2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 35–51. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

MODELLING OF A COLLAGE PROBLEM

ABDELAZIZ AÏT MOUSSA, LOUBNA ZLAÏJI

Abstract. In this paper we study the behavior of elastic adherents connected with an adhesive. We use the Γ-convergence method to approximate the problem modelling the assemblage with density energies assumed to be quasiconvex. In particular for the adhesive problem, we assume periodic density energy and some growth conditions with respect to the spherical and deviational components of the gradient. We obtain a problem depending on small parameters linked to the thickness and the stiffness of the adhesive.

1. Introduction

The problem under investigation arises in the study of adhesive bonding of elastic bodies, and the question is how to model the behavior of the adhesive material interposed between the adherents. Such problems find their applications for example in aeronautics, in the study of composites, and in other fields of engineering. In general, the computation of the solution using numerical methods is very difficult. In one hand, this is because the thickness of the adhesive requires a fine mesh, which in turn implies an increase of the degrees of freedom of the system, and in the other, the adhesive is usually more flexible than the adherents, and this produces numerical instabilities in the stiffness matrix. To overcome this difficulties, thanks to Goland and Reissner [12], it is usual to find a limit problem in which the adhesive is treated as a material surface; it disappears from a geometrical point of view, but it is represented by the energy of adhesion. In this framework, we find many works investigated on this theoretical approach; see for example Moussa [19], Suquet [20], Ganghoffer, Brillard and Schultz [10], Geymonat, Krasucki and Lenci [11], Licht and MiChaille [15], Brezis, Caffarelli and Friedman [4], Acerbi, Buttazo and Percivale [1], Klarbring [14], Caillerie [5].

This work is specially interested in approximating a minimization problem (P_{r}), where r is a small parameter linked to the thickness and the stiffness of the adhesive. In particular, we associate to each component of gradient (spherical or deviational) an independent stiffness parameter. We use the method described in [15] to find a certain limit problem denoted (P). Precisely, by the Γ-convergence method (introduced in a paper by De Giorgi and Franzoni in 1975 [9]), we look for a weak limit of a (P_{r})-minimizing sequence which is a solution of (P). The outline of the paper is the following.

2000 Mathematics Subject Classification. 35K22, 58D25, 73D30.

Key words and phrases. Adhesive; spherical; deviational; Γ-convergence; homogenization;

quasiconvexity; subadditivity.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

Section 2 contains some notation and a brief summary of results related to notions of Γ-convergence, quasiconvexity and subadditivity . Section 3 is devoted to Problem statement, hypothesis which we assume on his different components and existence of solutions. In section 4, we discuss topology that we shall consider for limit problem study, we compute Γ-limit of the stored strain energies represented by the functionals (F_{r})_{r} and we deduce the limit problem.

2. Notation and preliminaries

We begin by introducing some notation which is used throughout the paper. First, let O_{1} and O_{2} be two open subsets of ℝ^{N} with interface S. For a function v defined on O_{1} ∪ O_{2}, we call the jump of v across S the function defined on S by [v]_{S} = v_{/O_{1}} − v_{/O_{2}}. Let M^{N} be the space of N × N real matrices endowed with the Hilbert-Schmidt scalar product A : A′ = trace(A^{t}A′). For a given A ∈ M^{N}, we call spherical part of A the matrix A_{s} = \frac{trace A}{N}I_{N}, where I_{N} is the unit matrix of ℝ^{N}. The deviational part will be A_{d} = A − A_{s}. In mechanics, the spherical part of the deformation tensor changes the volume without changing the shape whereas the deviational tensor changes the shape preserving the same volume (the trace is void, therefore there is no relative variation of volume). On the space M^{N}, operators A ↦ A_{s} and A ↦ A_{d} are linear continuous for matrix norm |A| = Σ_{1≤i,j≤N}|A_{i j}|, where A = (A_{i j})_{1≤i,j≤N}. Definition. A a Carathéodory function f : ℝ^{N} ×M^{N} → ℝ satisfies condition (C_{p}) if there exists α_{p}, β_{p}, c ∈ ℝ^{N}, such that for x ∈ ℝ^{N} and all (A, A′) ∈ (M^{N})^{2}, we have

\begin{matrix}α_{p}|A|^{p} ≤ f(x, A) ≤ β_{p}(1 + |A|^{p}) \\ |f(x, A) − f(x, A′)| ≤ c|A − A′|(1 + |A|^{p−1} + |A′|^{p−1}). \end{matrix}  (2.1)

As we have already mentioned, our method will be based on the notion of Γconvergence. Let (X, τ) be a metrisable topological space, and for every n ∈ ℕ let F_{n}, F : X → \overline{ℝ} be functions defined on X. For every x ∈ X, the Γ(τ)-liminf F_{n} (respectively, Γ(τ)-limsup F_{n} ) are defined as:

Γ(τ) − lim inf F_{n}(x) = inf{lim inf F_{n}(x_{n}) : x_{n} →^{τ} x}
Γ(τ) − lim inf F_{n}(x) = inf{lim sup F_{n}(x_{n}) : x_{n} →^{τ} x}

If the two expressions are equal to F (x), then we say that the sequence (F_{n}) Γ(τ)- converges to F on X and we write F = Γ(τ)-limF_{n}. An other way to define F=Γ-limF_{n} is the following:

(∀x ∈ X)(∃x_{0,n} ∈ X) such that x_{0,n} →^{τ} x and lim sup_{n→+∞} F_{n}(x_{0,n}) ≤ F (x)

(∀x ∈ X)(∀x_{n} ∈ X) such that x_{n} →^{τ} x, lim inf_{n→+∞} F_{n}(x_{n}) ≥ F (x)

The Γ-convergence method is made precise in item (1) below.

Proposition 2.1. Suppose that (F_{n})_{n} Γ-converges to F.

(1) [2, Theorem 2.11]. Let x_{n} ∈ X be such that F_{n}(x_{n}) ≤ inf{F_{n}(x) : x ∈ X} + ε_{n}, where ε_{n} > 0, ε_{n} → 0. We assume furthermore that {x_{n}, n ∈ ℕ} is τ-relatively compact, then any cluster point \overline{x} of \{x_{n}, n ∈ ℕ\} is a minimizer of F and

lim inf_{n→+∞}\{F_{n}(x) : x ∈ X\} = F (\overline{x}).

(2) [2, Theorem 2.15]. If L : X → ℝ is continuous, then (F_{n} + L)_{n} Γ-converges to F+L.

For details about Γ-convergence, we refer the reader to [2, 7]. To establish existence of solutions for our initial problem, it will be useful to consider quasiconvex energy densities. So if f is a Borel measurable and locally integrable function defined on M^{N}, we say that f is quasiconvex if

f(A) ≤ \frac{1}{meas D} ∫_{D} f(A + ∇φ)dx

where D is a bounded domain of ℝ^{N}, A ∈ M^{N} and φ ∈ W_{0}^{1, ∞}(D, ℝ^{N}). If f is not quasiconvex, his quasiconvex envelope is given as

Qf = sup{g ≤ f : g is quasiconvex }

If f is locally bounded, then the definition of Qf can be expressed as [6, Page 201]

Qf(A) = inf\{\frac{1}{measD} ∫_{D} f(A + ∇φ)dx : φ ∈ W_{0}^{1, ∞}(D, ℝ^{N})\}

The following proposition establish sufficiency of quasiconvexity to obtain weak lower semicontinuity in W^{1,p}

Proposition 2.2. Let O be an open bounded subset of ℝ^{N} and f : O × M^{N} → ℝ a continuous quasiconvex function satisfying condition (2.1), for p ≥ 1. Then, the functional F : u → ∫_{O} f(x, ∇u(x))) dx is weakly lower semicontinuous on W^{1,p}(O, ℝ^{N}).

For the proof of the above proposition, see [6, Theorem 2.4 and Remark iv]. To describe a global subadditive theorem, we consider B_{b}(ℝ^{d}) the family of

Borel bounded subsets of ℝ^{d} and δ Euclidean distance in ℝ^{d}. for every A ∈ B_{b}(ℝ^{d}), ρ(A) = sup\{r ≥ 0 : ∃\overline{B}_{r}(x) ⊂ A\}, where \overline{B}_{r}(x) = \{y ∈ ℝ^{d} : δ(x, y) ≤ r\}. A sequence (B_{n})_{n∈ℕ} ⊂ B_{b}(ℝ^{d}) is called regular if there exist an increasing sequence of intervals (I_{n})_{n} ⊂ ℤ^{d} and a constant C independent of n such that B_{n} ⊂ I_{n} and meas(I_{n}) ≤ C meas(B_{n}), ∀n. The global subadditive theorem is essentially based on subadditive ℤ^{d}-periodic functions . A function S : A ∈ B_{b}(ℝ^{d}) → S_{A} ∈ ℝ is called subadditive ℤ^{d}-periodic if it satisfy the following conditions:

(i) For all A, B ∈ B_{b}(ℝ^{d}) such that A ∩ B = ∅, S_{A∪B} ≤ S_{A} + S_{B}.

(ii) For all A ∈ B_{b}(ℝ^{d}), all z ∈ ℤ^{d}, S_{A+z} = S_{A}.

Now, we shall see the global subadditive theorem, firstly used in the setting of the calculus of variation by Dal Maso and Modica [8], and generalized to sequences indexed by convex sets by Licht and Michaille [15]

Theorem 2.3. Let S be a subadditive ℤ^{d}-periodic function such that

γ(S) = inf\{\frac{S_{I}}{meas I} : I = [a, b[, a, b ∈ ℤ^{d} and a_{i} < b_{i} ∀1 ≤ i ≤ d\} > −∞

In addition, we suppose that S satisfies the dominant property: There exists C(S), for every Borel convex subset A ⊂ [0, 1[^{d}, |S_{A}| ≤ C(S). Let (A_{n})_{n} be a regular sequence of Borel convex subsets of B_{b}(ℝ^{d}) with lim_{n→+∞} ρ(A_{n}) = +∞. Then lim_{n→+∞} \frac{S_{An}}{meas A_{n}} exists and is equal to

lim_{n→+∞} \frac{S_{A_{n}}}{meas A_{n}} = inf_{m∈ℕ^{∗}}\{\frac{S_{[0,m[^{d}}}{m^{d}} \} = γ(S)

For the proof of the above theorem see [16, page 24].

3. Statement of the problem

Let O be a domain of ℝ^{N} with Lipschitz boundary, divided in two parts O^{±} by the plane {x_{N} = 0}. The common interface is noted S. The structure under study contains two adherents filling O_{ε} = O_{ε}^{+} ∪ O_{ε}^{−}, glued perfectly with an adhesive occupying B_{ε} = \{x = (x̃, x_{N}) ∈ O : ±x_{N} ≤ ε γ^{±}(\frac{x̃}{ε})\} = O \ \overline{O}_{ε}, along the common surfaces S_{ε}^{±} = \{x ∈ O : ±x_{N} = εγ^{±}(\frac{x̃}{ε})\}, where ε is a small parameter intended to tend toward 0, γ^{±} : ℝ^{N−1} ⟶ ℝ^{+} are two C^{1} Ỹ-periodic functions, Ỹ =]0, 1[^{N−1}. The maximum (respectively Minimum) of γ^{±} on Ỹ is noted γ_{M}^{±}(respectively γ_{m}^{±}). Surface forces are applied on a portion Γ_{1} of ∂O with surface measure supposed to be positive, and the structure is clamped on his complementary Γ_{0}. The illustration of the domain is shown in Figure 1.

B_{ε}→  O_{ε}^{+} O_{ε}^{−}S_{ε}^{+} S_{ε}^{−} Γ_{1}  ←S O^{+} O^{−}  Γ_{1}

Figure 1. Initial problem (left). Limit problem (right)

Our study is focused on the minimization problem (P_{r}): Find u ∈ V_{ε} such that:

I_{r}(u) = min_{v∈V_{ε}} I_{r}(v) = min_{v∈V_{ε}} F_{r}(v) − L(v)  (3.1)

where

• r = (ε, μ, η), the three parameters are positive intended to tend to 0. The first concern the thickness of adhesive and the others the stiffness connected respectively to spherical and deviational components of ∇.

• V_{ε} = {v ∈ W^{1,q}(O_{ε}) × W^{1,p}(B_{ε}) : ∇_{s} v ∈ L^{p_{s}}(B_{ε}, M^{N}) and ∇_{d} v ∈ L^{p_{d}}(B_{ε}, M^{N}), [v]_{S_{ε}^{±}} = 0, v = 0 sur Γ_{0}}, ∇_{s} and ∇_{d} are respectively spherical and deviational components of ∇. p_{s}, p_{d} and q are constants with 1 < p_{s}, p_{d} ≤ q and p = min(p_{s}, p_{d}).

• For v ∈ V_{ε},

F_{r}(v) = ∫_{O_{ε}} h(x, ∇v) dx + ∫_{B_{ε}} μ b_{s}({x \atop ε}, ∇_{s} v) + η b_{d}({x \atop ε}, ∇_{d} v) dx

L(v) = ∫_{O} f(x) v(x) dx + ∫_{Γ_{1}} g(x) v(x) dσ(x)

In the following, we denote w = s or w = d, and we make the hypotheses:

(H1) b_{w} and h are Carathéodory functions defined on ℝ^{N} × M^{N}. In particular, b_{w} is Ỹ-periodic with respect to first variable and satisfies the condition (C_{w}): There exists α_{w}, β_{w}, c_{w} ∈ ℝ_{∗}^{+} such that for all x ∈ ℝ^{N} and all

(Q, Q′) ∈ (M^{N})^{2} we have

\begin{matrix}α_{w}|Q_{w}|^{p_{w}} ≤ b_{w}(x, Q_{w}) ≤ β_{w}(1 + |Q_{w}|^{p_{w}}) \\ |b_{w}(x, Q_{w}) − b_{w}(x, Q′_{w})| ≤ c_{w}|Q_{w} − Q′_{w}|(1 + |Q_{w}|^{p_{w}−1} + |Q′_{w}|^{p_{w}−1}). \end{matrix}  (3.2)

The function h satisfies the conditions C_{q} (2.1). (H2) There exist a function b^{∞,w} such that Q ↦ b^{∞,w}(x, Q) is positively p_{w}-

homogeneous, a positive constant c′_{w} and 0 < m_{w} < p_{w} so that for all (x, Q) ∈ ℝ^{N} × M^{N}4,

|b^{∞,w}(x, Q_{w}) − b_{w}(x, Q_{w})| ≤ c′_{w}(1 + |Q_{w}|^{p_{w}−m_{w}})

(H3) (f, g) ∈ L^{q′}(O, ℝ^{N}) × L^{q′}(Γ_{1}, ℝ^{N}), where q′ is the conjugate exponent of q,

and there exists ε_{0} > 0 such that for all ε ≤ ε_{0} : (supp f ∪ Γ_{1}) ∩ B_{ε} = ∅. To lighten notation, we shall often use const to designate different constants

(independent of r) in a same proof.

Remark 3.1. (1) If we consider the following norm on V_{ε},

∥v∥_{V_{ε}} = ∥v∥_{W^{1,q}(O_{ε},ℝ^{N})} + ∥∇_{s}v∥_{L^{ps} (B_{ε},M^{N})} + ∥∇_{d}v∥_{L^{p_{d}} (B_{ε},M^{N})}.

Then V_{ε} will be a reflexive Banach space (because 1 < p_{s}, p_{d}), and L is a linear continuous mapping on (V_{ε}, ∥·∥_{V_{ε}}). (2) Hypothesis (H2) implies that for all (x, Q) ∈ ℝ^{N} × M^{N}, lim_{t→+∞} \frac{b_{w}(x,tQ_{w})}{t^{pw}} = b^{∞,w}(x, Q_{w}).

Proposition 3.2. Let b_{w} and h be quasiconvex, and in particular b_{w} is continuous on ℝ^{N} × M^{N}. Then, under (H1) and (H3), problem (3.1) admits at least one solution.

Proof. (V_{ε}, ∥ · ∥_{V_{ε}}) is a reflexive Banach space (Remark 3.1), then using the well known theorem [13, Page 135], it suffices to establish that I_{r} is weakly lower semicontinuous and coercive on (V_{ε}, ∥ · ∥_{V_{ε}}). So let v ∈ V_{ε}, in one hand we are V_{ε} ↪ W^{1,q}(O_{ε}) which implies by Proposition 2.2 that functional v ∈ V_{ε} ↦ ∫_{O_{ε}} h(x, ∇v) dx is weakly lower semicontinuous on V_{ε}. It is the same for functional v ↦ ∫_{B_{ε}} b_{w}(x, ∇_{w}v) dx. Indeed, using embedding V_{ε} ↪ L^{p_{w}}(B_{ε}), the fact that operators Q ∈ M^{N} ↦ Q_{w} are linear continuous (section 2), then it suffices to adapt proof of [6, Theorem 2.4] by replacing ∇ with ∇_{w}. We Conclude using linearity and continuity of L on V_{ε}. For coercivity, it’s easily seen according to (H1) and (H3) that on V_{ε}, lim_{∥v∥_{Vε} →+∞} I_{r}(v) = +∞. □

Remark 3.3. In general, if a function f : ℝ^{N} ×M^{N} ↦ \overline{ℝ} satisfying condition (2.1) is not quasiconvex, then for an open bounded subset O of ℝ^{N}, inf F (u)_{W^{1,p}(O)} = inf ∫_{Ω} f(x, ∇u)dx can be not existent. In return, if we take his quasiconvex envelope Qf we can study existence of solutions of the problem inf QF = inf ∫_{Ω} Qf(x, ∇.)dx noticing that infF = infQF (in the sense described in [6, Corollary 2.3], and that QF is weakly lower semicontinuous on W^{1,p}(O).

4. Limit Problem

In order to determine the limit problem, we first identify the topological space that we shall consider in the following. In one hand, the space must be big enough not depending on the parameter r to include the spaces V_{ε} defined in section 3. In the other, topology must provide the relative compactness of a (3.1)-minimizers sequence. Let X = W_{loc}^{1,q}(O \ S) and τ his weak topology. Let us consider the subset

V = {v ∈ X : v ∈ W^{1,q}(O \ S), v = 0 on Γ_{0}}

Proposition 4.1. If (v_{r})_{r} is a sequence in X verifying F_{r}(v_{r}) ≤ C, then there exist v ∈ V and a subsequence such that v_{r} →^{τ} v in X. Moreover,

(1) X_{O_{ε}}∇v_{r} ⇀ ∇v in L^{q}(O).

(2) ∫_{ℝ^{N−1}} |v_{r}(x̃, ±εγ(\frac{x̃}{ε})) − v^{±}(x̃)|^{q}dx̃ →^{r} 0.

For the proof of the above proposition see [15, page 9].

Remark 4.2. (1) Let (\overline{u}_{r})_{r} be a (3.1)-minimizer sequence, i.e.

lim_{r→0} I_{r}(\overline{u}_{r}) − inf\{I_{r}(v) : v ∈ V_{ε}\} = 0,

then (\overline{u}_{r})_{r} is relatively compact in (X, τ). It suffices to show lim inf_{r→0} I_{r}(\overline{u}_{r}) < +∞, which implies according to conditions (3.2) and (3.2) with w replaced by q that lim inf_{r→0} F_{r}(\overline{u}_{r}) < +∞, and we apply Proposition 4.1.

(2) Let p = min(p_{s}, p_{d}). In accordance with results of [15], we obtain (X, τ) so that:

• If lim sup_{(ε,μ)} \frac{ε}{μ^{1/ps}} and lim sup_{(ε,η)} \frac{ε}{η^{1/p_{d}}} < +∞, then X = L^{α}(O) and τ is his strong topology for any α ∈ [1, p[.

• If lim sup_{(ε,μ)} \frac{ε}{μ^{1/ps}} = lim sup_{(ε,η)} \frac{ε}{η^{1/p_{d}}} = 0, then X = L^{p}(O) and τ is his strong topology.

Now, we look for the Γ-limit of functionals I_{r}. First we have to remark that The functional L is linear continuous on (X, τ) (for the proof, is a straightforward consequence of (H3) and the compact embedding W_{loc}^{1,q}(O \ S, ℝ^{N}) ↪ L^{q}(S)), then according to Proposition 2.1, it suffices to study Γ-limit for functionals F_{r}. To this end, we extend F_{r} on the space (X, τ) as

F_{r}(v) = \begin{cases}∫_{O_{ε}} h(x, ∇v) dx + ∫_{B_{ε}} μb_{s}(\frac{x}{ε}, ∇_{s} v) + η b_{d}(\frac{x}{ε}, ∇_{d} v) dx & if v ∈ V_{ε} \\ +∞ & if v ∉ V_{ε} \end{cases}

we recall that V_{ε} = {v ∈ W^{1,q}(O_{ε}) × W^{1,p}(B_{ε}) : ∇_{s} v ∈ L^{p_{s}}(B_{ε}, M^{N}) and ∇_{d} v ∈ L^{p_{d}}(B_{ε}, M^{N}), [v]_{S_{ε}^{±}} = 0, v = 0 on Γ_{0}}. Let

l_{s} = lim_{(ε,μ)} \frac{μ}{2(2ε)^{p_{s}−1}} and l_{d} = lim_{(ε,η)} \frac{η}{2(2ε)^{p_{d}−1}} .

We define functional F on X as follows:

(i) If l_{s}, l_{d} ∈ [0, +∞[:

F (v) = \begin{cases}∫_{O} Qh(x, ∇v) dx + ∫_{S}\{l_{s} (b^{∞,s})^{hom} + l_{d} (b^{∞,d})^{hom}\}[v]dx̃ & if v ∈ V \\ +∞ & if v ∉ V \end{cases}

we recall that V = {v ∈ X : v ∈ W^{1,q}(O \ S, ℝ^{N}) and v = 0 on Γ_{0}}. (ii) If l_{s} = +∞ and l_{d} < +∞:

F (v) = \begin{cases}∫_{O} Qh(x, ∇v) dx + l_{d} ∫_{S}(b^{∞,d})^{hom}[v_{T}]dx̃ & if v ∈ V_{0,N} \\ +∞ & if v ∉ V_{0,N} \end{cases}

where V_{0,N} = {v ∈ V : [v_{N}] = 0}, v_{N} = (v.e_{N}) and v_{T} = v − v_{N}e_{N}.

(iii) If l_{d} = +∞:

F (v) = \begin{cases}∫_{O} Qh(x, ∇v) dx & if v ∈ V_{0} \\ +∞ & if v ∉ V_{0}, \end{cases}

where V_{0} = {v ∈ V : [v] = 0}. For the three functionals, Qh is the quasiconvex envelope of h, [v] the jump of v across S and (b^{∞,w})^{hom} is the function defined on ℝ^{N} as

(b^{∞,w})^{hom}(a) = inf_{k} \frac{1}{k^{N−1}} inf\{∫_{B_{k}} b^{∞,w}(y, ∇_{w}φ)dy : φ ∈ Ψ^{γ}a + W_{0}^{1,p_{w}}(B_{k}, ℝ^{N})\}

where w = s or w = d, B_{k} = {x ∈ ℝ^{N} : x̃ ∈ kỸ, ±x_{N} ≤ γ^{±}(x̃)}, for x ∈ ℝ^{N}

Ψ^{γ}(x) = sign(x_{N})Ψ(\frac{|x_{N}|}{γ^{±}(x̃)})

with

Ψ(t) = \begin{cases}0 & if t < 0 \\ t & if 0 ≤ t < 1 \\ 1 & if t ≥ 1 \end{cases}

Without loss of generality, we suppose in the following that b_{s}(., 0) = 0 on B_{ε}. The principal result of this section is in the following proposition

Proposition 4.3. Γ(τ) − lim F_{r} = F

To establish this result, we need some lemmas. Let Ã ∈ B_{b}(ℝ^{N−1}), a ∈ ℝ^{N} and we take p = p_{w}. We define

SÃ(a) = inf{∫_{A} b^{∞,w}(y, ∇_{w}φ)dy : φ ∈ Ψ^{γ}a + W_{0}^{1,p}(A, ℝ^{N})}  (4.1)

where

A = {x ∈ ℝ^{N} : x̃ ∈ Ã, ±x_{N} ≤ γ^{±}(x̃)}  (4.2)

Lemma 4.4. Let Ã be a convex open bounded subset of ℝ^{N−1}. Then for a sequence (ε_{n})_{n} of real positive, ε_{n} → 0 we have

lim_{n→+∞} \frac{S_{_{\frac{1}{εn}} Ã}(a)}{meas(\frac{1}{ε_{n}} Ã)} = (b^{∞,w})^{hom}(a)

Proof. Let Ã ∈ B_{b}(ℝ^{N−1}), and the function S : Ã ↦ SÃ. Then S is a subadditive ℤ^{N−1}-periodic function:

(i) Let Ã, B̃ ∈ B_{b}(ℝ^{N−1}) such that Ã∩ B̃ = ∅, then S_{Ã∪B̃} ≤ SÃ + SB̃. To establish this, we take φ_{A} ∈ Ψ^{γ}(a) + W_{0}^{1,p}(A, ℝ^{N}) and φ_{B} ∈ Ψ^{γ}(a) + W_{0}^{1,p}(B, ℝ^{N}), where A and B ( in (4.2) we replace Ã by B̃ ) are defined from Ã and B̃ by (4.2).

Let us take

Φ = \begin{cases}φ_{A} & on A \\ φ_{B} & on B \end{cases}

Since Ã ∩ B̃ = ∅, A ∩ B = ∅. Thus Φ ∈ Ψ^{γ}a + W_{0}^{1,p}(A ∪ B), and

S_{Ã∪B̃} ≤ ∫_{A∪B} b^{∞,w}(y, ∇_{w}Φ)dy = ∫_{A} b^{∞,w}(y, ∇_{w}φ_{A})dy + ∫_{B} b^{∞,w}(y, ∇_{w}φ_{B})dy

for all φ_{A} and φ_{B}. Thus

S_{Ã∪B̃} ≤ SÃ + SB̃

(ii) Let Ã ∈ B_{b}(ℝ^{N−1}) and z ∈ ℤ^{N−1}. Let A and A_{z} subsets associated respectively to Ã and Ã + z by relation (4.2), and φ ∈ W_{0}^{1,p}(A_{z}). Since b_{w} is Ỹ-periodic, it’s the same for b^{∞, w}. Thus

∫_{A_{z}} b^{∞, w}(x, ∇_{w}φ)dx = ∫_{Ã+z} ∫_{{x_{N}:±x_{N} ≤γ^{±}(x̃)}} b^{∞, w}(x, ∇_{w}φ)dx_{N} dx̃

= ∫Ã ∫_{{x_{N}:±x_{N} ≤γ^{±}(x̃+z)}} b^{∞, w}(x̃ + z, x_{N}, ∇_{w}φ)dx_{N} dx̃

= ∫_{A} b^{∞, w}(x, ∇_{w}φ)dx_{N} dx̃

Subadditivity and ℤ^{N−1}-periodicity being proved for S, we have to show dominant property (Theorem 2.3). So, let Ã ∈ B_{b}(ℝ^{N−1}) be a convex included in [0, 1[^{N−1}, and A, B subsets associated respectively with Ã and [0, 1[^{N−1} by (4.2), A ⊂ B. Let Φ_{0} ∈ W_{0}^{1,p}(B, ℝ^{N}) and Φ = Ψ^{γ}a + Φ_{0}. We take φ = Ψ^{γ}a + φ_{0}, where φ_{0} = ηΦ_{0} and η ∈ D(A), then φ ∈ Ψ^{γ}a + W_{0}^{1,p}(A). If we use Remark 3.1 and condition (3.2),

SÃ ≤ ∫_{A} b^{∞,w}(y, ∇_{w}φ)dy ≤ ∫_{B} b^{∞,w}(y, ∇_{w}φ)dy ≤ β_{w} ∫_{B} |∇_{w}φ|^{p}dy

And we have

|∇_{w}φ|^{p} = |(∇Ψ^{γ} ⊗ a)_{w} + ∇_{w}φ_{0}|^{p} ≤ const(|(∇Ψ^{γ} ⊗ a)_{w}|^{p} + |∇_{w}φ_{0}|^{p})

By the fact that |∇Ψ^{γ}| ≤ 1 + \frac{const}{γ_{m}^{±}} and η ∈ D(A), we have

|∇_{w}φ|^{p} ≤ const(1 + |∇_{w}Φ_{0}|^{p} + |Φ_{0}|^{p}).

According to Poincaré inequality, we obtain

SÃ ≤ const(meas B + ∫_{B} |∇_{w}Φ_{0}|^{p} + ∫_{B} |Φ_{0}|^{p})

≤ const(meas B + ∥Φ_{0}∥_{W_{0}^{1,p}(B)}^{p}) , ∀Φ_{0} ∈ W_{0}^{1,p}(B)

≤ const(meas B + inf_{W_{0}^{1,p}(B)} ∥Φ_{0}∥_{W_{0}^{1,p}(B)}^{p})

which establish the dominant property. In the other hand, b^{∞,w} ≥ 0 ⇒ γ(S) ≥ 0 > −∞ (see Theorem 2.3 for γ(S) definition). Let Ã be a convex open bounded subset of ℝ^{N−1}, and Ã_{n} = \frac{1}{ε_{n}} Ã. (Ã_{n})_{n} is a regular sequence. Indeed, since Ã is a bounded subset, we can find a cube Ĩ ⊂ ℤ^{N−1} such that Ã ⊂ Ĩ and α small enough so that αĨ ⊂ Ã. If we take Ĩ_{n} = \frac{1}{ε_{n}} Ĩ we obtain regularity. Now, let \overline{B}_{m}(x) = \{y ∈ ℝ^{N−1} : δ(x, y) ≤ m\} where m ≥ 0, δ euclidian distance in ℝ^{N−1} and t ≥ 0. Since t\overline{B}_{m}(x) = \overline{B}_{tm}(tx), then for A ⊂ ℝ^{N−1} we have ρ(tA) = tρ(A) where ρ(A) = sup\{m ≥ 0 : ∃\overline{B}_{m}(x) ⊂ A\}. Thus

ρ(Ã_{n}) = ρ(\frac{1}{ε_{n}} Ã) = \frac{1}{ε}ρ(Ã) →^{n} +∞

Conditions of Theorem 2.3 are then satisfied for Ã_{n}, which prove lemma. □

Lemma 4.5. If a sequence (v_{r})_{r} ⊂ X satisfies F_{r}(v_{r}) ≤ c, then

μ ∫_{B_{ε}} |∇_{s}v_{r}|^{p_{s}}dx ≤ C_{1}  (4.3)
η ∫_{B_{ε}} |∇_{d}v_{r}|^{p_{d}}dx ≤ C_{2}  (4.4)
∫_{O_{ε}} |∇v_{r}|^{q}dx ≤ C_{3}  (4.5)

The proof of the above lemma is a straightforward consequence of (H1). Now, we consider the following regularity condition.

(H4) l_{s} and l_{d} ∈ [0, +∞[, u ∈ V′, v_{r} → u in (X, τ) and lim inf F_{r}(v_{r}) < +∞. where V′ = {v ∈ V : v^{±} = v_{/O^{±}} ∈ C^{1}(O^{±}, ℝ^{N})}. We define on V, the application

R_{ε}u(x) = \frac{u(|x_{N}|) − u(−|x_{N}|)}{2} Ψ_{ε}(x) + \frac{u(|x_{N}|) + u(−|x_{N}|)}{2},  (4.6)

where Ψ_{ε}(x) = Ψ^{γ}(\frac{x}{ε}). We take θ = ∇Ψ^{γ} ∈ L^{∞}(ℝ^{N}). We also consider

\begin{matrix}t^{±}(ε) = (∫_{ℝ^{N−1}} | (v_{r} − u)(x̃, ±εγ^{±}(\frac{x̃}{ε})) |^{p}dx̃)^{\frac{1}{p}} \\ B′ε = \{x ∈ O : ±x_{N} ≤ (1 + t^{±}(ε))εγ^{±}(\frac{x̃}{ε})\} \\ φ_{ε}(x̃, ±x_{N}) = 1 − Ψ( \frac{\frac{|x_{N} |}{εγ^{±}( \frac{x̃}{ε})} − 1}{t^{±}(ε)}) \end{matrix}  (4.7)

Let α > 0 such that α → 0. Let (S_{i})_{i∈I(α)} be a family of open bounded disconnected cubes of ℝ^{N−1} with diameter α so that meas(ℝ^{N−1} \ ∪_{i∈I(α)}S_{i}) = 0, and B′_{ε}_{,i} = B′_{ε} ∩ (S_{i} × ℝ). we denote by (λ, w) pair (μ, s) or (η, d) and b = b_{w}. Lemma 4.6. With condition (H4), for ω_{r} = φ_{ε}(v_{r} − R_{ε}u) we have

lim inf_{r→0} λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w} + ∇_{w}ω_{r})dx

≥ lim inf_{r→0} λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w})dx − o(α)

Proof. We have B′_{ε}_{,i} = B′_{ε} ∩ (S_{i} × ℝ), then

meas(B′_{ε,i}) ≤ α^{N−1}ε(1 + t^{±}(ε))γ_{M}^{±} (± in sense of maximum )  (4.8)

If we take p = min(p_{s}, p_{d})

t^{±}(ε)^{p} = ∫_{ℝ^{N−1}} | (v_{r} − u)(x̃, ±εγ^{±}(\frac{x̃}{ε})) |^{p}dx̃

≤ const(∫_{ℝ^{N−1}} | v_{r}(x̃, ±εγ^{±}(\frac{x̃}{ε})) − u(x̃, 0)|^{p}dx̃

+ ∫_{ℝ^{N−1}} | u(x̃, ±εγ^{±}(\frac{x̃}{ε})) − u(x̃, 0)|^{p}dx̃).

Since v_{r} ⇀ u in W_{loc}^{1,q}(O \ S), v_{r} → u in L_{loc}^{q}(O \ S). According to Proposition 4.1, embedding L^{q} ↪ L^{p} (p ≤ q) and regularity of u, t^{±}(ε) →^{r} 0. Applying this result on (4.8),

meas(B′_{ε,i}) ≤ const α^{N−1}ε.  (4.9)

Since b satisfies condition (3.2) and using (4.9),

λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w})dx ≤ constλ(1 + \frac{1}{(2ε)^{p}}) meas(B′_{ε,i})

≤ const(ελ + \frac{λ}{(2ε)^{p−1}} ) α^{N−1}

Thus

lim inf_{r→0} λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w})dx ≤ const l α^{N−1} = o(α)

Since b ≥ 0,

lim inf_{r→0} λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w} + ∇_{w}ω_{r})dx

≥ 0 ≥ lim inf_{r→0} λ ∫_{B′_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w})dx

−o(α)
□

Now we are ready to establish Proposition 4.3

Proposition 4.7. For every sequence (v_{r})_{r} ⊂ X and every u ∈ X such that v_{r} →^{τ} u in X, we have

F (u) ≤ lim inf_{r→0} F_{r}(v_{r}).

Proof. Let (v_{r})_{r} be a sequence in X and u ∈ X so that v_{r} →^{τ} u in X. If lim inf_{r→0} F_{r}(v_{r}) = +∞, then proposition is proved. If not, by Proposition 4.1 u ∈ V.

(i) Case l_{s} and l_{d} are finite. We begin by treating regular case; i.e., when condition (H4) is satisfied. Then, by adaptation of [15, Lemmas 4.4, 4.5, 4.6, 4.9] and by application of Lemma 4.6, we have for ω_{r} = φ_{ε}(v_{r} − R_{ε}u),

lim inf_{r→0} λ ∫_{B_{ε}} b(\frac{x}{ε}, ∇_{w}v_{r})dx

= lim inf_{r→0} λ ∫_{B′_{ε}} b(\frac{x}{ε}, \frac{1}{2ε}((u(|x_{N}|) − u(−|x_{N}|)) ⊗ θ(\frac{x}{ε}) )_{w} + ∇_{w}ω_{r})dx

≥ lim inf_{r→0} λ ∑_{i∈I(α)} ∫_{B′_{ε}_{,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w} + ∇_{w}ω_{r})dx − o(α)

≥ lim inf_{r→0} λ ∑_{i∈I(α)} ∫_{B′_{ε}_{,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a_{i}) ⊗ θ(\frac{x}{ε}))_{w}) − o(α)

≥ l_{w} ∑_{i∈I(α)} meas(S_{i})(b^{∞,w})^{hom}([u](a_{i})) − o(α).

As α → 0, we obtain

lim inf_{r→0} λ ∫_{B_{ε}} b(\frac{x}{ε}, ∇_{w}v_{r})dx ≥ l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃.  (4.10)

By the characterization of quasiconvex envelope (see section 2), we have

Qh(x, ∇v_{r}(x)) = inf\{\frac{1}{meas D} ∫_{D} h(x, ∇v_{r}(x) + ∇φ(y)) : φ ∈ W_{0}^{1,∞}(D)\}

where D is a bounded domain of ℝ^{N}. If we take φ = 0, then

Qh(x, ∇v_{r}(x)) ≤ \frac{1}{meas D} ∫_{D} h(x, ∇v_{r}(x))dy = h(x, ∇v_{r}(x))

Let δ be a fixed real less than 1. For a given ε small enough, O_{δ} ⊂ O_{ε}. Thus

∫_{O_{δ}} Qh(x, ∇v_{r}(x))dx ≤ ∫_{O_{ε}} h(x, ∇v_{r}(x))dx

Since the sequence (F_{r}(v_{r}))_{r} is bounded, and according to (4.5), the fact that O_{δ} ⊂ O_{ε}, then v_{r} ⇀ u in W^{1,q}(O_{δ}, ℝ^{N}). Qh being quasiconvex, by Proposition 2.2 the functional I(v) = ∫_{O_{δ}} Qh(x, ∇v(x))dx is then weakly lower semicontinuous on W^{1,q}(O_{δ}, ℝ^{N}). Thus

lim inf_{r→0} ∫_{O_{ε}} h(x, ∇v_{r}(x))dx ≥ ∫_{O_{δ}} Qh(x, ∇u(x))dx

tending δ toward 0

lim inf_{r→0} ∫_{O_{ε}} h(x, ∇v_{r}(x))dx ≥ ∫_{O} Qh(x, ∇u(x))dx  (4.11)

According to (4.10) and (4.11)

lim inf_{r→0} F_{r}(v_{r}) ≥ F (u), for u regular  (4.12)

If u is not regular, we consider a regular vector valued function u_{δ} so that ∥u − u_{δ}∥_{W^{1,q}(O\S,ℝ^{N})} ≤ δ and we take v_{δ,r} = v_{r} − R_{ε}u + R_{ε}u_{δ}. Now, let us verify that R_{ε}u →^{τ} u. Since R_{ε}u = u on O_{ε} (see (4.6)) and ψ_{ε} ≤ 1, it follows that for p = min(p_{s}, p_{d}),

∫_{O} |R_{ε}u − u|^{p}dx = ∫_{B_{ε}} |R_{ε}u − u|^{p}dx

≤ const{∫_{B_{ε}} |R_{ε}u|^{p}dx + ∫_{B_{ε}} |u|^{p}dx}

≤ const ∫_{B_{ε}} |u|^{p}dx

→^{ε} 0

(4.13)

So we have the result for u and u_{δ}, thus v_{δ,r} →^{τ} u_{δ}. Using (4.12),

lim inf_{r→0} F_{r}(v_{δ,r}) ≥ F (u_{δ})  (4.14)

According to conditions (3.2) and (3.2) with w replaced by q, we have

\begin{aligned}F_{r}(v_{r}) & = F_{r}(v_{δ,r} + R_{ε}u − R_{ε}u_{δ}) \\ & ≥ F_{r}(v_{δ,r}) − const \{∫_{O_{ε}} |∇ (u_{δ} − u)|(1 + |∇v_{r}|^{q−1} + |∇v_{δ,r}|^{q−1} ) \\ & + ∑_{(λ,w)} λ ∫_{B_{ε}} |∇_{w} R_{ε}(u_{δ} − u)|(1 + |∇_{w}v_{r}|^{p_{w}−1} + |∇_{w}v_{δ,r}|^{p_{w}−1} )\} \end{aligned} (4.15)

Let v_{δ} = u_{δ} − u, v_{δ}^{ε} = R_{ε}(v_{δ}) and

A_{1} = ∫_{O_{ε}} |∇ v_{δ}|(1 + |∇v_{r}|^{q−1} + |∇v_{δ,r}|^{q−1})

A_{2} = ∑_{(λ,w)} λ ∫_{B_{ε}} |∇_{w}v_{δ}^{ε}|(1 + |∇_{w}v_{r}|^{p_{w}−1} + |∇_{w}v_{δ,r}|^{p_{w}−1} )dx .

By Holder inequality

A_{1} ≤ (∫_{O_{ε}} |∇v_{δ}|^{q})^{\frac{1}{q}} (∫_{O_{ε}}(1 + |∇v_{r}|^{q−1} + |∇v_{δ,r}|^{q−1})^{q′}dx)^{1/q′}

≤ const∥v_{δ}∥_{W^{1,q}(O\S)}.(∫_{O_{ε}} 1 + |∇v_{r}|^{q} + |∇u|^{q} + |∇u_{δ}|^{q}dx)^{1/q′}

(q′ is the conjugate exponent of q). We have

∫_{O_{ε}} |∇u_{δ}|^{q}dx ≤ ∥u_{δ}∥_{W^{1,q}(O\S)}^{q}

≤ const (∥v_{δ}|_{|W^{1,q}(O\S)}^{q} + ∥u∥_{W^{1,q}(O\S)}^{q})

≤ const (1 + ∥u∥_{W^{1,q}(O\S)}^{q}).

Using this result and (4.5)

A_{1} ≤ const ∥v_{δ}∥_{W^{1,q}(O\S)}(1 + ∥u∥_{W^{1,q}(O\S)}^{q})^{1/q′}.  (4.16)

On the other hand, by Holder inequality,

A_{2} ≤ const ∑_{(λ,w)} λ(∫_{B_{ε}} |∇v_{δ}^{ε}|^{p_{w}})^{\frac{1}{pw}} (∫_{B_{ε}} 1 + |∇_{w}v_{r}|^{p_{w}} + |∇v_{δ}^{ε}|^{p_{w}}dx)^{\frac{pw−1}{pw}} .

We have

∫_{B_{ε}} |∇v_{δ}^{ε}|^{p_{w}}dx = ∫_{B_{ε}} |∇R_{ε}v_{δ}|^{p_{w}}dx

= ∫_{B_{ε}} |\frac{1}{2ε}(v_{δ}|x_{N}| − v_{δ}(−|x_{N}|)) ⊗ θ(\frac{x}{ε})

+ \frac{1}{2}(∇v_{δ}|x_{N}| − ∇v_{δ}(−|x_{N}|))ψ_{ε}(x)|^{p_{w}}dx .

Using θ ∈ L^{∞}(ℝ^{N}), ψ_{ε} ≤ 1 and a change of variable,

∫_{B_{ε}} |∇u_{δ}^{ε}|^{p_{w}}dx ≤ const(\frac{1}{(2ε)^{p_{w}}} ∫_{B_{ε}} |v_{δ}|^{p_{w}} + ∫_{B_{ε}} |∇v_{δ}|^{p_{w}}dx).  (4.17)

Since v_{δ} ∈ V, by [15, Lemma 3.1],

\frac{λ}{(2ε)^{p_{w}}} ∫_{B_{ε}} |v_{δ}|^{p_{w}}dx ≤ const(λ ∫_{B_{ε}} |∇v_{δ}|^{p_{w}}dx + \frac{λ}{ε^{p_{w}−1}} ∥v_{δ}∥_{W^{1,q}(O\S)}^{p_{w}})

≤ const(o(r) + \frac{λ}{ε^{p_{w}−1}} ∥v_{δ}∥_{W^{1,q}(O\S)}^{p_{w}}).

By (4.17), we have

λ ∫_{B_{ε}} |∇u_{δ}^{ε}|^{p_{w}} ≤ const(o(r) + \frac{λ}{ε^{p_{w}−1}} ∥v_{δ}∥_{W^{1,q}(O\S)}^{p_{w}}).

Using this result, (4.3) and (4.4),

\begin{matrix}A_{2} \\ ≤ const ∑_{(λ,w)}(o(r) + \frac{λ}{ε^{p_{w}−1}} ∥v_{δ}∥_{W^{1,q}(O\S)}^{p_{w}})^{\frac{1}{pw}} (1 + o(r) + \frac{λ}{ε^{p_{w}−1}} ∥v_{δ}∥_{W^{1,q}(O\S)}^{p_{w}})^{\frac{pw−1}{pw}} \end{matrix}

(4.18)

Applying (4.14), (4.15), (4.16) and (4.18), we obtain

lim inf_{r→0} F_{r}(v_{r}) ≥ F (u_{δ}) − C(u)∥v_{δ}∥_{W^{1,q}(O\S)}  (4.19)

where C(u) is a constant depending on u. On the other hand, since b^{∞,w} and h satisfies respectively conditions (3.2) and (3.2) with w replaced by q, (b^{∞,w})^{hom} and Qh are lipshitz functions (the proof is an adaptation of the proof of [18, Proposition 2.1]). Then

F (u_{δ}) ≥ F (u) − const{∫_{O} |∇v_{δ}|(1 + |∇u|^{q−1} + |∇u_{δ}|^{q−1})
+ ∫_{S} |[v_{δ}]|(1 + |[u]|^{p_{w}−1} + |[u_{δ}]|^{p_{w}−1})dx̃}.

Using the fact that p_{w} ≤ q, Holder inequality, continuity of the jump, the compact embeding W^{1,q}(O \ S) ↪ L^{q}(S) and that ∥u_{δ}∥_{W^{1,q}(O\S)} ≤ ∥u∥_{W^{1,q}(O\S)} + 1, we have

F (u_{δ}) ≥ F (u) − C(u)∥v_{δ}∥_{W^{1,q}(O\S)}.

We then use this result and (4.19), and we let δ approach 0. Thus

lim inf_{r→0} F_{r}(v_{r}) ≥ F (u)

(ii) Case l_{s} = +∞ and l_{d} < +∞: We have [u_{N}] = 0. Indeed, let σ ∈ D(O, M^{N}). By Green formula and Proposition 4.1

∫_{B_{ε}} σ : ∇v_{r} dx = ∫_{O} σ : ∇v_{r} dx − ∫_{O} σ : (X_{O_{ε}}∇v_{r}) dx

= − ∫_{O} divσ : v_{r} dx − ∫_{O} σ : (X_{O_{ε}}∇v_{r}) dx

→^{r} ∫_{S} σn.[u] dx̃,

where n is the unit vector normal exterior to O^{+} . If we take σ = ϕ.I_{N}, where I_{N} is the unit matrix of ℝ^{N} and ϕ ∈ D(O), we have

lim_{r} ∫_{B_{ε}} ϕ divv_{r} dx = ∫_{S} ϕ.[u_{N}] dx̃  (4.20)

According to (4.5) and that l_{s} = +∞

| ∫_{B_{ε}} ϕ divv_{r} dx| ≤ ∥ϕ∥_{L^{p′_{s}} (B_{ε})}∥divv_{r}∥_{L^{ps} (B_{ε})}

≤ const(\frac{ε^{p_{s}−1}}{μ})^{\frac{1}{ps}} →^{r} 0

where p′_{s} is the conjugate exponent of p_{s}. By (4.20), we obtain [u_{N}] = 0. Thus, u ∈ V_{0,N}. And we have (b^{∞,s})^{hom}[u] = 0. Indeed let us take (b^{∞,s})^{hom}(a) = (b^{∞,s})^{hom}(a, γ). According to [15, Proposition 3.8], we have

(b^{∞,s})^{hom}([u]) ≤ (b^{∞,s})^{hom}([u], γ_{m}).

Let k ∈ ℕ and φ = Ψ^{γ_{m}}.[u], where for a given y ∈ B_{k}, Ψ^{γ_{m}}(y) = sign(y_{N})Ψ(\frac{|y_{N} |}{γ_{m}^{±}}) = ±\frac{y_{N}}{γ^{±}_{m}}. Definition of (b^{∞,s})^{hom} implies

(b^{∞,s})^{hom}([u]) ≤ \frac{1}{k^{N−1}} ∫_{B_{k}} b^{∞,s}(y, ∇_{s}φ)dy.

Since [u_{N}] = 0, ∇_{s}φ(y) = (∇Ψ^{γ_{m}}(y) ⊗ [u])_{s} = ±\frac{1}{γ^{±}_{m}} (e_{N} ⊗ [u])_{s} = 0. By the fact that b_{s}(x, 0) = 0, we deduce that (b^{∞,s})^{hom}([u]) = 0 and we conclude using result of case (i). (iii) case l_{d} = +∞: In this case, [u] = 0. Indeed, let σ ∈ D(O, M^{N}). We have

lim_{r} ∫_{B_{ε}} σ : ∇_{d}v_{r}dx = ∫_{S} σ_{d}n.[u]dx̃

for all σ ∈ D(O, M^{N}). Thus u ∈ V_{0}, and φ = Ψ^{γ_{m}}.[u] = 0. Consequently

(b^{∞,w})^{hom}([u]) ≤ \frac{1}{k^{N−1}} ∫_{B_{k}} b^{∞,w}(y, ∇_{w}φ)dy = 0

for w = s and w = d. The result is then proved. □

Proposition 4.8. If u ∈ X, then there exist a sequence (v_{r})_{r} ⊂ X such that v_{r} →^{τ} u and

lim sup_{r→0} F_{r}(v_{r}) ≤ F (u)

Proof. (i) Case l_{s} and l_{d} are finite: Let u ∈ X. If u ∉ V, F (u) = +∞, and the result is established taking for example v_{r} = u. If not, we first take u regular. Let (S_{i}) be the family of open bounded disconnected cubes of ℝ^{N−1} with diameter α so that meas(ℝ^{N−1} \ ∪_{i∈I(α)}S_{i}) = 0, and v_{r} = R_{ε}u (4.6). By (4.13), v_{r} →^{τ} u. Let a ∈ ℝ^{N}, for (λ, w) = (μ, s) or (η, d) we have

∑_{i∈I(α)} l_{w} meas(S_{i})(b^{∞,w})^{hom}([u](a)) ≥ lim_{r→0} λ ∫_{B_{ε}} b(\frac{x}{ε}, ∇_{w}v_{r})dx − o(α). (4.21)

Indeed, let u_{ε,i} be an ε-minimizer of S_{\frac{1}{ε} S_{i}}(a) defined by

S_{\frac{1}{ε} S_{i}}(a) = inf\{∫_{\frac{1}{ε} B_{ε,i}} b^{∞,w}(y, ∇_{w}φ)dy : φ ∈ Ψ^{γ}a + W_{0}^{1,p_{w}}(\frac{1}{ε}B_{ε,i})\},

where B_{ε,i} = B_{ε} ∩ (S_{i} × ℝ). Let θ = ∇Ψ^{γ} ∈ L^{∞}(ℝ^{N}). Using lemma 4.4 and the change of variable x = εy, we have

l_{w} meas(S_{i})(b^{∞,w})^{hom}([u](a))

= lim_{r→0} ε^{N−1} \frac{λ}{2(2ε)^{p_{w}−1}} ∫_{\frac{1}{ε} B_{ε,i}} b^{∞,w}(y, ([u](a) ⊗ θ(x))_{w} + ∇_{w}u_{ε,i})dy

= lim_{r→0} \frac{λ}{(2ε)^{p_{w}}} ∫_{B_{ε,i}} b^{∞,w}(\frac{x}{ε}, ([u](a) ⊗ θ(\frac{x}{ε}))_{w} + (∇_{w}u_{ε,i})(\frac{x}{ε}))dx

(4.22)

According to (H2) and the inequalities meas(B_{ε,i}) ≤ γ_{M}^{±}α^{N−1}ε and

∫_{B_{ε,i}} |(∇_{w}u_{ε,i})(\frac{x}{ε})|^{p_{w}}dx ≤ const ε(α^{N−1} + ε^{N})

(this last result is obtained using the u_{ε,i} definition and condition (3.2) satisfied by b^{∞,w}), (4.22) becomes

l_{w} meas(S_{i})(b^{∞,w})^{hom}([u](a))

= lim_{r→0} λ ∫_{B_{ε,i}} b(\frac{x}{ε}, \frac{1}{2ε}([u](a) ⊗ θ(\frac{x}{ε}))_{w} + \frac{1}{2ε}(∇_{w}u_{ε,i})(\frac{x}{ε}))dx . (4.23)

By Holder inequality, condition (3.2) and the result |∇R_{ε}u| ≤ const(1 + \frac{1}{ε}), we deduce

l_{w} meas(S_{i})(b^{∞,w})^{hom}([u](a))

≥ lim_{r→0} λ ∫_{B_{ε,i}} b(\frac{x}{ε}, ∇_{w}R_{ε}u + \frac{1}{2ε}(∇_{w}u_{ε,i})(\frac{x}{ε}))dx − o(α)

≥ lim_{r→0} λ ∫_{B_{ε,i}} b(\frac{x}{ε}, ∇_{w}R_{ε}u)dx − o(α)

Summing over I(α) and tending α towards 0, we deduce that

l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃ ≥ lim_{r→0} λ ∫_{B_{ε}} b(\frac{x}{ε}, ∇_{w}v_{r})dx .

Since v_{r} = u on O_{ε},

lim_{r→0}\{∫_{O_{ε}} h(x, ∇v_{r})dx + ∑_{λ,w} λ ∫_{B_{ε}} b(\frac{x}{ε}, ∇_{w}v_{r})dx\}

≤ ∫_{O} h(x, ∇u)dx + ∑_{w} l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃

Thus

G(u) = inf{lim sup_{r} F_{r}(v_{r}) : v_{r} →^{τ} u}

≤ ∫_{O} h(x, ∇u)dx + ∑_{w} l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃

If we take the weak lower semicontinuous envelope on W^{1,q}(O \ S) denoted Γ_{τ} for the two members, we obtain

Γ_{τ} G(u) ≤ ∫_{O} Qh(x, ∇u)dx + ∑_{w} l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃

(we use the integral representation of quasiconvex envelope for the first integral term and compact embeeding W^{1,q}(O \ S) ↪ L^{p_{w}}(S) for the second, noticing that function (b^{∞,w})^{hom} is convex [17, Proposition 2.6]. Since G is the Γ-limsup of F_{r}, it will be τ-lower semicontinuous [2, Theorem 2.1]; thus

G(u) = Γ_{τ} G(u) ≤ ∫_{O} Qh(x, ∇u)dx + ∑_{w} l_{w} ∫_{S}(b^{∞,w})^{hom}([u])dx̃ ≤ F (u).

We conclude noticing the infimum in the definition of G is attained. If u is not regular, we use a density argument like in Proposition 4.7. (ii) Case l_{s} = +∞ and l_{d} < +∞: If u ∉ V_{0,N}, F (u) = +∞ and we take for example v_{r} = u. If not, u ∈ V_{0,N} ⊂ V and it suffice to apply results of case (i) noticing that (b^{∞,s})^{hom}([u]) = 0.

(iii) Case l_{d} = +∞: It is deduced from the fact that (b^{∞,w})^{hom}([u]) = 0 for w = s and w = d. □

The proof of Proposition 4.3 is a direct consequence of Propositions 4.7 and 4.8. Recall the functional I_{r} = F_{r} − L is defined on the space (X, τ) and take I =

F − L. Let

W = \begin{cases}V & if l_{s} and l_{d} are finite \\ V_{0,N} & if l_{s} = +∞ and l_{d} is finite \\ V_{0} & if l_{d} = +∞ \end{cases}

Corollary 4.9. Let (\overline{u}_{r})_{r} be a (3.1)-minimizing sequence. Thus (\overline{u}_{r})_{r} is relatively compact in (X, τ). Moreover, for every cluster point \overline{u} and a subsequence, we have

lim_{r→0} I_{r}(\overline{u}_{r}) = I(\overline{u}) = inf\{I(v) : v ∈ W \}.

The proof of this corollary is a straightforward application of Remark 4.2, propositions 2.1 and 4.3.

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Abdelaziz Aït Moussa Université Mohamed Premier, Faculté des sciences, Département de Mathématiques et Informatique, Oujda, Maroc

   E-mail address: moussa@sciences.univ-oujda.ac.ma

Loubna Zlaïji Université Mohamed Premier, Faculté des sciences, Département de Mathématiques et Informatique, Oujda, Maroc

   E-mail address: l.zlaiji@yahoo.fr