Bending Moment in Membrane Theory

GUY BOUCHITTÉ

Département de Mathématiques, Université de Toulon et du Var-BP 132 83957 La Garde Cedex, France bouchitte@univ-tln.fr

IRENE FONSECA

Department of Mathematical Sciences, Carnegie Mellon University Pittsburgh, PA 15213, USA fonseca@andrew.cmu.edu

M.LUÍSA MASCARENHAS

C.M.A.F., Universidade de Lisboa, Av. Prof. Gama Pinto 2 1649-003 Lisboa Codex, Portugal mascar@ptmat.fc.ul.pt

Abstract. A model of nonlinear membrane is proposed where the external loading induces a density of bending moments. The mathematical justification of this model is obtained through a 3D-2D dimension reduction via a Γ-convergence analysis taking into account the Cosserat vector field.

AMS classification numbers : 35E99, 35M10, 49J45, 74B20, 74K15, 74K20, 74K35

Key words : Dimension reduction, Γ-convergence, thin films, relaxation, quasiconvexity.

1. Introduction and main results.

In this paper we deduce, via Γ-convergence, a 2-D membrane model from a 3-D nonlinear elasticity framework where we consider a class of surface forces generating the bending moment.

Let ω be an open bounded subset of ℝ^{2} and let I be the interval (−1/2, 1/2). Define Ω := ω × I, Σ^{±} := ω × {±1/2}, Γ := ∂ω × I and, for each ε > 0, Ω_{ε} := ω × εI, Σ_{ε}^{±} := ω × {±ε}, Γ_{ε} := ∂ω × εI.

In what follows L^{N} stands for the N-dimensional Lebesgue measure in ℝ^{N}, N = 2, 3, and H^{2} denotes the 2-dimensional Hausdorff measure in ℝ^{3}. Greek indexes will be used to distinguish the first two components of a tensor, for instance (x_{α}) and (x_{α}, x_{3}), designates (x_{1}, x_{2}) and (x_{1}, x_{2}, x_{3}), respectively.

We write M^{3×2} to denote the vector space of 3 × 2 real-valued matrices, and for \overline{F} ∈ M^{3×2} and b ∈ ℝ^{3}, let (\overline{F} |b) denote the 3 × 3 matrix whose first two columns are those of \overline{F} and the last one is b.

As it is usual in the study of thin structures (e.g. [BB],[BF],[LDR1],[LDR2], [FF], [BJ], [BFF], [BrFF] ; see also [ABP],[FMJ]), the rescaled total energy of a deformation U of Ω_{ε} is given by

\frac{1}{ε} ∫_{Ω_{ε}} W (DU) dx̃ − < F_{ε} U >

where DU = (D_{α}U|D_{3}U) is the strain of a deformation U ∈ W^{1,p}(Ω_{ε}; ℝ^{3}), W satisfies some suitable growth hypotheses (see (H1)) and F_{ε} stands for the external loading. Here we take F_{ε} as the sum of a bulk density and a surface density on Σ_{ε}^{±}, precisely,

F_{ε} := f_{ε} L^{3}⌊Ω_{ε} + (g_{ε}^{+} H^{2}⌊Σ_{ε}^{+} − g_{ε}^{−} H^{2}⌊Σ_{ε}^{−}),  (1.1)

where

f_{ε} := \frac{1}{ε}f (x̃_{α}, \frac{x̃_{3}}{ε} ) , f ∈ L^{p′}(Ω; ℝ^{3})

and

g_{ε}^{±} := g_{0}^{±} + \frac{1}{ε}g , g_{0}^{±}, g ∈ L^{p′}(ω; ℝ^{3}),

where p′ = p/(p − 1), for a fixed p ∈ (1, +∞).

In the classical approach for recovering the membrane energy from a 3D model, the surface load term

scaled with factor \frac{1}{ε} is not present so that F_{ε} converges weakly-⋆ to a surface load of density f̄+ g_{0}^{+} − g_{0}^{−} on ω × {0}. In our case, the limit of F_{ε} includes, in addition, a first order distribution which represents a density of bending moments.

Let W_{Γ_{ε}}^{1,p}(Ω_{ε}; ℝ^{3}) (respectively W_{Γ}^{1,p}(Ω; ℝ^{3})) denote the space of functions in W^{1,p}(Ω_{ε}; ℝ^{3}) (respectively W^{1,p}(Ω; ℝ^{3})) that vanish on Γ_{ε} (respectively on Γ). Assuming that the deformations of the body satisfy a boundary condition of place on Γ_{ε}, the equilibrium problem under the load F_{ε} given in (1.1) can be formulated as a minimization problem

inf_{U−x̃∈W_{Γε}^{1,p}(Ω_{ε};ℝ^{3})} \{\frac{1}{ε} ∫_{Ω_{ε}} W (DU) dx̃ − < F_{ε}, U >\} .  (1.2)

In the sequel we will assume that the potential W is a Borel function satisfying the p-growth conditions

(H1)  \frac{1}{C} |ξ|^{p} − C ≤ W (ξ) ≤ C(1 + |ξ|^{p}),

for a certain positive constant C and for all ξ ∈ M^{3×3}.

The existence of a solution for problem (1.2) can be obtained via the direct method of the calculus of variations under the additional hypothesis that W is quasiconvex.

In order to pass to the limit in problem (1.2) as the thickness ε goes to zero, we perform the usual change of variables that transforms the problem from the thin, varying domain Ω_{ε}, into the fixed domain Ω : to

each x̃ = (x̃_{α}, x̃_{3}) ∈ Ω_{ε} we associate x = (x_{α}, x_{3}) = (x̃_{α}, \frac{1}{ε}x̃_{3}) ∈ Ω, and define u, u^{±} and u_{0,ε} by

u(x_{α}, x_{3}) := U(x̃_{α}, x̃_{3}) , u^{±}(x_{α}) := u (x_{α}, ±\frac{1}{2}) , u_{0,ε}(x_{α}, x_{3}) := (x_{α}, εx_{3}).

Taking into account (1.1), we may rewrite (1.2) as

(P_{ε})  inf_{u−u_{0,ε}∈W_{Γ}^{1,p}(Ω;ℝ^{3})} \{∫_{Ω} W (D_{α}u | \frac{1}{ε}D_{3}u) dx − L_{ε}(u)\} ,

where now the work L_{ε}(u) of the external loads is given by

L_{ε}(u) := ∫_{Ω} f u dx + ∫_{ω}(g_{0}^{+} u^{+} − g_{0}^{−} u^{−})dx_{α} + ∫_{ω} g (\frac{u^{+} − u^{−}}{ε} ) dx_{α}.

¿From (H1), together with the boundary conditions u_{ε} = u_{0,ε} on Γ, it follows that any minimizing sequence {u_{ε}} satisfies the bound

sup_{ε} \{∫_{Ω} |D_{α}u_{ε}|^{p}dx + ∫_{Ω} \frac{1}{ε^{p}} |D_{3}u_{ε}|^{p}dx\} < +∞.

This allow us to infer the weak compactness of \{(u_{ε}, b_{ε})\} in W^{1,p}(Ω; ℝ^{3}) × L^{p}(Ω; ℝ^{3}), where b_{ε} := \frac{1}{ε}D_{3}u_{ε}, and therefore it is natural to expect a limit problem involving the corresponding weak limit (u, b). In fact, as it turns out, u will depend only on x_{α} (see Section 3), whereas the vector field b is, in general, no longer related with the bidimensional tangential strain tensor of u. In other words, u and b have to be considered as independent macroscopic entities.

We remark that, since our model includes a bending moment of density g, the limiting energy involves explicitly the average b̄ of b, with respect to x_{3}. More precisely, recalling that u = u(x_{α}), we have lim_{ε→0} L_{ε}(u_{ε}) = L(u, b̄) , where L is the linear function given by

L(u, b̄) := ∫_{ω} f̄ u dx_{α} + ∫_{ω}(g_{0}^{+} − g_{0}^{−}) u dx_{α} + ∫_{ω} g b̄ dx_{α}  (1.3)

and where

b̄(x_{α}) := ∫_{I} b(x_{α}, x_{3}) dx_{3} and f̄(x_{α}) := ∫_{I} f(x_{α}, x_{3}) dx_{3}.  (1.4)

We notice that the first equality in (1.4) is an immediate consequence of the fact that b is the weak limit of

b_{ε} = \frac{1}{ε}D_{3}u_{ε} in L^{p}(Ω; ℝ^{3}) and b̄ is the weak limit in L^{p}(ω; ℝ^{3}) of the sequence \frac{u_{ε}^{+} − u_{ε}^{−}}{ε}.

In order to individualize this new variable b̄ in the principal part of the total energy, we introduce the following functional E_{ε} : W^{1,p}(Ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}) → \overline{ℝ}, defined by

E_{ε}(u, b̄) := \{ \begin{matrix}∫_{Ω} W (D_{α}u |\frac{1}{ε}D_{3}u) dx & if \frac{1}{ε} ∫_{I} D_{3}u(x_{α}, x_{3}) dx_{3} = b̄(x_{α}), \\ +∞ & otherwise, \end{matrix}  (1.5)

Our aim is to prove that this functional Γ-converges, with respect to the weak topology of W^{1,p}(Ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}), to some functional E(u, b̄). As shown in Theorem 1.2 below, this limit energy admits an integral representation which we propose as an alternative model for non linear membranes with bending moments density. This new model is described by introducing the potential Q^{∗}W (\overline{F} |b) defined, for every \overline{F} ∈ M^{3×2} and b ∈ ℝ^{3}, as

Q^{∗}W (\overline{F} |b) := inf_{(φ,λ)} \{∫_{Q} W (\overline{F} + D_{α}φ|λ D_{3}φ) dx : λ ∈ ℝ ,
φ ∈ W^{1,p}(Q; ℝ^{3}), φ(·, x_{3}) is Q′ -periodic L^{1} a.e. x_{3} ∈ I, λ ∫_{Q} D_{3}φ dx = b} ,

(1.6)

with Q := I^{3}, Q′ := I^{2}.

The main properties of the potential Q^{∗}W are summarized in the proposition below, proved in Section 2.

Proposition 1.1. Let Q^{∗}W be defined by (1.6). Under hypothesis (H1) :

i) The following inequality holds

CW ≤ Q^{∗}W ≤ QW,

where CW and QW are the convex and quasiconvex envelopes of W, respectively. ii) There exists a positive constant C such that for all \overline{F} ∈ M^{3×2}

\frac{1}{C} (|\overline{F} |^{p} + |b|^{p}) − C ≤ Q^{∗}W (\overline{F} |b) ≤ C (1 + |\overline{F} |^{p} + |b|^{p}) .  (1.7)

iii) There holds

Q^{∗}(QW) = Q^{∗}W.  (1.8)

iv) Let W_{0} : M^{3×2} → ℝ be given by

W_{0}(\overline{F}) := inf_{b∈ℝ^{3}} \{W (\overline{F} |b) : b ∈ ℝ^{3}\}

and denote by Q_{2}W_{0} its quasiconvex envelope. Then we have

inf_{b∈ℝ^{3}} Q^{∗}W (\overline{F} |b) = Q_{2}W_{0}(\overline{F}).  (1.9)

Let us introduce the space V, that we will identify with W^{1,p}(ω; ℝ^{3}),

V := {u ∈ W^{1,p}(Ω; ℝ^{3}) | D_{3}u(x) = 0 a.e. in x ∈ Ω},

and consider the functional

E(u, b̄) := ∫_{ω} Q^{∗}W (D_{α}u | b̄)dx_{α} , if (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}).

We may now state the main results of this paper, which will be proved in Section 3 :

Theorem 1.2. Under the hypothesis (H1), the sequence {E_{ε}} Γ-converges to E, as ε → 0, precisely,

i) if u_{ε} ⇀ u in W^{1,p}(Ω; ℝ^{3}) and \frac{u_{ε}^{+} − u_{ε}^{−}}{ε} ⇀ b̄ in L^{p}(ω; ℝ^{3}) then

lim inf_{ε→0} ∫_{Ω} W (D_{α}u_{ε} |\frac{1}{ε}D_{3}u_{ε}) dx ≥ E(u, b̄) ;

ii) for every pair (u, b̄) in V × L^{p}(ω; ℝ^{3}), there exists a sequence {u_{ε}} such that

(u_{ε}, \frac{u_{ε}^{+} − u_{ε}^{−}}{ε} ) ⇀ (u, b̄) , lim_{ε→0} ∫_{Ω} W (D_{α}u_{ε} |\frac{1}{ε}D_{3}u_{ε}) dx = E(u, b̄) .

Corollary 1.3. Let W satisfy hypothesis (H1). Let f ∈ L^{p′}(Ω, ℝ^{3}), g_{0}^{±}, g ∈ L^{p′}(ω, ℝ^{3}), and let u_{ε} be a

minimizer for (P_{ε}). Then the sequence \{(u_{ε}, \frac{u_{ε}^{+} − u_{ε}^{−}}{ε} )\} is weakly relatively compact in W^{1,p}(Ω, ℝ^{3}) ×

L^{p}(ω, ℝ^{3}). Furthermore, any cluster point (u, b̄) of this sequence belongs to V × L^{p}(ω, ℝ^{3}) and is a solution of the problem

(P)  min_{{u−x∈W{1,p \atop 0}(ω;ℝ^{3}) \atop b̄∈L^{p}(ω,ℝ^{3})}} \{∫_{ω} Q^{∗}W (D_{α}u | b̄)dx_{α} − L(u, b̄)\} ,

where L(u, b̄) is defined by (1.3).

Remark 1.4. The boundary condition considered in (P_{ε}) were those treated in [LDR1] and [BJ]. It is easy to see that all the conclusions of this work still stand with obvious adaptations, if instead of taking u_{0,ε}(x_{α}, x_{3}) := (x_{α}, εx_{3}) for (x_{α}, x_{3}) ∈ Ω, we assume that {u_{0,ε}} is weakly-⋆ converging to some function u_{0} in W^{1,∞}(Ω; ℝ^{3}) and ∥D_{3}u_{0,ε}∥_{∞} ≤ Cε for some C > 0. Note that in the statement of problem (P), in Corollary 1.3, the Dirichlet boundary condition will now read u − u_{0} ∈ W_{0}^{1,p}(Ω; ℝ^{3}). Furthermore the above uniform bounds may be replaced by the condition of equi-integrability of the sequence \{|D_{α}u_{0,ε}|^{p} + |\frac{1}{ε}D_{3}u_{0,ε}|^{p}\}.

Remark 1.5. By Theorem 1.2 and by virtue of the coercivity condition in (1.7), it turns out that the functional

(u, b̄) ∈ V × L^{p}(ω; ℝ^{3}) ↦ ∫_{ω} Q^{∗}W (D_{α}u | b̄) dx_{α}

is weakly lower semicontinuous. Therefore, by [FM] (see also [FKP]), we conclude that Q^{∗}W (· | b̄) is quasiconvex and that Q^{∗}W (\overline{F} |·) is convex.

Remark 1.6. If no bending moments are present, that is if g = 0, then L(u, b̄) does not depend on b̄ and we may perform explicitly the infimum in (P) with respect to b̄, by passing it under the integral sign. By (1.9), we are reduced to solving the following minimum problem in u

(\overline{P}),  min_{u−x∈W_{0}^{1,p}(ω;ℝ^{3})} \{∫_{ω} Q_{2}W_{0}(D_{α}u)dx_{α} − (∫_{ω} f̄ u dx_{α} + ∫_{ω}(g_{0}^{+} − g_{0}^{−}) u dx_{α})\} .

and thus we have recovered, as a particular case, the membrane problem treated in [LDR1].

Remark 1.7. We stress the fact that strict inequalities between CW, Q^{∗}W and QW may occur in part i) of Proposition 1.1. In particular, we may have Q^{∗}W < QW even if W is quasiconvex. In other words, our model, which differs from the classical membrane model discussed in [LDR1] as it involves the Cosserat vector field b, cannot be derived by simply replacing the third column of the deformation tensor by b.

To show that we may have Q^{∗}W < QW , we consider as in [LDR1] a quasiconvex function W which is frame invariant and such that W (𝕀) = 0, where 𝕀 is the identity matrix in ℝ^{3}, whereas W (F) > 0 if F ∉ SO(3). As an example, we may take W (F) := ∑_{i=1}^{i=3}(v_{i}(F)^{2} − 1)_{+}^{2} + | det F − 1| , where v_{i}(F),i = 1, 2, 3, denote the the singular values of F. If Q^{∗}W ≡ QW, then by taking the infimum with respect to the third column and applying (1.9) we would obtain that Q_{2}W_{0} ≡ W_{0}, and this is incompatible with [Corollary 11, [LDR1]). Precisely, Q_{2}W_{0}(\overline{F}) = 0 < W_{0}(\overline{F}) holds for all \overline{F} ∈ M^{3×2} such that the eigenvalues v_{1}(\overline{F}), v_{2}(\overline{F})

of \sqrt{\overline{F}^{T} \overline{F}} satisfy 0 ≤ v_{1}(\overline{F}) < v_{2}(\overline{F}) ≤ 1.

The strict inequality between CW and Q^{∗}W can be obtained by considering the potential introduced in [KoS]

W (F) := \{ \begin{matrix}1 + |F − I|^{2} & if F ≠ 𝕀, \\ 0 & otherwise, \end{matrix}

where | · | denotes the Euclidean norm. Recall that W is not quasiconvex. The identity Q^{∗}W ≡ CW would imply, once again by taking the infimum with respect to the third column and applying (1.9), that Q_{2}W_{0} ≡ (CW)_{0}. A straightforward computation leads to

W_{0}(\overline{F}) = \{ \begin{matrix}1 + |\overline{F} − \overline{I}|^{2} & if \overline{F} ≠ \overline{I}, \\ 0 & otherwise, \end{matrix} and (CW)_{0}(\overline{F}) = \{ \begin{matrix}1 + |\overline{F} − \overline{I}|^{2} & if |\overline{F} − \overline{I}| ≥ 1, \\ 2|\overline{F} − \overline{I}| & if |\overline{F} − \overline{I}| ≤ 1 , \end{matrix}

where \overline{𝕀} the 3 × 2 matrix associated with 𝕀. For every Z = (z_{ij}) ∈ M^{2×2} set

\overline{F} (Z) = ( \begin{matrix}1 + z_{11} & z_{12} \\ z_{21} & 1 + z_{22} \\ 0 & 0 \end{matrix} ) .

Introduce also the 2D version of Kohn and Strang potential and its convexification

f(Z) := \{ \begin{matrix}1 + |Z|^{2} & if Z ≠ 0, \\ 0 & otherwise, \end{matrix} and f^{∗∗}(Z) = \{ \begin{matrix}1 + |Z|^{2} & if |Z| ≤ 1, \\ 2|Z| & if |Z| ≥ 1. \end{matrix}

Then one has (CW)_{0}(\overline{F} (Z)) = f^{∗∗}(Z). On the other hand, it is easy to check that Q_{2}W_{0}(\overline{F} (Z)) = Qf(Z) where, according to [KoS], the quasiconvex envelope Qf is given by

Qf(Z) = \{ \begin{matrix}1 + (tr Z)^{2} − 2 det Z & if |tr Z| ≥ 1, \\ 2 tr Z − 2 det Z & otherwise, \end{matrix}

and, consequently, it differs from f^{∗∗}(Z).

2. Preliminary results.

2.1 Proof of Proposition 1.1.

(i) Recall that, for \overline{F} ∈ M^{3×2} and b ∈ ℝ^{3},

QW (\overline{F} |b) = inf_{ψ} \{∫_{Q} W (\overline{F} + D_{α}ψ |b + D_{3}ψ) dx : ψ ∈ W_{#}^{1,p}(Q; ℝ^{3})\} ,  (2.1)

where W_{#}^{1,p}(Q; ℝ^{3}) stands for the space of all W^{1,p}(Q; ℝ^{3}) functions that are Q-periodic.

For each ψ ∈ W_{#}^{1,p}(Q; ℝ^{3}), define φ(x) := ψ(x) + bx_{3}. Since D_{α}φ = D_{α}ψ, D_{3}φ = D_{3}ψ + b and ∫_{I} D_{3}ψdx_{3} = 0, we obtain, in view of (1.6) with λ = 1,

\begin{aligned}Q^{∗}W (\overline{F} |b) & ≤ ∫_{Q} W (\overline{F} + D_{α}φ | D_{3}φ) dx \\ & = ∫_{Q} W (\overline{F} + D_{α}ψ | b + D_{3}ψ) dx. \end{aligned}

Taking the infimum in all ψ we get from (2.1)

Q^{∗}W (\overline{F} |b) ≤ QW (\overline{F} |b).

Consider now (φ, λ) such that φ ∈ W^{1,p}(Q; ℝ^{3}), φ(·, x_{3}) is Q′-periodic for L^{1} a.e. x_{3} ∈ I and λ ∫_{Q} D_{3}φdx_{3} = b. Define ψ(x) := φ(x) − bx_{3}/λ. We have D_{α}ψ = D_{α}φ, D_{3}ψ = D_{3}φ − b/λ and ∫_{Q} D_{α}ψdx = ∫_{Q} λD_{3}ψdx = 0. Then, by Jensen inequality,

∫_{Q} W (\overline{F} + D_{α}φ | λD_{3}φ) dx = ∫_{Q} W (\overline{F} + D_{α}ψ | b + λD_{3}ψ) dx
≥ ∫_{Q} CW (\overline{F} + D_{α}ψ | b + λD_{3}ψ) dx
≥ CW (\overline{F} , b).

Taking the infimum in all (φ, λ), we deduce that CW (\overline{F} , b) ≤ Q^{∗}W (\overline{F} |b).

(ii) The bounds in (1.7) are a straightforward consequence of hypothesis (H1), of the convexity of |ξ|^{p}, and of the inequalities

CW ≤ Q^{∗}W ≤ QW ≤ W.

(iii) To prove (1.8), first note that trivially Q^{∗}W (\overline{F} |b) ≥ Q^{∗}(QW)(\overline{F} |b).

To show the converse inequality we prove the auxiliary result

(QW_{ε})(ξ) = (QW)_{ε}(ξ), with W_{ε}(ξ) := W (ξ_{α}| \frac{1}{ε} ξ_{3}) .  (2.2)

Recall that, under the growth hypothesis (H1), the quasiconvex envelop of W is given by

QW (ξ) = inf \{\frac{1}{L^{3}(U)} ∫_{U} W (ξ + Dφ) dx | φ ∈ W_{0}^{1,p}(U; ℝ^{3})\} ,  (2.3)

provided L^{3}(∂U) = 0, where U is any open bounded subset of ℝ^{3}.

Let δ > 0 and choose φ ∈ W_{0}^{1,p}(Q; ℝ^{3}) such that

\begin{aligned}(QW_{ε})(ξ) & ≥ ∫_{Q} W_{ε} (ξ + Dφ) dx − δ \\ & = ∫_{Q} W (ξ_{α} + D_{α}φ | \frac{1}{ε}ξ_{3} + \frac{1}{ε}D_{3}φ) dx − δ. \end{aligned}

Changing variables (y_{α} = x_{α}, y_{3} = εx_{3}) and letting ψ(y) := φ(y_{α}, ε^{−1} y_{3}), U_{ε} := Q′ × εI, we obtain

(QW_{ε})(ξ) ≥ \frac{1}{ε} ∫_{Q′×εI} W (ξ_{α} + D_{α}ψ | \frac{1}{ε}ξ_{3} + D_{3}ψ) dy − δ

= \frac{1}{L^{3}(U_{ε})} ∫_{U_{ε}} W (ξ_{α} + D_{α}ψ | \frac{1}{ε}ξ_{3} + D_{3}ψ) dy − δ.

≥ (QW)_{ε}(ξ) − δ.

Letting δ → 0 we conclude that (QW_{ε})(ξ) ≥ (QW)_{ε}(ξ). The opposite inequality is proven in a similar way.

Fix δ > 0 and let φ ∈ W^{1,p}(Q; ℝ^{3}), φ(·, x_{3}) Q′− periodic, and λ ∈ ℝ, satisfying λ ∫_{Q} D_{3}φ dx = b, and

Q^{∗}(QW)(\overline{F} |b) ≥ ∫_{Q}(QW)(\overline{F} + D_{α} φ | λ D_{3}φ) dx − δ.  (2.4)

It is easy to show that for any φ ∈ W^{1,p}(Q; ℝ^{3}), φ(·, x_{3}) Q′− periodic, there exists a sequence of piecewise affine functions {φ_{n}} such that φ_{n}(·, x_{3}) is Q′− periodic and

lim_{n} ∥φ_{n} − φ∥_{W^{1,p}(Q;ℝ^{3})} = 0.

Let {φ_{n}} be such a sequence, with respect to φ, and consider

ψ_{n}(x) := φ_{n}(x) + (\frac{b}{λ} − ∫_{Q} D_{3}φ_{n} dx) x_{3}.

Then ψ_{n}(·, x_{3}) is still Q′− periodic, λ ∫_{Q} D_{3}ψ_{n} dx = b and lim_{n} ∥ψ_{n} −φ∥_{W^{1,p}(Q;ℝ^{3})} = 0. Up to a subsequence,

and almost everywhere in Q, we have

C(1 + |∇ψ_{n}|^{p}) − QW (\overline{F} + D_{α}ψ_{n} |λD_{3}ψ_{n}) → C(1 + |∇φ|^{p}) − QW (\overline{F} + D_{α}φ |λD_{3}φ),

where we have used the fact that, by, (H1), QW is p-Lipschitz continuous (see [M]), and using Fatou’s Lemma we obtain

lim sup_{n} ∫_{Q} QW (\overline{F} + D_{α}ψ_{n} |λD_{3}ψ_{n}) ≤ ∫_{Q} QW (\overline{F} + D_{α}φ |λD_{3}φ).  (2.5)

From (2.4) and (2.5) we get, for some n big enough,

Q^{∗}(QW)(\overline{F} |b) ≥ ∫_{Q}(QW)(\overline{F} + D_{α} ψ_{n} | λ D_{3}ψ_{n}) dx − 2δ.  (2.6)

Let Dψ_{n} = ∑_{i=1}^{k} χ_{U_{i}} (A_{i} | η_{i}), where χ_{U_{i}} denotes the characteristic function of U_{i}, L^{N}(∂U_{i}) = 0, A_{i} ∈ M^{3×2}

and η_{i} ∈ ℝ^{3}, for all i = 1, ⋯, k. Recalling (2.2) and (2.3), inequality (2.6) becomes, for suitable θ_{i} ∈ W_{0}^{1,p}(U_{i}; ℝ^{3}),

Q^{∗}(QW)(\overline{F} |b) ≥ ∑_{i=1}^{k} L^{N}(U_{i}) QW (\overline{F} + A_{i} |λ η_{i}) − 2δ

= ∑_{i=1}^{k} L^{N}(U_{i}) (QW)_{λ^{−1}}(\overline{F} + A_{i} | η_{i}) − 2δ

= ∑_{i=1}^{k} L^{N}(U_{i}) (QW_{λ^{−1}})(\overline{F} + A_{i} | η_{i}) − 2δ

≥ ∑_{i=1}^{k} ∫_{U_{i}} W_{λ^{−1}}(\overline{F} + A_{i} + D_{α} θ_{i} | η_{i} + D_{3}θ_{i}) dx − 3δ

= ∫_{Q} W (\overline{F} + D_{α} Λ | λ D_{3} Λ) dx − 3δ ≥ Q^{∗}W (\overline{F} |b) − 3δ,

where Λ := ψ_{n} + ∑_{i=1}^{k} χ_{U_{i}}θ_{i}. Since θ_{i}|_{∂U_{i}} = 0, Λ(·, x_{3}) is Q′− periodic and λ ∫_{Q} D_{3}Λdx = b. Letting δ → 0,

we conclude that Q^{∗}(QW)(\overline{F} |b) ≥ Q^{∗}W (\overline{F} |b).

(iv) Using the definition of quasiconvex envelope of W_{0}, we have

Q_{2}W_{0}(\overline{F}) = inf_{φ∈W^{1,p}(Q′;ℝ^{3}) φ is Q′−periodic} ∫_{Q′} inf_{c∈ℝ^{3}} W (\overline{F} + D_{α}φ(x_{α})|c) dx_{α}

= inf_{φ∈W^{1,p}(Q′;ℝ^{3}) φ is Q′−periodic b∈L^{p}(Q′;ℝ^{3})} ∫_{Q′} W (\overline{F} + D_{α}φ(x_{α})|b(x_{α})) dx_{α},

(2.7)

where the last equality is justified using the Measurable Selection Theorem (see [CV]) and the coercivity hypothesis (H1). The second equality in (2.7) still holds if we consider test function depending in x_{3}, and thus

Q_{2}W_{0}(\overline{F}) = inf_{φ∈W^{1,p}(Q;ℝ^{3}) φ is Q′−periodic b∈L^{p}(Q;ℝ^{3})} ∫_{Q′×I} W (\overline{F} + D_{α}φ(x_{α}, x_{3})|b(x_{α}, x_{3})) dx_{α} dx_{3}

≤ inf_{(λ,c)∈ℝ×ℝ^{3} φ∈W^{1,p}(Q;ℝ^{3}) φ is Q′−periodic λ∫_{Q} D_{3}φdx=c} ∫_{Q} W (\overline{F} + D_{α}φ(x)|λ D_{3}φ(x)) dx

= inf_{c∈ℝ^{3}} Q^{∗}W (\overline{F} |c).

(2.8)

Then Q_{2}W_{0}(\overline{F}) ≤ inf_{c∈ℝ^{3}} Q^{∗}W (\overline{F} |c). To prove the opposite inequality, in view of (1.8) we may admit,

without loss of generality, that W is quasiconvex and this, together with hypothesis (H1), implies that W is locally Lipschitz (see [M]). In fact, if the inequality holds for the quasiconvex envelope of W, we will have

inf_{c∈ℝ^{3}} Q^{∗}W (\overline{F} |c) = inf_{c∈ℝ^{3}} Q^{∗}(QW)(\overline{F} |c) ≤ Q_{2}(QW)_{0} (\overline{F} |c) ≤ Q_{2}W_{0}(\overline{F} |c).

Let φ ∈ W^{1,p}(Q′; ℝ^{3}) and b ∈ L^{p}(Q′; ℝ^{3}) be admissible functions for the infimum in the right hand side of (2.7). Let (b_{n}) be a sequence in W_{0}^{1,p}(Q′; ℝ^{3}), such that b_{n} → b in L^{p}(Q′; ℝ^{3}) such that there exists

lim_{n} ∥D_{α}b_{n}∥_{L^{p}(Q′;ℝ^{3})} and define

φ̃_{n}(x) := φ(x_{α}) + \frac{x_{3}}{λ_{n}} b_{n}(x_{α}),

where

λ_{n} := \{ \begin{matrix}n & if lim_{n} ∥D_{α}b_{n}∥_{L^{p}(Q′;ℝ^{3})} < +∞, \\ n ∥D_{α}b_{n}∥_{L^{p}(Q′;ℝ^{3})} & otherwise. \end{matrix}

We easily see that φ̃_{n} ∈ W^{1,p}(Q; ℝ^{3}) is Q′-periodic, λ_{n} ∫_{Q} D_{3}φ̃_{n}dx = ∫_{Q′} b_{n}dx_{α} =: c_{n} ∈ ℝ^{3} and

W (\overline{F} + D_{α}φ̃_{n}(x)|λ_{n}D_{3}φ̃_{n}(x)) = W (\overline{F} + D_{α}φ(x_{α}) + \frac{x_{3}}{λ_{n}} D_{α}b_{n}(x_{α})|b_{n}(x_{α})) .

Since \frac{x_{3}}{λ_{n}} D_{α}b_{n}(x_{α}) → 0 and b_{n} → b in L^{p}(Q; ℝ^{3}), the p-Lipschitz continuity of W yields

∫_{Q′} W (\overline{F} + D_{α}φ(x_{α}) |b(x_{α})) dx_{α} = lim_{n} ∫_{Q} W (\overline{F} + D_{α}φ̃_{n}(x)|λ_{n}D_{3}φ̃_{n}(x)) dx

≥ lim sup_{n} Q^{∗}W (\overline{F} |c_{n})

≥ inf_{c∈ℝ^{3}} Q^{∗}W (\overline{F} |c).

(2.9)

Taking the infimum in the left hand side of (2.9), over all the admissible φ in W^{1,p}(Q′; ℝ^{3}) and b in L^{p}(Q′; ℝ^{3}), and using (2.7), by (2.9) yields

Q_{2}W_{0}(\overline{F}) ≥ inf_{c∈ℝ^{3}} Q^{∗}W (\overline{F} |c),

which, together with (2.8), completes the proof of (1.9).

2.2. Properties of the Γ-limit.

We start by localizing the functionals E_{ε} introduced in (1.5). Representing by A(ω) the family of all open subsets of ω, define

E_{ε} : W^{1,p}(Ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}) × A(ω) → \overline{ℝ},

E_{ε}(u, b̄, A) := \{ \begin{matrix}∫_{A×I} W (D_{α}u |\frac{1}{ε}D_{3}u) dx & if \frac{1}{ε} ∫_{I} D_{3}u(x_{α}, x_{3}) dx_{3} = b̄(x_{α}), \\ +∞ & otherwise. \end{matrix}

In order to prove that the sequence {E_{ε}(·, ·, A)} Γ-converges to some functional E_{0}(·, ·, A) for all A ∈ A(ω), it is enough to show that any given sequence {ε_{k}} converging to 0^{+} admits a subsequence {ε_{k_{n}}} such that the Γ-lower limit of E_{ε_{kn}} given by

E^{−}(u, b̄, A) := inf {lim inf_{n} ∫_{A×I} W (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx | u_{n} ⇀ u in w - W^{1,p}(A × I; ℝ^{3}),

λ_{n} ∫_{I} D_{3}u_{n} dx_{3} ⇀ b̄, in w - L^{p}(A; ℝ^{3})} ,

(2.10)

where for simplicity of notations we write λ_{n} := (ε_{k_{n}})^{−1}, coincides with E_{0}(·, ·, A) for all (u, b̄) in V × L^{p}(ω; ℝ^{3}).

We will denote E^{−}(u, b̄) := E^{−}(u, b̄, ω).

Lemma 2.1. Let W be a Borel function, satisfying hypothesis (H1). Any sequence {ε_{k}} converging to 0^{+} admits a subsequence {ε_{k_{n}}} such that for each (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}), the set function E^{−}(u, b̄, ·), defined in (2.10) (with λ_{n} := (ε_{k_{n}})^{−1}), is the trace of a Radon measure, absolutely continuous with respect to the 2-dimensional Lebesgue measure.

Proof. We first prove that

E^{−}(u, b̄, A) ≤ C (L^{2}(A) + ∫_{A} |D_{α}u|^{p} dx_{α} + ∫_{A} |b̄|^{p} dx_{α})  (2.11)

for some positive constant C and for all (u, b̄, A) ∈ V × L^{p}(ω; ℝ^{3}) × A(ω).

Suppose that b̄ ∈ W^{1,p}(ω; ℝ^{3}) and λ_{n} → +∞, where as in (2.10) we set λ_{n} := (ε_{k_{n}})^{−1}. Define u_{n}(x) := u(x_{α}) + \frac{x_{3}}{λ_{n}} b̄(x_{α}). The sequence \{u_{n}\} satisfies u_{n} → u in W^{1,p}(Ω; ℝ^{3}) and λ_{n} D_{3}u_{n} = b̄.

Consequently

\begin{matrix}∫_{A×I} |D_{α}u_{n}|^{p} dx → ∫_{A} |D_{α}u|^{p} dx_{α} , \\ λ_{n}^{p} ∫_{A×I} |D_{3}u_{n}|^{p} dx = ∫_{A} |b̄|^{p} dx_{α} . \end{matrix}  (2.12)

By hypothesis (H1) we get

lim sup_{n} ∫_{A×I} W (D_{α}u_{n} |λ_{n} D_{3}u_{n})dx

≤ lim_{n} C (L^{2}(A) + ∫_{A×I} |D_{α}u_{n}|^{p} dx_{α} + λ_{n}^{p} ∫_{A×I} |D_{3}u_{n}|^{p} dx_{α})

which, together with (2.12), yields (2.11) for b̄ ∈ W^{1,p}(ω; ℝ^{3}). In view of the coercivity condition in (H1) together with the Dirichlet condition we deduce that the Γ- lower limit E^{−}(u, ·, A) is lower semicontinuous (see [DM]) and by density we obtain (2.11) for all b̄ ∈ L^{p}(ω; ℝ^{3}).

Fix a sequence {ε_{k}} converging to 0^{+}, and let R(ω) be the countable subfamily of A(ω) obtained by taking all finite unions of open squares in ω with faces parallel to the axes, centered at x_{α} ∈ ω ∩ ℚ^{2} and with rational edge length. Since L^{p}(Ω; ℝ^{3}) is a separable metric space, using Kuratowski’s Compactness Theorem (see [DM]) and a diagonal argument, we may find a subsequence {ε_{k_{n}}} of {ε_{k}}, such that Γ−lim_{n} E_{ε_{kn}} (u, b̄, A) exists for all u ∈ V and for all A ∈ R(ω).

Let E^{−}(u, b̄, ·) be defined by (2.10) (with λ_{n} := (ε_{k_{n}})^{−1}) . To prove that E^{−}(u, b̄, ·) is the trace of a Borel measure, we begin by showing that

E^{−}(u, b̄, A) ≤ E^{−}(u, b̄, B) + E^{−}(u, b̄, A \ \overline{U})  (2.13)

for all A, B, U ∈ A(ω), with U ⊂⊂ B ⊂⊂ A, and for all u ∈ V. Fix η > 0 and find {v_{n}} ⊂ W^{1,p}(Ω; ℝ^{3}) such

that v_{n} → u in L^{p}((A \ \overline{U}) × I; ℝ^{3}), λ_{n} ∫_{I} D_{3}v_{n} ⇀ b̄, and

lim inf_{n} ∫_{(A\\overline{U})×I} W (D_{α}v_{n}|λ_{n}D_{3}v_{n}) dx ≤ E^{−}(u, b̄, A \ \overline{U}) + η.  (2.14)

Extract a subsequence {n_{j}} for which

lim_{j} ∫_{(A\\overline{U})×I} W (D_{α}v_{n_{j}}|λ_{n_{j}}D_{3}v_{n_{j}}) dx = lim inf_{n} ∫_{(A\\overline{U})×I} W (D_{α}v_{n}|λ_{n}D_{3}v_{n}) dx.  (2.15)

Let B_{0} ∈ R(ω) be such that U ⊂⊂ B_{0} ⊂⊂ B. Then E^{−}(u, b̄, B_{0}) = Γ − lim_{j} E_{λ_{n_{j}}^{−1}}(u, b̄, B_{0}), and thus there

exists a sequence {u_{j}} ⊂ W^{1,p}(Ω; ℝ^{3}) such that u_{j} → u in L^{p}(B_{0} × I; ℝ^{3}), λ_{n_{j}} ∫_{I} D_{3}u_{j} ⇀ b̄, and

lim_{j} ∫_{B_{0}×I} W (D_{α}u_{j}|λ_{n_{j}}D_{3}u_{j}) dx = E^{−}(u, b̄, B_{0}).  (2.16)

Consider G_{n}(v, A) := ∫_{A×I} (1 + |D_{α}v|^{p} + λ_{n}^{p} |D_{3}v|^{p}) dx. Due to the coercivity hypothesis (H1), we

may extract a bounded subsequence, that we will not relabel, from the sequences of measures ν_{j} := G_{n_{j}}(v_{n_{j}}, ·) + G_{n_{j}}(u_{j}, ·) restricted to B_{0} \ \overline{U}, converging in the sense of distributions to some Radon measure ν defined on B_{0} \ \overline{U}.

For every t > 0 let B_{t} := {x_{α} ∈ B_{0}| dist(x_{α}, ∂B_{0}) > t}. Define, for 0 < δ < η, the subsets L_{δ} := B_{η−2δ} \ \overline{B_{η+δ}}. Consider a smooth cut-off function φ_{δ} ∈ C_{0}^{∞}(B_{η−δ}; [0, 1]) such that φ_{δ}(x_{α}) = 1 on B_{η}. As the thickness of the strip L_{δ} is of order δ, we have an upper bound of the form ∥Dφ_{δ}∥_{L^{∞}(B_{η−δ})} ≤ C/δ. Define

w_{j}(x) := u_{j}(x)φ_{δ}(x_{α}) + v_{n_{j}}(x)(1 − φ_{δ}(x_{α})).

Clearly this sequence converges to u in L^{p}(A × I; ℝ^{3}) as j → ∞, satisfies

D_{α}w_{j} = φ_{δ}D_{α}u_{j} + (1 − φ_{δ})D_{α}v_{n_{j}} + D_{α}φ_{δ} ⊗ (u_{j} − v_{n_{j}}), D_{3}w_{j} = φ_{δ}D_{3}u_{j} + (1 − φ_{δ})D_{3}v_{n_{j}}

and λ_{n_{j}} ∫_{I} D_{3}w_{j} ⇀ b̄ . By the growth condition (H1), we have the estimate

∫A×IW (D_{α}w_{j}|λ_{n_{j}}D_{3}w_{j}) dx
≤ ∫_{B_{η}×I} W (D_{α}u_{j}|λ_{n_{j}}D_{3}u_{j}) dx + ∫_{(A\\overline{B_{η−δ}})×I} W (D_{α}v_{n_{j}}|λ_{n_{j}}D_{3}v_{n_{j}}) dx

+ C (G_{n_{j}}(v_{n_{j}}, L_{δ}) + G_{n_{j}}(u_{j}, L_{δ})) + \frac{1}{δ^{p}} ∫_{L_{δ}×I} |v_{n_{j}} − u_{j}|^{p} dx

≤ ∫_{B_{0}×I} W (D_{α}u_{j}|λ_{n_{j}}D_{3}u_{j}) dx + ∫_{(A\\overline{U})×I} W (D_{α}v_{n_{j}}|λ_{n_{j}}D_{3}v_{n_{j}}) dx

+ C (G_{n_{j}}(v_{n_{j}}, L_{δ}) + G_{n_{j}}(u_{j}, L_{δ})) + \frac{1}{δ^{p}} ∫_{L_{δ}×I} |v_{n_{j}} − u_{j}|^{p} dx

Passing to the limit and using (2.14), (2.15) and (2.16), we have

\begin{aligned}E^{−}(u, b̄, A) & ≤ E^{−}(u, b̄, B_{0}) + E^{−}(u, b̄, A \ \overline{U}) + η + Cν(\overline{L_{δ}}) \\ & ≤ E^{−}(u, b̄, B) + E^{−}(u, b̄, A \ \overline{U}) + η + Cν(\overline{L_{δ}}), \end{aligned}

where we have used the fact that

E^{−}(u, b̄, A) ≤ lim inf_{n} E_{λ_{n}^{−1}}(\overline{w}_{n}, b̄_{n}, A) ≤ lim inf_{j} ∫_{A×I} W (D_{α}w_{j}|λ_{n_{j}}D_{3}w_{j}) dx,

where (\overline{w}_{n}, b̄_{n}) := (w_{j}, λ_{n_{j}} ∫_{I} D_{3}w_{j} dx_{3}) if n = n_{j}, and (\overline{w}_{n}, b̄_{n}) := (u, b̄) if n ≠ n_{j}. We let δ go to zero to obtain

E^{−}(u, b̄, A) ≤ E^{−}(u, b̄, B) + E^{−}(u, b̄, A \ \overline{U}) + η + Cν(∂B_{η}).

It suffices to choose a subsequence {η_{n}} such that η_{n} → 0^{+} and ν(∂B_{η_{n}}) = 0, to conclude the proof of (2.13).

Finally we prove that E^{−}(u, b̄, ·) is the trace of a Borel measure. Fix (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}) and let {u_{n}} satisfy, up to a subsequence of {λ_{n}} that we will not relabel,

E^{−}(u, b̄) = lim_{n} ∫_{Ω} W (D_{α}u_{n} |λ_{n} D_{3}u_{n})dx.  (2.17)

Let μ_{n} := (∫_{I} W (D_{α}u_{n} |λ_{n}D_{3}u_{n}) dx_{3}) L^{2}⌊Ω, and let μ be defined, up to a subsequence, as the limit of

{μ_{n}} in the sense of distributions . We prove that E^{−}(u, b̄, A) = μ(A) for all A ∈ A(ω).

We have, from the definition of E^{−},

E^{−}(u, b̄, A) ≤ lim inf_{n} ∫_{A×I} W (D_{α}u_{n} |λ_{n}D_{3}u_{n})dx ≤ μ(\overline{A}).  (2.18)

Fix ε > 0 and let U_{ε} and B_{ε} be open sets such that U_{ε} ⊂⊂ B_{ε} ⊂⊂ A and, invoking (2.11), E^{−}(u, b̄, A\\overline{U}_{ε}) < ε. Inequalities (2.13) and (2.18) yield

\begin{aligned}E^{−}(u, b̄, A) & ≤ ε + E^{−}(u, b̄, B_{ε}) \\ & ≤ ε + μ(\overline{B}_{ε}) ≤ ε + μ(A). \end{aligned}

Letting ε → 0 we obtain

E^{−}(u, b̄, A) ≤ μ(A).  (2.19)

Conversely, for ε > 0 let A_{ε} ⊂⊂ A be such that μ(A \ \overline{A}_{ε}) < ε. Since by (2.17) μ(ω) = E^{−}(u, b̄), by (2.13) and (2.19) we obtain

μ(A) ≤ μ(\overline{A}_{ε}) + ε
= μ(ω) − μ(ω \ \overline{A}_{ε}) + ε
≤ E^{−}(u, b̄) − E^{−}(u, b̄, ω \ \overline{A}_{ε}) + ε
≤ E^{−}(u, b̄, A) + ε.

Letting ε → 0 we conclude that μ(A) ≤ E^{−}(u, b̄, A).

The following lemma is based on a standard argument due to E. De Giorgi and allow us to restrict ourselves, in (2.10), to sequences {u_{n}} having the same trace as its limit u.

Lemma 2.2. Let W be a Borel function, satisfying hypothesis (H1), let A be an open subset of ω, and let L ∈ ℝ. Consider u ∈ V, a sequence {u_{n}} in W^{1,p}(A × I; ℝ^{3}) and a sequence {λ_{n}} in ℝ such that u_{n} → u in L^{p}(A × I; ℝ^{3}), λ_{n} → +∞, λ_{n} ∫_{I} D_{3}φ_{n} dx_{3} ⇀ b̄, weakly in L^{p}(A; ℝ^{3}) and

lim_{n} ∫_{A×I} W (D_{α}u_{n}(x) | λ_{n} D_{3}u_{n}(x)) dx = L.  (2.20)

Then, there exist a subsequence {λ_{n_{k}}} of {λ_{n}} and a sequence {ũ_{k}} in W^{1,p}(A × I; ℝ^{3}) satisfying ũ_{k} = u

on a neighborhood of ∂A × I, ũ_{k} → u in L^{p}(A × I; ℝ^{3}), λ_{n_{k}} ∫_{I} D_{3}ũ_{k} dx_{3} ⇀ b̄ weakly in L^{p}(A; ℝ^{3}), and

lim sup_{k} ∫_{A×I} W (D_{α}ũ_{k}(x) | λ_{n_{k}} D_{3}ũ_{k}(x)) dx ≤ L.

Proof. For v ∈ W^{1,p}(Ω; ℝ^{3}) and A ∈ A(ω) define, as in Lemma 2.1,

G_{n}(v, A) := ∫_{A×I} (1 + |D_{α}v|^{p} + λ_{n}^{p} |D_{3}v|^{p}) dx.

Let {u_{n}} satisfy (2.20).

Due to the coercivity hypothesis (H1), from the sequences of measures ν_{n} := G_{n}(u_{n}, ·) + G_{n}(u, ·) we may extract a bounded subsequence {ν_{n_{j}}} converging in the sense of distributions to some Radon measure ν.

For A ∈ A(ω) and t > 0, let A_{t} := {x_{α} ∈ A| dist(x_{α}, ∂A) > t}. Fix η > 0 and for every 0 < δ < η define the subsets L_{δ} := A_{η−2δ} \ \overline{A_{η+δ}}. Consider a smooth cut-off function φ_{δ} ∈ C_{0}^{∞}(A_{η−δ}; [0, 1]) such that φ_{δ}(x_{α}) = 1 on A_{η}. As the thickness of the strip L_{δ} is of order δ, we have an upper bound of the form ∥Dφ_{δ}∥_{L^{∞}(A_{η−δ})} ≤ C/δ. Define

w_{j}^{η,δ}(x) := u_{n_{j}}(x)φ_{δ}(x_{α}) + u(x_{α})(1 − φ_{δ}(x_{α})).

We have

D_{α}w_{j}^{η,δ} = φ_{δ}D_{α}u_{n_{j}} + (1 − φ_{δ})D_{α}u + D_{α}φ_{δ} ⊗ (u_{n_{j}} − u), D_{3}w_{j}^{η,δ} = φ_{δ}D_{3}u_{n_{j}}

and, passing to the limit in j,

lim_{j} w_{j}^{η,δ} = u in L^{p}(A × I; ℝ^{3}) , λ_{n_{j}} ∫_{I} D_{3}w_{j}^{η,δ} dx_{3} ⇀ φ_{δ}b̄ weakly in L^{p}(A; ℝ^{3}).  (2.21)

By the growth condition (H1), we have the estimate

∫A×IW (D_{α}w_{j}^{η,δ}|λ_{n_{j}}D_{3}w_{j}^{η,δ}) dx
≤ ∫_{A×I} W (D_{α}u_{n_{j}}|λ_{n_{j}}D_{3}u_{n_{j}}) dx + ∫_{(A\\overline{A_{η−δ}})×I} W (D_{α}u |0) dx

+ C (G_{n_{j}}(u_{n_{j}}, L_{δ}) + G_{n_{j}}(u, L_{δ})) + \frac{1}{δ^{p}} ∫_{L_{δ}×I} |u_{n_{j}} − u|^{p} dx.

Passing to the limit in j, we get

\begin{aligned}lim sup_{j} ∫_{A×I} W (D_{α}w_{j}^{η,δ}|λ_{n_{j}}D_{3}w_{j}^{η,δ}) dx ≤ L & + ∫_{(A\\overline{A_{η−δ}})×I}W (D_{α}u|0) dx \\ & + Cν(\overline{L}_{δ}), \end{aligned}

and then, passing to the limit as δ goes to zero,

lim sup_{δ} lim sup_{j} ∫_{A×I} W (D_{α}w_{j}^{η,δ}|λ_{n_{j}}D_{3}w_{j}^{η,δ}) dx ≤ L + ∫_{(A\\overline{A_{η}})×I}W (D_{α}u|0) dx
+ Cν(∂A_{η}).

(2.22)

Choosing a subsequence η_{k} → 0^{+} such that ν(∂A_{η_{k}}) = 0, and δ_{k} < η_{k}, we obtain, from (2.22),

lim sup_{k} lim sup_{j} ∫_{A×I} W (D_{α}w_{j}^{η_{k}δ_{k}}|λ_{n_{j}}D_{3}w_{j}^{η_{k}δ_{k}}) dx ≤ L.  (2.23)

Since the weak topology is metrizable in bounded sets and, by (H1) and (2.23), {λ_{n} ∫_{I} D_{3}w_{j}^{η_{k}δ_{k}} dx} is bounded in L^{p}(A; ℝ^{3}), (2.21) and (2.23) yield the existence of subsequence {j_{k}} such that ũ_{k} := w_{j_{k}}^{η_{k}δ_{k}}

satisfies ũ_{k} = u on a neighborhood of ∂A × I, ũ_{k} → u in L^{p}(A × I; ℝ^{3}), λ_{n_{j_{k}}} ∫_{I} D_{3}ũ_{k} dx_{3} ⇀ b̄ weakly in

L^{p}(A; ℝ^{3}), and

lim sup_{k} ∫_{A×I} W (D_{α}ũ_{k}(x) | λ_{n_{j_{k}}} D_{3}ũ_{k}(x)) dx ≤ L.

Lemma 2.3. The infimum in (2.10) remains unchanged if we replace W by its quasiconvex envelope QW.

Proof. Fix (u, b̄, A) ∈ V × L^{p}(ω; ℝ^{3}) × A(ω) and define

E′(u, b̄, A) := inf{ lim inf_{n} ∫_{A×I} QW (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx|
u_{n} ⇀ u in W^{1,p}(A × I; ℝ^{3}), λ_{n} ∫_{I} D_{3}u_{n}(·, x_{3}) dx_{3} ⇀ b̄, in L^{p}(A; ℝ^{3})}

Since, by the definition of quasiconvex envelope, W ≥ QW, it is obvious that E^{−}(u, b̄, A) ≥ E′(u, b̄, A). We prove the opposite inequality. For fixed δ > 0, let u_{n} ⇀ u weakly in W^{1,p}(A × I; ℝ^{3}) and

λ_{n} ∫_{I} D_{3}u_{n}(x_{α}, x_{3}) dx_{3} ⇀ b̄ weakly in L^{p}(A; ℝ^{3}), such that

E′(u, b̄, A) ≥ lim_{n} ∫_{A×I} QW (D_{α}u_{n} |λ_{n}D_{3}u_{n}) dx − δ.  (2.24)

In view of (2.2) and by the relaxation Theorem of Acerbi and Fusco (see [AF]), for each n, there exists a sequence {u_{n,k}} converging to u_{n} weakly in W^{1,p}(A × I; ℝ^{3}) such that

∫_{A×I} QW (D_{α}u_{n} |λ_{n}D_{3}u_{n}) dx = lim_{k} ∫_{A×I} W (D_{α}u_{n,k} |λ_{n}D_{3}u_{n,k}) dx.  (2.25)

¿From (2.24) and (2.25) we have

E′(u, b̄, A) ≥ lim_{n} lim_{k} ∫_{A×I} W (D_{α}u_{n,k} |λ_{n}D_{3}u_{n,k}) dx − δ  (2.26)

with

lim_{n} lim_{k} ∥u_{n,k} − u∥_{L^{p}(A×I;ℝ^{3})}  (2.27)

and, for the weak topology of L^{p}(A; ℝ^{3}),

lim_{n} lim_{k} ∫_{I} λ_{n} D_{3}u_{n,k} dx_{3} = b̄.  (2.28)

In view of hypothesis (H1) and the convexity of | · |^{p} we have

sup_{n} sup_{k} ∥∫_{I} λ_{n} D_{3}u_{n,k} dx_{3}∥_{L^{p}(A;ℝ^{3})} < +∞.  (2.29)

Since the weak topology is metrizable in bounded sets of L^{p}(A; ℝ^{3}), (2.26), (2.27), (2.28) and (2.29) yield the existence of a diagonal sequence {u_{n,k_{n}}} satisfying u_{n,k_{n}} → u in L^{p}(A×I; ℝ^{3}) (or weakly in W^{1,p}(A×I; ℝ^{3}),

in view of the coercivity hypothesis (H1)), ∫_{I} λ_{n} D_{3}u_{n,k_{n}} dx_{3} ⇀ b̄ weakly in L^{p}(A; ℝ^{3}), and realizing the

double limit in the right hand side of (2.26). Consequently we have

\begin{aligned}E′(u, b̄, A) & ≥ lim_{n} ∫_{A×I} W (D_{α}u_{n,k_{n}} |λ_{n} D_{3}u_{n,k_{n}}) dx − δ \\ & ≥ E^{−}(u, b̄, A) − δ. \end{aligned}

Letting δ go to zero, the conclusion follows .

3. Proofs of the main results.

Proof of Theorem 1.2. First we notice that, by Lemma 2.3 and (1.8) we may assume with no loss of generality that W is quasiconvex. Then by (H1) (see [M]) we may also assume that W satisfies

(H2)  |W (ξ) − W (ξ′)| ≤ C (1 + |ξ|^{p−1} + |ξ′|^{p−1})|ξ − ξ′|,

for certain C > 0 and for all ξ, ξ′ ∈ M^{3×3}.

Now let {ε_{k}} be a sequence converging to 0^{+} and choose a subsequence {ε_{k_{n}}} in agreement with Lemma 2.1. Recalling (2.10) we have

E^{−}(u, b̄) =E^{−}(u, b̄, ω)
= inf {lim inf_{n} ∫_{Ω} W (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx | u_{n} ⇀ u in W^{1,p}(Ω; ℝ^{3}),
λ_{n} ∫_{I} D_{3}u_{n} dx_{3} ⇀ b̄, in L^{p}(ω; ℝ^{3})} ,

where λ_{n} = (ε_{k_{n}})^{−1}. As we have noticed in the beginning of Subsection 2.2, in order to establish the Γ-convergence of E_{ε} to E it is enough to show that

E^{−} ≡ E.

This identity is a consequence of the Lemmas 3.1 and 3.2 below, applied to A = ω.

Lemma 3.1. Let W be a Borel function, satisfying hypotheses (H1) and (H2). Consider the functional E^{−}, defined in (2.10) for the sequence {λ_{n}} given by Lemma 2.1, and Q^{∗}W defined in (1.6). Then,

E^{−}(u, b̄, A) ≥ ∫_{A} Q^{∗}W (D_{α}u |b̄) dx_{α},  (3.1)

for all (u, b̄, A) ∈ V × L^{p}(ω; ℝ^{3}) × A(ω).

Proof. Step 1. We begin by proving the Lemma for A = Q′ and for u(x_{α}) := \overline{F} x_{α} + u_{0}, with \overline{F} ∈ M^{3×2} and u_{0}, b̄ ∈ ℝ^{3}.

Assume that E^{−}(u, b̄, Q′) < +∞. By Lemma 2.2 we may restrict ourselves, in (2.10), to sequences having the same trace as its limit. Then consider {φ_{n}} in W^{1,p}(Q; ℝ^{3}) such that

φ_{n} = 0 on ∂Q′ × I, φ_{n} ⇀ 0 in W^{1,p}(Q; ℝ^{3}), λ_{n} ∫_{I} D_{3}φ_{n} dx_{3} ⇀ b̄ in L^{p}(Q′; ℝ^{3}).

Define

φ̃_{n}(x) := φ_{n}(x) + x_{3} (\frac{b̄}{λ_{n}} − ∫_{Q} D_{3}φ_{n} dx) .

By (H1) {φ_{n}} and, consequently, {φ̃_{n}} are bounded in W^{1,p}(Q; ℝ^{3}), ∫_{Q} λ_{n} D_{3}φ̃_{n} dx = b̄, D_{α}φ_{n} = D_{α}φ̃_{n}

and φ̃_{n}(·, x_{3}) is Q′-periodic. Then,

∫_{Q} W (\overline{F} + D_{α}φ̃_{n}|λ_{n} D_{3}φ̃_{n}) dx ≥ Q^{∗}W (\overline{F} |b).  (3.2)

On the other hand,

lim_{n} ∥(\overline{F} + D_{α}φ_{n}|λ_{n} D_{3}φ_{n}) − (\overline{F} + D_{α}φ̃_{n}|λ_{n} D_{3}φ̃_{n})∥_{L^{∞}(Q;ℝ^{3})} = lim_{n} |∫_{Q} λ_{n} D_{3}φ_{n} dx − b̄| = 0,

which yields, in view of (H2) and since {φ_{n}} and {φ̃_{n}} are bounded in W^{1,p}(Q; ℝ^{3}),

lim_{n} ∫_{Q} W (\overline{F} + D_{α}φ̃_{n}|λ_{n} D_{3}φ̃_{n}) dx = lim_{n} ∫_{Q} W (\overline{F} + D_{α}φ_{n}|λ_{n} D_{3}φ_{n}) dx.  (3.3)

¿From (3.2) and (3.3) we conclude that

lim inf_{n} ∫_{Q} W (\overline{F} + D_{α}φ_{n}|λ_{n} D_{3}φ_{n}) dx ≥ Q^{∗}W (\overline{F} |b).  (3.4)

We complete the proof of step 1 by taking the infimum over all the admissible sequences {φ_{n}} in (3.4).

Step 2. Now we establish the general case.

Fix (u, b̄, A) ∈ V × L^{p}(ω, ℝ^{3}) × A(ω). Let {u_{n}} be such that u_{n} ⇀ u weakly in W^{1,p}(A × I; ℝ^{3}), λ_{n} ∫_{I} D_{3}u_{n} dx_{3} ⇀ b̄ weakly in L^{p}(A; ℝ^{3}), and

E(u, b̄, A) = lim_{n} ∫_{A×I} W (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx.

Define the sequence of measures μ_{n} := (∫_{I} W (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx_{3}) L^{2}⌊A. Since {μ_{n}} is bounded, it

converges weakly-⋆, up to a subsequence, to some measure μ. Represent by ρ the absolutely continuous part of μ with respect to the 2-dimensional Lebesgue measure. To prove (3.1) it suffices to show that, for a.e. x_{0} ∈ A,

ρ(x_{0}) ≥ Q^{∗}W (D_{α}u(x_{0}), b̄(x_{0}))  (3.5)

It is known that, for a.e. x_{0} ∈ A,

ρ(x_{0}) = lim_{ε→0} \frac{μ(x_{0} + εQ′)}{ε^{2}},  (3.6)
lim_{ε→0} \frac{1}{ε^{p+2}} ∫_{x_{0}+εQ′} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})|^{p} dx_{α} = 0,  (3.7)

lim_{ε→0} \frac{1}{ε^{2}} ∫_{x_{0}+εQ′} |b̄(x) − b̄(x_{0})|^{p} dx_{α} = 0.  (3.8)

Let x_{0} satisfy (3.6), (3.7) and (3.8). Let {ε} represent a sequence converging to zero such that, for all ε,

μ(∂(x_{0} + εQ′)) = 0.  (3.9)

Using (3.6), the definition of μ and (3.9), we have

ρ(x_{0}) = lim_{ε→0} lim_{n} \frac{1}{ε^{2}} ∫_{(x_{0}+εQ′)×I} W (D_{α}u_{n} |λ_{n} D_{3}u_{n}) dx

= lim_{ε→0} lim_{n} ∫_{Q} W (D_{α}u_{n}(x_{0} + εy_{α}, y_{3}) |λ_{n} D_{3}u_{n}(x_{0} + εy_{α}, y_{3})) dy

= lim_{ε→0} lim_{n} ∫_{Q} W (D_{α}u_{n,ε} |λ_{n} D_{3}u_{n,ε}) dy,

(3.10)

where u_{n,ε}(y) := \frac{u_{n}(x_{0} + εy_{α}, y_{3}) − u(x_{0})}{ε}.

Since u_{n} → u in L^{p}(A × I; ℝ^{3}), (3.7) yields

lim_{ε→0} lim_{n} ∥u_{n,ε}(·) − ∇u(x_{0}) · ∥_{L^{p}(Q;ℝ^{3})}^{p} = 0.  (3.11)

We also have for φ ∈ L^{p′}(Q′; ℝ^{3}), and as λ_{n} ∫_{I} D_{3}u_{n} dx_{3} ⇀ b̄ weakly in L^{p}(A; ℝ^{3}),

lim_{ε→0} lim_{n} ∫_{Q} λ_{n} D_{3}u_{n,ε}(y) φ(y) dy = lim_{ε→0} lim_{n} \frac{1}{ε^{2}} ∫_{(x_{0}+εQ′)×I} λ_{n}D_{3}u_{n}(x)φ (\frac{x_{α} − x_{0}}{ε} ) dx

= lim_{ε→0} \frac{1}{ε^{2}} ∫_{x_{0}+εQ′} b̄(x_{α}) φ (\frac{x_{α} − x_{0}}{ε} ) dx_{α}

= b(x_{0}) ∫_{Q′} φ(y_{α}) dy_{α},

(3.12)

where we have used (3.8).

By means of a standard diagonalization process, from (3.10), (3.11) and (3.12), we construct ũ_{k} := u_{ε_{k},n_{k}} and λ_{n_{k}} such that

ũ_{k}(y) → D_{α}u(x_{0})y in L^{p}(Q; ℝ^{3}) , λ_{n_{k}} ∫_{I} D_{3}ũ_{k} dy_{3} ⇀ b̄(x_{0}) weakly in L^{p}(Q′; ℝ^{3})

and

ρ(x_{0}) = lim_{k} ∫_{Q} W (D_{α}ũ_{k} |λ_{n_{k}} D_{3}ũ_{k}) dy.  (3.13)

Since

lim_{k} ∫_{Q} W (D_{α}ũ_{k} |λ_{n_{k}} D_{3}ũ_{k}) dy ≥ E^{−}(D_{α}u(x_{0})y, b̄(x_{0}), Q′),

(3.5) follows from (3.13) and Step 1.

Lemma 3.2. Under the hypotheses of Lemma 3.1, we have

E^{−}(u, b̄, A) ≤ ∫_{A} Q^{∗}W (D_{α}u |b̄) dx_{α}  (3.14)

for all (u, b̄, A) ∈ V × L^{p}(ω; ℝ^{3}) × A(ω).

Proof. In view of Proposition 1.1 iii) and of Lemma 2.3, without loss of generality, we may assume in (3.14) that the density W is a quasiconvex function. Therefore, by (H1) W is p-Lipschitz which easily entails the continuity of the function

W^{(λ)}(\overline{F} |b) := inf_{φ∈W^{1,p}(Q;ℝ^{3}) φ is Q′−periodic} \{∫_{Q} W (\overline{F} + D_{α}φ|λ D_{3}φ) dx : λ ∫_{Q} D_{3}φ dx = b\} ,

where λ ∈ ℝ and (\overline{F} , b) ∈ M^{3×2} × ℝ. Note that

Q^{∗}W (\overline{F} | b) = inf_{λ∈ℝ} W^{(λ)}(\overline{F} |b).  (3.15)

We claim that for each fixed λ and for all (u, b̄, A) ∈ W^{1,p}(ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}) × A(ω),

E^{−}(u, b̄, A) ≤ ∫_{A} W^{(λ)}(D_{α}u | b̄) dx_{α}.  (3.16)

Suppose that (3.16) holds. Since, by Lemma 2.1, E^{−}(u, b̄, ·) is a measure absolutely continuous with respect to the 2-dimensional Lebesgue measure, we can write E^{−}(u, b̄, ·) = ρL^{2}⌊ω for some function ρ ∈ L^{1}(ω). Let x_{0} ∈ ω be a Lebesgue point for ρ, D_{α}u and b̄. Using (3.15), we may find a sequence {λ_{j}} such that

Q^{∗}W (D_{α}u(x_{0})|b̄(x_{0})) = inf_{j} W^{(λ_{j})}(D_{α}u(x_{0})|b̄(x_{0})).

Now (3.16) yields

ρ(x_{α}) ≤ W^{(λ_{j})}(D_{α}u(x_{α}) | b̄(x_{α}))

for L^{2} a.e. x_{α} ∈ ω and for all j ∈ ℕ, and in turn we conclude that

ρ(x_{0}) ≤ Q^{∗}W (\overline{F} |b̄)(x_{0}).

Given the arbitrariness of x_{0}, we deduce that

E^{−}(u, b̄, A) = ∫_{A} ρ(x_{α}) dx_{α} ≤ ∫_{A} Q^{∗}W (D_{α}u(x_{α}) | b̄(x_{α})) dx_{α},

which establishes (3.14).

We now prove claim (3.16) in three steps. Step 1. First we consider the case where u is affine, u = \overline{F} x_{α}, and b̄ constant.

Let φ be admissible for W^{(λ)}(\overline{F} |b̄). Using the Q′-periodicity of φ, we define

φ_{n} : ℝ^{2} × I → ℝ^{3} by φ_{n}(x) := \frac{λ}{λ_{n}} φ (\frac{λ_{n}}{λ} x_{α}, x_{3}), where \{λ_{n}\}, λ_{n} → +∞, is the sequence given

by Lemma 2.1.

We have φ_{n} ∈ W^{1,p}(A × I; ℝ^{3}), φ_{n} → 0 in L^{p}(A × I; ℝ^{3}) and, by the Riemann-Lebesgue Lemma,

λ_{n} ∫_{I} D_{3}φ_{n} dx_{3} = λ ∫_{I} D_{3}φ (\frac{λ_{n}}{λ} x_{α}, x_{3}) dx_{3} ⇀ λ ∫_{I} ∫_{Q′} D_{3}φ(y_{α}, x_{3}) dy_{α} dx_{3} = b̄,

\begin{matrix}E^{−}(u, b̄, A) ≤ & lim_{n} ∫_{A×I} W (\overline{F} + D_{α}φ_{n} | λ_{n}D_{3}φ_{n}) dx \\ = & lim_{n} ∫_{A} ∫_{I} W (\overline{F} + D_{α}φ (\frac{λ_{n}}{λ} x_{α}, x_{3}) | λD_{3}φ (\frac{λ_{n}}{λ} x_{α}, x_{3})) dx_{3} dx_{α} \\ = & ∫_{A} ∫_{Q′} ∫_{I} W (\overline{F} + D_{α}φ(y_{α}, x_{3}) | λD_{3}φ(y_{α}, x_{3})) dx_{3} dy_{α} dx_{α} \\ = & L^{2}(A) ∫_{Q} W (\overline{F} + D_{α}φ | λD_{3}φ) dx. \end{matrix}

Taking the infimum over all admissible φ , we obtain

E^{−}(u, b̄, A) ≤ L^{2}(A)W^{(λ)}(\overline{F} | b̄) = ∫_{A} W^{(λ)}(\overline{F} | b̄) dx_{α}.

Step 2. We prove the claim for u piecewise affine and b̄ piecewise constant.

Let {A_{j}}_{j=1,···,k} be a finite and measurable partition of A, such that u and b̄ are affine and constant, respectively, on each A_{j}, j = 1, ⋯, k. By Step 1 we have, for all j = 1 ⋯, k,

E^{−}(u, b̄, A_{j}) ≤ ∫_{A_{j}} W^{(λ)}(D_{α}u | b̄) dx_{α}.

By Lemma 2.1 E^{−}(u, b̄, ·) is a measure, thus

E^{−}(u, b̄, A) = ∑_{j=1}^{k} E^{−}(u, b̄, A_{j}) ≤ ∑_{j=1}^{k} ∫_{A_{j}} W^{(λ)}(D_{α}u | b̄) dx_{α} = ∫_{A} W^{(λ)}(D_{α}u | b̄) dx_{α}.

Step 3. We prove the claim for an arbitrary (u, b̄, A) ∈ W^{1,p}(ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}) × A(ω).

Let {(u_{n}, b̄_{n})} be a sequence such that u_{n} are piecewise affine functions, b̄_{n} are constants and {(u_{n}, b̄_{n})} converges to (u, b̄) in W^{1,p}(A; ℝ^{3}) × L^{p}(A; ℝ^{3}). Since E^{−}(·, ·, A) is lower semicontinuous, we have

E^{−}(u, b̄, A) ≤ lim inf_{n} E^{−}(u_{n}, b̄_{n}, A) ≤ lim inf_{n} ∫_{A} W^{(λ)}(D_{α}u_{n} | b̄_{n}) dx_{α}.  (3.17)

Since (\overline{F} |b̄) ↦ W^{(λ)}(\overline{F} |b̄) is continuous and satisfies p-growth conditions, we have

lim_{n} ∫_{A} W^{(λ)}(D_{α}u_{n} | b̄_{n}) dx_{α} = ∫_{A} W^{(λ)}(D_{α}u | b̄) dx_{α},

which, together with (3.17), concludes the proof of (3.16).

Proof of Corollary 1.3. Consider the problem (P_{ε}). Taking u_{0,ε} as an admissible test function, we see that the infimum of (P_{ε}) is uniformly bounded in ε, so any sequence of minimizers {u_{ε}} will satisfy

sup_{ε} ∫_{Ω} W (D_{α}u_{ε} |\frac{1}{ε}D_{3}u_{ε}) dx − L_{ε}(u_{ε}) < +∞ and u_{ε} = u_{0,ε} on Γ.

By the coercivity assumption in (H_{1}) we have

sup_{ε} \{∫_{Ω} |D_{α}u_{ε}|^{p}dx + ∫_{Ω} \frac{1}{ε^{p}} |D_{3}u_{3}|^{p}dx\} < +∞.

Therefore {u_{ε}} is bounded in W^{1,p}(Ω; ℝ^{3}) and D_{3}u_{ε} → 0 in L^{p}(Ω; ℝ^{3}), as ε → 0. On the other hand,

\{b_{ε}\}, b_{ε} := \frac{1}{ε}D_{3}u_{ε}, is bounded in L^{p}(Ω; ℝ^{3}), therefore we conclude that the sequence \{(u_{ε}, b_{ε})\} is

weakly compact in W^{1,p}(Ω; ℝ^{3}) × L^{p}(Ω; ℝ^{3}), and all cluster points (u, b) belong to V × L^{p}(Ω; ℝ^{3}). Since u_{0,ε}|_{Γ} converges uniformly to u_{0}|_{Γ}, u_{0}(x) := (x_{α}, 0), the limit u satisfies u = u_{0} on Γ. Also, with

b̄ := ∫_{I} b(x_{α}, x_{3}) dx_{3}, it follows that \overline{b}_{ε} := ∫_{I} \frac{1}{ε}D_{3}u_{ε}dx_{3} ⇀ b̄ weakly in L^{p}(ω; ℝ^{3}) and L_{ε}(u_{ε}) → L(u, \overline{b}),

as ε → 0.

Introduce the functionals Φ_{ε} and Φ given by

Φ_{ε}(u, b̄) := \{ \begin{matrix}E_{ε}(u, b̄) − L_{ε}(u) & if u = u_{0,ε} on Γ, \\ +∞ & otherwise, \end{matrix}

and

Φ(u, b̄) := \{ \begin{matrix}∫_{ω} Q^{∗}W (D_{α}u |b̄) dx_{α} − L(u, b̄) & if u = u_{0} on Γ, \\ +∞ & otherwise. \end{matrix}

We will show that Φ_{ε} Γ-converges to Φ, therefore any cluster point of a minimizing sequence of (P_{ε}) will be a solution of (P) (see [DM]), and this will complete the proof of Corollary 1.3. By the Kuratowski’s Compactness Theorem (see [DM]), it is enough to prove that the Γ-lower limit of any subsequence of Φ_{ε} coincides with Φ.

Let (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}) and (u_{ε}, b̄_{ε}) ⇀ (u, b̄) weakly in W^{1,p}(Ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}). If lim inf_{ε→0} Φ_{ε}(u_{ε}, b̄_{ε}) <

+∞, then u_{ε} = u_{0,ε} on Γ for infinitely many ε. Since u_{ε}|_{Γ} → u|_{Γ} weakly in L^{p}(Γ; ℝ^{3}) and u_{0,ε}|_{Γ} → u_{0}|_{Γ} uniformly, we have u|_{Γ} = u_{0}|_{Γ} and, in view of Theorem 1.2, we deduce that

lim inf_{ε→0} Φ_{ε}(u_{ε}, b̄_{ε}) ≥ Φ(u, b̄).  (3.18)

To complete the proof we need to construct, for each (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}) such that u = u_{0} on Γ, a sequence (ū_{ε}, b̄_{ε}) ⇀ (u, b̄) weakly in W^{1,p}(Ω; ℝ^{3}) × L^{p}(ω; ℝ^{3}), satisfying

lim inf_{ε→0} Φ_{ε}(u_{ε}, b̄_{ε}) = Φ(u, b̄),

or, equivalently, since lim_{ε→0} L_{ε}(u_{ε}) = L(u, b̄) will trivially hold, we must find (u_{ε}, b̄_{ε}) such that

lim inf_{ε→0} E_{ε}(u_{ε}, b̄_{ε}) = ∫_{A} Q^{∗}W (D_{α}u |b̄) dx_{α} , u_{ε} = u_{0,ε} on Γ.  (3.19)

Let (u, b̄) ∈ V × L^{p}(ω; ℝ^{3}) such that u = u_{0} on Γ. In view of Theorem 1.2, there exists a sequence {u_{ε}} in

W^{1,p}(Ω; ℝ^{3}) such that u_{ε} ⇀ u in W^{1,p}(Ω; ℝ^{3}), ∫_{I} \frac{1}{ε}D_{3}u_{ε}dx_{3} ⇀ b̄ weakly in L^{p}(ω; ℝ^{3}) and

∫_{A} Q^{∗}W (D_{α}u |b̄) dx_{α} = lim_{n} ∫_{Ω} W (D_{α}u_{ε} |\frac{1}{ε}D_{3}u_{ε}) dx.  (3.20)

In order to obtain the boundary conditions in (3.20), we associate to each deformation u_{ε} the displacement v_{ε} := u_{ε} − u_{0,ε}. We then rewrite (3.20) as

∫_{A} Q^{∗}W (D_{α}u |b̄) dx_{α} = lim_{n} ∫_{Ω} W (e_{α} + D_{α}v_{ε} |e_{3} + \frac{1}{ε}D_{3}v_{ε}) dx,  (3.21)

where v_{ε} ⇀ u − u_{0} weakly in W^{1,p}(Ω; ℝ^{3}) and ∫_{I} \frac{1}{ε}D_{3}v_{ε} ⇀ b̄ − e_{3} weakly in L^{p}(ω; ℝ^{3}). By Lemma 2.2,

applied to W̃(ξ) := W (𝕀 + ξ) and to \{v_{ε}\}, there exist two subsequences, ε_{k} → 0 and \{\overline{v}_{k}\} in W^{1,p}(Ω; ℝ^{3}),

satisfying \overline{v}_{k} = 0 on Γ, \overline{v}_{k} ⇀ u − u_{0} weakly in W^{1,p}(Ω; ℝ^{3}), ∫_{I} \frac{1}{ε_{k}} D_{3}\overline{v}_{k} dx_{3} ⇀ b̄ − e_{3} weakly in L^{p}(ω; ℝ^{3})

and

lim inf_{k} ∫_{Ω} W (e_{α} + D_{α}\overline{v}_{k} |e_{3} + \frac{1}{ε_{k}} D_{3}\overline{v}_{k}) dx ≤ lim_{n} ∫_{Ω} W (e_{α} + D_{α}v_{ε} |e_{3} + \frac{1}{ε}D_{3}v_{ε}) dx.

Then, defining \overline{u}_{k} := \overline{v}_{k} + u_{0ε_{k}}, from (3.18) and (3.21), we conclude (3.19).

Acknowledgments. The research of G. Bouchitté was partially supported by ICCTI-MENESR 166-B2, the research of I. Fonseca by the National Science Foundation under Grants No. DMS–9731957, DMS–0103798, and the research of M.L. Mascarenhas by ICCTI-PRAXIS XXI, FEDER-PRAXIS/2/2.1/MAT/125/94, PRAXIS-FEDER/3/3.1/CTM/10/94 and ICCTI-MENESR 166-B2. The authors acknowledge the hospitality of the Universities of Toulon (ANLA), Carnegie Mellon and its Center for Nonlinear Analysis (CNA, Grant No. DMS–9803791), and Lisbon (CMAF), where this work was undertaken.

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