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

www.elsevier.com/locate/anihpc

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

aLMGC, UMR-CNRS 5508 and ACSIOM, UMR-CNRS 5149, Université Montpellier II, Case courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

bACSIOM, UMR-CNRS 5149, Université Montpellier II, Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

cACSIOM and AVA, UMR-CNRS 5149, Université Montpellier II et Université de Nîmes, Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Received 1 September 2008; accepted 14 January 2009 Available online 18 January 2010

Abstract

The gradient displacement field of a micro-structured strong interface 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 orderεand ¹_ε 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.

©2010 Elsevier Masson SAS. All rights reserved.

MSC:49J45; 74N15; 35B40

Keywords:Micro structures; Young measures; Variational convergences;Γ-convergence

1. Introduction

In [1] and [10] a variational model of multi-material with a very rigid interface is obtained by identifying the classicalΓ-limit of the stored strain energy functional when the magnitude orderεof 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 mechanical features are high strain of the soft material and oscillations of gradient displacement in the layer of hight stiffness, we deal with

* Corresponding author.

E-mail addresses:[email protected] (A.L. Bessoud), [email protected] (F. Krasucki), [email protected] (G. Michaille).

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.01.007

the asymptotic analysis of the problem by means of a new variational convergence where competing objects 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 formulation (u,¯ ν)¯ ∈arg minF(u, ν)−L(u)of the problem by identifying the limit(u, ν)→F(u, ν)of the stored strain energy functionalu→F_ε(u)rewritten as a bifunctional(u, μ)→Fε(u, μ)(we writeL(u)for the exterior loading).

Let Ω be the reference configuration occupied by the material and S× ]0, ε[the thin inclusion. The limit en- ergy functional of F_ε obtained in [10] is of the form u→F (u):=

ΩQf (∇u) dx+

SQg₀(ˆ∇γ_S(u)) dxˆ for all Sobolev-functionsuwith smooth traceγ_S(u)on the two-dimensional interfaceS, whereQf andQg₀denote the qua- siconvexifications off andg₀. As a straightforward consequence of our formulation we find the stored strain energy F as to be the marginal mapu→infνF(u, ν)of the energy functionalF when the Young measureν=ν_xˆ⊗dxˆ is then regarded as an internal state variable. By comparing the two variational formulationsu¯∈arg min(F−L(u))and (u,¯ ν)¯ ∈arg min(F−L(u)), we obtain an integral representation with respect to the probability measureν¯_xˆon the set M^3×2of 3×2-matrices, of the significant macroscopic quantities ˆ∇γ_S(u)¯ andQg₀(ˆ∇γ_S(u)). In some sense we may¯ think the variableν¯as the microscopic description of ˆ∇γ_S(u)¯ andQg₀(ˆ∇γ_S(u)).¯

Another way for obtaining a variational formulation of the problem is to consider the marginal map G of F−Lwhen the displacement fieldu is now regarded as an internal variable. We show that the energy functional ν →G(ν):=infu(F(u, ν)−L(u)) is a variational limit of μ→infu(Fε(u, μ)−L(u)) so that ν¯∈arg minG is a new formulation of the problem in terms of gradient Young measures parametrized on the interfaceS. By compar- ing it with the formulation(u,¯ ν)¯ ∈arg min(F−L), we show thatu¯is a solution of the nonlinear Dirichlet problem min(

Ω\SQf (∇u) dx−L(u)) subjected to the boundary condition u(¯ x)ˆ = ˆ∇⁻¹(

M^3×2λ dˆ ν¯_xˆ)on the interface S.

Consequently, one may think the surface energy

SQg₀(ˆ∇γ_S(u)) dxˆobtained in [10] as a relaxation of the boundary condition above (notice the analogy with the relaxation of boundary conditions inBV-spaces).

This paper illustrates, in the modeling of multi-materials, the following general strategy: in order to capture various convergence phenomena on minimizing sequences regarded as Sobolev variables of a problem(Pε), one defines a suit- able measure state variableμconnected to the Sobolev state variableu(Young measure, concentration measure. . .), an energy bifunctional(u, μ)→Fε modeling(Pε), a suitable variational convergence process, and identify its limitF.

We recover the limit energy in terms of Sobolev variables as the marginal functional of F when μis regarded as an internal variable. This idea as already been used in the framework of relaxation theory for a reduction dimension 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 ofFwhenuis an internal variable.

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 Sec- tion 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 bifunctionalFεto the bifunctionalF(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 A.

2. A variational convergence of sequences of functionals defined on topological product spaces 2.1. The abstract setting

Given three first countable topological spaces,X,Y,Yˆ, a mapAfromY toYˆ, and extended real-valued function- alsFn:X×Y →R∪ {+∞},F:X× ˆY →R∪ {+∞}, our purpose is to define a variational convergence of the sequence(Fn)_n∈Ntoward the functionalFso that, under a suitable convergence process associated withAand some compactness hypotheses, the following implication holds true whenn→ +∞:

Fn→F⇒

infX×YFn→min_{X× ˆY}F;

arg minX×YFn(xn, yn)→(x,y)ˆ ∈arg min_{X× ˆY}Fat least for a subsequence.

We begin by introducing a new weak notion of convergence between elements ofY andYˆ, and next, between elements ofX×Y andX× ˆY.

Definition 1.Let(y_n)_n∈Nbe a sequence inY andyˆinYˆ. We say thaty_nA-converges toyˆand we write y_n^A yˆ

iff there existsyinY such thaty_n→y andyˆ=A(y).

Let((x_n, y_n))_n∈N be a sequence inX×Y and(x,y)ˆ inX× ˆY. We say that(x_n, y_n)converges to(x,y)ˆ and we write

(x_n, y_n)^I (x,^×A y)ˆ

iffx_nconverges tox, andy_nA-converges toyˆ.

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

Definition 2.We say thatFnΓ_X,Y,Yˆ-converges toFand we write Fn

Γ_X,Y,Yˆ

−→ F

iff for all(x,y)ˆ inX× ˆY, both following assertions hold:

(i) ∀(xn, yn)∈X×Y s.t.(xn, yn)^I (x,^×A y),ˆ F(x,y)ˆ lim infn→+∞Fn(xn, yn), (ii) ∃(x_n, y_n)∈X×Y s.t.(x_n, y_n)^I (x,^×A y),ˆ F(x,y)ˆ lim sup_n→+∞Fn(x_n, y_n).

Note that this convergence is closely related to theΓ-convergence. WhenX= {0}or, which is equivalent, whenFn

andFdo not depend onx, we denote it briefly byΓ_Y,Yˆ. WhenY= {0}andYˆ= {ˆ0}i.e. whenFnandFdo not depend onyandyˆ, we will write it simplyΓ_Xand our definition agrees with the classicalΓ-convergence (for more details on Γ-convergence, see [5,16]). Note also that whenY = ˆY andAis the identity map,Γ_X,Y,Y is theΓ_X×Y-convergence.

The proposition below expresses the variational nature of theΓ_X,Y,Yˆ-convergence.

Proposition 1.Let us assume that(Fn)_n∈N Γ_X,Y,Yˆ-converges toF and let ((x_n, y_n))_n∈N be a sequence ofX×Y satisfying

Fn(x_n, y_n) inf

(x,y)∈X×YFn(x, y)+1 n.

Assume furthermore that {(x_n, y_n): n∈N}is relatively compact for the convergence ^I×Adefined above. Then any cluster point(x,¯ y)¯ˆ ∈X× ˆY is a minimizer ofFand

n→+∞lim inf

Fn(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 2.2. The variational convergence of marginal functionals

Let us consider the following marginal functionalsFn, F:X→R∪ {+∞},Gn:Y →R∪ {+∞}, andG:Yˆ → R∪ {+∞}defined by:

F_n(x)=inf

y∈YFn(x, y), F (x)=inf

ˆ y∈ ˆY

F(x,y),ˆ G_n(y)=inf

x∈XFn(x, y), G(y)ˆ = inf

x∈XF(x,y).ˆ

The variational convergence of the functionalsFnyields the variational convergence of their marginal maps, precisely:

Theorem 1. Let us assume that (Fn)_n∈N Γ_X,Y,Yˆ-converges to F. Assume furthermore that the following inf- compactness property holds: for every sequence ((x_n, y_n))_n∈N satisfying sup_n∈NFn(x_n, y_n) <+∞, there exists a subsequence(x_{σ (n)}, y_{σ (n)})inX×Y and(x,y)ˆ inX× ˆY such that(x_{σ (n)}, y_{σ (n)})^I (x,^×A y). Thenˆ

(i) F_n−→^ΓX 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 ofF_n, one can extract a subsequence whichΓ_X-converges toF. Let(x_n)_n∈Nbe a sequence converging toxinXand consider a sequence(y_n)_n∈NinY such that

Fn(x_n, y_n)−1 ninf

y∈YFn(x_n, y):=F_n(x_n). (1)

We can assume that sup_n∈NF_n(x_n) <+∞(otherwise there is nothing to prove) so that sup_n∈NFn(x_n, y_n) <+∞. Thus, according to the inf-compactness assumption, there exist a subsequence (x_{σ (n)}, y_{σ (n)})andyˆ inYˆ such that x_{σ (n)}converges tox, andy_{σ (n)}A-converges toyˆ. Furthermore, since(Fn)_n∈NΓ_X,Y,Yˆ-converges toF, one has

F(x,y)ˆ lim inf

n→+∞Fσ (n)(xσ (n), yσ (n)).

Since from (1) one hasF_{σ (n)}(x_{σ (n)})Fσ (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 writeninstead ofσ (n). Let(yˆ_p)_p∈Nbe a sequence inYˆ satisfying F (x)=inf

ˆ y∈ ˆY

F(x,y)ˆ = lim

p→+∞F(x,yˆ_p). (2)

Combining (i) and (ii) of Definition 2, for every fixedpthere exists a sequence((x_n^p, yp,n))n∈NinX×Y such that x_n^p→x,

yp,n A

yˆp

and satisfying

F(x,yˆ_p)= lim

n→+∞Fn

x_n^p, y_p,n

. (3)

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

p→+∞ lim

n→+∞Fn

x_n^p, yp,n

.

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

F (x)= lim

n→+∞Fn

x_n^p(n), y_p(n),n

lim sup

n→+∞F_n x_n^p(n)

.

The sequence defined byxn=x_n^p(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

2.3. A concrete example

In this section, we present a concrete example entering within the general framework 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 byM^3×3andM^3×2respectively. Con- sidering the spaceM^3×3as the productM^3×2×R³, we will denote byλˆ the first coordinate inM^3×2of any element λofM^3×3. We writeP_M3×2 to denote the projection mapping M^3×3 toM^3×2. In what follows, we use notation of Theorem 4 in Appendix A.

LetΩ,Bbe two open bounded subsets ofR³, andS⊂R²such thatB=S×(0,1), we define the sets of Young measuresY3×3(B)andY3×2(S)as follow:

μ∈Y3×3(B) ⇔ μ∈M⁺

B×M^3×3

and PB#μ=L, ν∈Y3×2(S) ⇔ ν∈M⁺

S×M^3×2

and PS#ν= ˆL

whereP_B#μ(resp.P_S#ν) denotes the image of the measureμ(resp.ν) by the projectionP_B:B×M^3×3→B(resp.

P_S:S×M^3×2→S) andL(resp.L) the Lebesgue measure onˆ B(resp.S). For every probability measurePonM^3×3 orM^3×2, we write bar(P)for its barycenter, i.e. bar(P)=

λ dP(λ).

The mapAis defined by:

A:Y3×3(B)→Y3×2(S), μ=μ_x⊗dx→ν=

1 0

ˆ μ_x,sˆ ds

⊗dxˆ

whereμˆ_x=P_M3×2#μx,x∈Ω, and₁

0μˆ_x,sˆ ds is the probability measure parametrized byxˆ∈S, which acts on all ϕ∈C0(M^3×2)as follows:

1 0

ˆ μ_x,sˆ ds, ϕ

:=

1 0

M^3×2

ϕ(ˆλ) dμˆ_x,sˆ ds.

Given a sequence(μ_ε)_ε>0in the spaceY3×3(B)equipped with the narrow convergence (see Appendix A) andν inY3×2(S), according to Definition 1 one has

μ_ε ν^A ⇔ ∃μ∈Y3×3(B) s.t.

μ_ε μ,^nar ν=A(μ)

and, for(v_ε, μ_ε)inL^p(Ω,R³)×Y3×3(B), and(v, ν)inL^p(Ω,R³)×Y3×2(S):

(v_ε, μ_ε)^I (v, ν)^×A ⇔

v_ε→vinL^p(Ω,R³), μ_ε ν.^A

Given two locally Lipschitz functionsf, g:M^3×3→R⁺ satisfying a growth condition of orderp, we consider the integral functionalFε defined by

Fε:L^p Ω,R³

×Y3×3(B)→R∪ {+∞}, Fε(u, μ):= ^Ωεf (∇u) dx+

B×M^3×3g(ˆλ|¹_ελ₃) dμ+I_ε(u, μ) ifu∈W_Γ^1,p

0 (Ω,R³),

+∞ otherwise,

where

I_ε(u, μ):=

0 ifμ=δ_∇v(x)⊗dx, v=1_Brεu, +∞ otherwise.

andr_εuis defined byr_εu(x, xˆ 3)=u(x, εxˆ 3).

Let ˆ∇Y3×2(S) denote the subset of Y3×2(S) made up of all Young measures generated by gradients of W^1,p(S,R³)-Sobolev functions,γ_S the trace operator fromW^1,p(Ω\S,R³)intoL^p(S,R³)and define the func- tion g₀:M^3×2→Rfor all λˆ in M^3×2 byg₀(ˆλ):=inf_ξ∈R3g(ˆλ|ξ ). SettingX=L^p(Ω,R³),Y =Y3×3(B) and

Yˆ:=Y3×2(S), in the next section we prove that the sequence(Fε)_ε>0Γ_X,Y,Yˆ-converges to the functionalF, defined by

F:L^p Ω,R³

×Y3×2(S)→R∪ {+∞}, F(u, ν):= ^ΩQf (∇u) dx+

S×M^3×2g0(ˆλ) dν+I (u, ν) ifu∈W_Γ^1,p

0 (Ω,R³), γSu∈W^1,p(S,R³),

+∞ otherwise,

where

I (u, ν):=

0 ifν∈ ˆ∇Y3×2(S), bar(ν_xˆ)= ˆ∇γ_S(u)(x)ˆ a.e. inS, +∞ 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 measures: Analysis of microstructures of the strong material In the three-dimensional Euclidean spaceE³referred to the orthonormal frame (0;e1, e2, e3), we consider a do- mainΩ with aC¹boundaryΓ. LetΩ^±=Ω∩ [±x3>0], the interiorS= {∂Ω⁺∩∂Ω⁻}^◦of the common part of the boundaries of Ω^± is assumed to have a positiveH²-measure and, to shorten the proofs, included in the plane [x₃=0]. The setΩis the physical reference configuration of the assembly of two materials. More precisely, given a small dimensionless parameterεand a global characteristic lengthh(for example the diameter ofΩ), the setB_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_εis the reference configuration of a material with stiffness of order 1. (see Fig. 1). The structure is clamped on a partΓ₀ofΓ with a positiveH²-measure, the complementary partΓ_ψ ofΓ₀is submitted to surface loadsψand we assume thatH¹(Γ₀∩ ¯S) >0. Obviously one can there consider other type of boundary conditions (e.g. a combina- tion of some components of the stress vector and of the displacement). Moreover the structure is submitted to applied body forcesΦ.

Letp1, we say that a Borel functionW:M^3×3→Rsatisfies a(C_p)condition if ∃α, β, C∈R⁺s. t.∀(λ, λ)∈M^3×3, α|λ|^pW (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 functionφ:M^3×3→R(resp.φ:M^3×2→R)satisfies a growth condition of orderp if there existsγ0 such that

∀λ∈M^3×3, φ(λ)γ

1+ |λ|^p

resp.∀ˆλ∈M^3×2, φ(ˆλ)γ

1+ |ˆλ|^p .

The soft and the strong materials are modeled as hyperelastic and the bulk energy densitiesf,gof the two materials occupyingΩ_εandB_εsatisfy a(C_p)condition withp >1. To shorten notation, we assume that(C_p)is satisfied with the same constantsα,β andC. We make the assumption that the strain of the soft material can be high and that the thin structureB_εis occupied by a material which undergoes reversible solid–solid phase transformation as for instance crystalline solids. In this context, the densitiesf andgare not convex andgentails 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 fielduis given by the functional

Fε(u):=

Ωε

f (∇u) dx+1 ε

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,R³

Fig. 1. Bonded assembly – left: the physical configuration – right: the rescaled layer – below: the limit configuration.

whereL(u)=

ΩΦ dx+

Γ_ψψdH². We want to analyze the behavior ofu¯εwhenεtends to zero and to identify the variational problem whose limit is a solution. But since the material in the layerB_εpossesses a fine micro-structure, the gradient minimizing sequence(∇ ¯u_ε)_ε>0develops oscillations we would like to integrate into the variational problem.

This is why we write the strain energy ¹_ε

Bεg(∇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 inΩ_εandB_ε, in a first stage, it is convenient to write the energy functionalF_ε in terms of two arguments, oneu, the displacement onΩ_ε, the otherv, the displacement on Bεoccupied by the strong material. On the other hand, in order to work in a fixed space for the variablev, the change of scale(x, xˆ 3)=(x, εyˆ 3)transforming(x, xˆ 3)∈Bε into(x, yˆ 3)∈B:=S×(0,1)leads to consider the following functional

Gε:L^p Ω,R³

×L^p B,R³

→R∪ {+∞}, Gε(u, v):= ^Ωεf (∇u) dx+

Bg(ˆ∇v|¹_ε∂x^∂v₃) dx+Iε(u, v) ifu∈W_Γ^1,p

0 (Ω,R³),

+∞ otherwise,

where, for all(u, v)∈W_Γ^1,p

0 (Ω,R³)×W^1,p(B,R³) I_ε(u, v):=

0 if 1_Br_εu=v, +∞ otherwise

andr_εu(x, yˆ ₃):=u(x, εyˆ ₃). Now we writeGε in terms of pairs of displacements-Young measures by defining the functionalFεas follows

Fε:L^p Ω,R³

×Y3×3(B)→R∪ {+∞},

Fε(u, μ):= ^Ωεf (∇u) dx+

B×M³×3g(ˆλ|¹_ελ₃) dμ+I_ε(u, μ) ifu∈W_Γ^1,p

0 (Ω,R³),

+∞ otherwise,

where

I_ε(u, μ):=

0 ifμ=δ_∇v(x)⊗dx, v=1_Br_εu, +∞ otherwise.

Clearly Fε is one way of writing Gε and from the strict variational point of view, it is equivalent to identify the variational limit ofF_εand that ofFεin the spirit of the previous section. Indeed we have

inf

u∈W_Γ^1,p

0 (Ω,R³)

Fε(u)−L(u)

= inf

(u,v)∈L^p(Ω,R³)×L^p(B,R³)

Gε(u, v)−L(u)

= inf

(u,μ)∈L^p(Ω,R³)×Y3×3(B)

Fε(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 layerB_εthanks to the Young measure state variable.

In order to apply Proposition 1 we begin by establishing the following compactness lemma

Lemma 1 (Compactness). Let ((u_ε, v_ε, μ_ε))_ε>0 be a sequence in L^p(Ω,R³)×L^p(B,R³)×Y3×3(B) satisfying sup_ε>0Gε(uε, vε)=sup_ε>0Fε(uε, με) <+∞. Then there exist(u, v)∈W_Γ^1,p

0 (Ω,R³)×W^1,p(B,R³),ν∈ ˆ∇Y3×2(S) and a subsequence not relabeled such that

(i) uε→u weakly in W_Γ^1,p

0 (Ω,R³)and strongly in L^p(Ω,R³),vε →v weakly in W^1,p(B,R³)and strongly in L^p(B,R³);

(ii) ^∂v_∂y^ε

3 →0strongly inL^p(B,R³), _∂y^∂v

3 =0andv∈W^1,p(S,R³);

(iii) γ_S(u)=vonSwhereγ_Sdenotes the trace operator fromW^1,p(Ω\S,R³)intoL^p(S);

(iv) (u_ε, μ_ε)^I (u, ν), with^×A ν=A(μ) where μ=(μˆ_x⊗δ₀

R3)⊗dx and μˆ := ˆμ_x⊗dx is the Young measure generated by(ˆ∇vε)ε>0. Moreoveruandνare connected as follows: bar(ν_xˆ)= ˆ∇γS(u)(x)ˆ for a.e.xˆ∈S.

Proof. Assertions (i) and (ii) are straightforward consequences of the coerciveness condition in (4), Poincaré’s inequality and the isometry between W^1,p(S,R³)and{v∈W^1,p(B,R³): _∂y^∂v

3 =0}. We are going to establish as- sertion (iii). For a.e.x∈Bwe have

v_ε(x, yˆ ₃)=u_ε(x, εyˆ ₃)=u_ε(x,ˆ 0)+

εy₃

0

∂u_ε

∂y₃(x, s) dsˆ

whereuε(x,ˆ 0)must be taken in the trace sense. LetQρ(xˆ0)= ˆQρ(xˆ0)×(0, ρ)be the cylinder ofR³whereQˆρ(xˆ0) is the ball ofR²centered atxˆ0∈Swithρ >0 small enough so thatQ_ρ(xˆ0)⊂B. Integrating the previous equality overQ_ρ(xˆ₀)yields

–

Qρ(xˆ0)

v_εdx=

–

Qρ(xˆ0)

u_ε(x,ˆ 0) dx+

–

Qρ(xˆ0) εy3

0

∂u_ε

∂y₃(x, s) ds dx.ˆ

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

–

ˆ Qρ(xˆ₀)

v dxˆ=

–

ˆ Qρ(xˆ₀)

γ_S(u) dxˆ+lim sup

ε→0

I_ρ,ε (5)

whereI_ρ,ε:=

–_Q

ρ(xˆ₀)

_εy3

0

∂uε

∂y3(x, s) ds dx. Let us estimateˆ I_ρ,ε. An easy calculation using Hölder’s inequality gives

|Iρ,ε|

–

Q_ρ(xˆ₀)

(εy3)^1−1p

εy3

0

∂u_ε

∂y₃(x, s)ˆ ^pds

¹

p

, dx dyˆ 3

(ερ)^1−p1

–

ˆ Q_ρ(xˆ₀)

ερ 0

∂u_ε

∂y₃(x, s)ˆ ^pds

¹

p

dxˆ

(ερ)^1−p1

ˆ Q_ρ(xˆ₀)

ερ 0

∂u_ε

∂y₃(x, s)ˆ ^pds dxˆ

¹

p

(ερ)¹⁻

1 p

B_ε

|∇uε|^pdx ¹

p

.

But since sup_ε>0Gε(u_ε, v_ε)=sup_ε>0F_ε(u_ε) <+∞, from the coerciveness property satisfied by g one has

Bε|∇u_ε|^pdx_α^ε so that the previous estimate yields|I_ρ,ε|C(ρ, α)εwhereC(ρ, α)is a positive constant de- pending only onρandα. From this estimate, (5) becomes

–

Qˆρ(xˆ0)

v dxˆ=

–

Qˆρ(xˆ0)

γS(u) dxˆ

for allρ >0. Lettingρ→0 finally givesγ_S(u)(x₀)=v(x₀)for a.e.x₀inS.

It remains to establish assertion (iv). Since Fε(u_ε, μ_ε) < +∞ we have μ_ε = δ_∇vε(x) ⊗ dx and sup_ε>0

B|∇v_ε|^pdx <+∞so that the Young measuresμ_ε andδ_ˆ∇v

ε(x)⊗dxare tight. According to the Prokhorov compactness theorem (Theorem 5 of Appendix A), there exist a subsequence that we do not relabel, and μ∈

∇Y3×3(B),μˆ∈Y3×2(B)such that με=δ_∇v_ε(x)⊗dx μ,^nar δ_ˆ∇v

ε(x)⊗dx^narμ.ˆ (6)

On the other hand, from assertion (ii) δ∂vε

∂x3(x)⊗dx^nar δ0_R3 ⊗dx. (7)

Combining (6) and (7) one easily deduce thatμ_x= ˆμ_x⊗δ₀

R3, and settingν=ν_xˆ⊗dxˆwhereν_xˆ:=₁

0μˆ_x,sˆ ds, we finally obtain thatμε A

ν.

We have to prove thatνbelongs to ˆ∇Y3×2(S). According to the Kinderlehrer–Pedregal characterization theorem (Theorem 6 of Appendix A), it is equivalent to establish the three following assertions:

(KP)1 there existsw∈W^1,p(S,R³)such that bar(νx_ˆ)= ˆ∇w(x)ˆ for a.e.x∈S;

(KP)2

M^3×2|ˆλ|^pdν_xˆ<+∞for a.e.x∈S;

(KP)3 for all quasiconvex functionφsatisfying a growth condition of orderp, φ

bar(ν_xˆ)

M^3×2

φ(ˆλ) dν_xˆ for a.e.xˆ∈S.

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

M^3×2λ dˆ μˆxfor a.e.x∈Band since_∂x^∂v

3 =0, ˆ∇v(x)ˆ =

1 0

M^3×2

ˆ

λ dμˆ_x,sˆ ds=

M^3×2

ˆ λ dν_xˆ

for a.e.xˆinSso thatvis the suitable Sobolev-functionw.

Proof of (KP)2: From the definition ofν_xˆand the lower semicontinuity property for Young measures (Proposition 5 of Appendix A) we have

S

M^3×2

|ˆλ|^pdν_xˆdxˆ=

B

M^3×2

|ˆλ|^pdμˆ_xdx

=

B

M^3×3

|ˆλ|^p(dμˆ_x⊗δ₀

R3) dx

=

B×M^3×3

|ˆλ|^pdμ

B×M^3×3

|λ|^pdμ

lim inf

ε→0

B

|∇vε|^pdxsup

ε>0

Fε(uε, με) <+∞

which proves that

M^3×2|ˆλ|^pdν_xˆ is finit for a.e.xˆinS.

Proof of (KP)3: Letφ:M^3×2→Rbe a quasiconvex function satisfying a growth condition of orderpand define the functionφ˜:M^3×3→Rbyφ(λ)˜ =φ(ˆλ). It is easy to check thatφ˜ is quasiconvex and clearly satisfies the same growth condition. Sinceμ∈ ∇Y3×3(B)we have for a.e.xinB

φ ˆ∇v(x)ˆ

= ˜φ

∇v(x)

M^3×3

φ(λ) dμ˜ _x

=

M^3×3

φ(ˆλ) dμˆ_x⊗δ₀

R3 =

M^3×2

φ(ˆλ) dμˆ_x

so that for a.e.x inS φ

ˆ∇v(x)ˆ

= 1 0

φ ˆ∇v(x)ˆ

ds 1 0

M^3×2

φ(ˆλ) dμˆ_x,sˆ ds=

M^3×2

φ(ˆλ) dν_xˆ. 2

Remark 1. In the proof above we established A(∇Y3×3(B)) ⊂ ˆ∇Y3×2(S). In fact it is easy to check that A(∇Y3×3(B))= ˆ∇Y3×2(S).

Consider the functional defined in Section 2.3:

F:L^p Ω,R³

×Y3×2(S)→R∪ {+∞}, F(u, ν):= ^ΩQf (∇u) dx+

S×M^3×2g0(ˆλ) dν+I (u, ν) ifu∈W_Γ^1,p

0 (Ω,R³), γSu∈W^1,p(S,R³),

+∞ otherwise.

We establish theΓ_X,Y,Yˆ-convergence of the sequence(Fε)ε>0to the functionalFby means of the two next proposi- tions.

Proposition 2(Lower bound). Let(u, ν)be any pair inL^p(Ω,R³)×Y3×2(S). Then, for all sequence((uε, με))ε>0

inL^p(Ω,R³)×Y3×3(B)converging to(u, ν), we have F(u, ν)lim inf

ε→0 Fε(uε, με).

Proof. One can assume lim infε→0Fε(uε, με) <+∞ otherwise there is nothing to prove. Consequently, from Lemma 1 we have

u∈W_Γ^1,p

0 (Ω,R³);

ν∈ ˆ∇Y3×2(S); bar(νx_ˆ)= ˆ∇γ_S(u)(x)ˆ for a.e.xˆ∈S.

This proves thatI (u, ν)=0 and it suffices to establish the two following estimates:

S×M³×2

g₀(ˆλ) dνlim inf

ε→0

B

g ˆ∇v_ε

1 ε

∂vε

∂x3

dx; (8)

Ω

Qf (∇u) dxlim inf

ε→0

Ω_ε

f (∇u_ε) dx. (9)

Proof of (8): From the lower semicontinuity property for Young measures (see Proposition 5 of Appendix A) and assertion (iv) of Lemma 1, it follows that

lim inf

ε→0

B×M^3×3

g λˆ

1 ελ₃

dμ_εlim inf

ε→0

B×M^3×3

g₀(ˆλ) dμ_ε

B×M^3×3

g₀(ˆλ) dμ

=

B×M^3×3

g0(ˆλ) d(μˆx⊗δ₀3 R)⊗dx

=

S×M^3×2

g₀(ˆλ) dν_xˆ⊗dxˆ=

S×M^3×2

g₀(ˆλ) dν.

Proof of (9): For fixedη > εwe have

Ωε

f (∇u_ε) dx

Ωη

f (∇u_ε) dx

Ωη

Qf (∇u_ε) dx

and, sincew→

ΩηQf (∇w) dxis lower semicontinuous for the weak convergence inW^1,p(Ω_η,R³), we deduce lim inf

ε→0

Ωε

f (∇u_ε) dx

Ωη

Qf (∇u) dx.

We end the proof by lettingη→0. 2

For establishing the upper bound in the definition of our variationel convergence, we need to prove the following relaxation result

Lemma 2. Let (u, ν) in W_Γ^1,p

0 (Ω,R³)× ˆ∇Y3×2(S) with ˆ∇γS(u)(x)ˆ =bar(νx_ˆ) for a.e. x in S and a sequence ((u_n, v_n))_n∈NinW_Γ^1,p

0 (Ω,R³)×W^1,p(S,R³)such thatu_nweakly converges touinW_Γ^1,p

0 (Ω,R³),δ_ˆ∇v

n(x)ˆ ⊗dx^nar ν inY3×2(S)and

n→+∞lim

Ω

f (∇u_n) dx=

Ω

Qf (∇u) dx,

n→+∞lim

S

g₀(ˆ∇v_n) dxˆ=

S×M^3×2

g₀(ˆλ) dν. (10)

Then there exists a sequence(u˜_n)_n∈Nsatisfying all the conditions fulfilled by(u_n)_n∈Nand furthermore which satisfies γ_S(u˜_n)=v_n.

Proof. Such a sequence((u_n, v_n))_n∈Nexists, consult for instance [6, Theorem 11.2.1 and Theorem 11.4.2]. Note that one may assume(|∇u_n|^p)_n∈Nuniformly integrable. Indeed, consider the sequence(u˜_n)_n∈Nwhose gradients generate