cylinder of R ³ where ˆ Q _ρ (ˆ x 0 ) is the ball of R ² centered at ˆ x 0 ∈ S. Write F _ε ^{ρ,ˆ x}

⁰

, F ^{ρ,ˆ x}

⁰

, H _ε ^{ρ,ˆ x}

⁰

, H ^{ρ,ˆ x}

⁰

for the functionals F ε , F, H _ε , H localized at Q _ρ (ˆ x 0 ) where the constraint u ∈ W _Γ ^1,p

₀

(Ω, R ³ ) is replaced by the constraint u ∈ u ¯ + W ₀ ^1,p (Q ρ (ˆ x 0 ), R ³ ). It is easy to check that (¯ u, ¯ ν) restricted to Q ρ (ˆ x 0 ) is also a minimizer of the localized problem

inf n

F ^{ρ,ˆ x}

⁰

(u, ν) : u ∈ L ^p (Q ρ (ˆ x 0 ), R ³ ), ν ∈ Y 3×2 ( ˆ Q _ρ (ˆ x 0 ) o .

On the other hand, analysis similar to that in the proofs of Theorem 2 and Theorem 3 shows that F _ε ^{ρ,ˆ x}

⁰

^Γ

^X,Y,

→

^Yˆ

F ^{ρ,ˆ x}

⁰

and H _ε ^{ρ,ˆ x}

⁰

^Γ →

^X

H ^{ρ,ˆ x}

⁰

. From the second variational convergence process ^Γ →, there exists a sequence (u

^X

ε ) ε>0 strongly converging to ¯ u in L ^p (Q ρ (ˆ x 0 ), R ³ ) such that

ε→0 lim

H _ε ^{ρ,ˆ x}

⁰

(u ε ) − L(u ε )

= Z

Q

ρ

(ˆ x

0

)

Qf(∇ u) ¯ dx + Z

Q ˆ

ρ

(ˆ x

0

)

Qg 0 ( ˆ ∇γ S (¯ u)) dˆ x − L(¯ u). (21)

Moreover, according to the Prokorov compactness theorem, µ ε = δ

∇uε

(x) ⊗dx ⇀

^A

¯¯

ν where ¯ ν ¯ is some Young measure in ˆ ∇Y 3×2 (S). Now, from the first variational