\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 35--51.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{35} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Modelling of a collage problem] {Modelling of a collage problem} \author[A. A. Moussa, L. Zla\"{\i}ji \hfil EJDE/Conf/14 \hfilneg] {Abdelaziz A\"{\i}t Moussa, Loubna Zla\"{\i}ji} % in alphabetical order \address{ Abdelaziz A\"{\i}t Moussa \newline Universit\'e Mohamed Premier \\ Facult\'e des sciences \\ D\'epartement de Math\'ematiques et Informatique \\ Oujda, Maroc} \email{moussa@sciences.univ-oujda.ac.ma} \address{Loubna Zla\"{\i}ji \newline Universit\'e Mohamed Premier \\ Facult\'e des sciences \\ D\'epartement de Math\'ematiques et Informatique \\ Oujda, Maroc} \email{l.zlaiji@yahoo.fr} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{35K22, 58D25, 73D30} \keywords{Adhesive; spherical; deviational; $\Gamma$-convergence; homogenization; \hfill\break\indent quasiconvexity; subadditivity} \begin{abstract} In this paper we study the behavior of elastic adherents connected with an adhesive. We use the $\Gamma$-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. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{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 \cite{g3}, 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 \cite{m2}, Suquet \cite{s1}, Ganghoffer, Brillard and Schultz \cite{g1}, Geymonat, Krasucki and Lenci \cite{g2}, Licht and MiChaille \cite{l1}, Brezis, Caffarelli and Friedman \cite{b1}, Acerbi, Buttazo and Percivale \cite{a1}, Klarbring \cite{k2}, Caillerie \cite{c1}. This work is specially interested in approximating a minimization problem $(\mathcal{P}_r)$, 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 \cite{l1} to find a certain limit problem denoted $(\mathcal{P})$. Precisely, by the $\Gamma$-convergence method (introduced in a paper by De Giorgi and Franzoni in 1975 \cite{d5}), we look for a weak limit of a $(\mathcal{P}_r)$-minimizing sequence which is a solution of $(\mathcal{P})$. The outline of the paper is the following. Section 2 contains some notation and a brief summary of results related to notions of $\Gamma$-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 $\Gamma$-limit of the stored strain energies represented by the functionals $(F_r)_r$ and we deduce the limit problem. \section{Notation and preliminaries} We begin by introducing some notation which is used throughout the paper. First, let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two open subsets of $\mathbb{R}^N$ with interface $S$. For a function v defined on $\mathcal{O}_1\cup\mathcal{O}_2$, we call the jump of v across $S$ the function defined on $S$ by $[v]_S=v_{/\mathcal{O}_1}-v_{/\mathcal{O}_2}$. Let $M^N$ be the space of $N\times N$ real matrices endowed with the Hilbert-Schmidt scalar product $A:A'=\mathop{\rm trace}(A^tA')$. For a given $A\in M^N$, we call spherical part of A the matrix $A_s=\frac{\mathop{\rm trace} A}{N} I_N$, where $I_N$ is the unit matrix of $\mathbb{R}^N$. 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 $A\mapsto A_s$ and $A\mapsto A_d$ are linear continuous for matrix norm $ |A|=\Sigma_{1\leq i,j\leq N}|A_{i\,j}|$, where $A=(A_{i\,j})_{1\leq i,j\leq N}$. \noindent\textbf{Definition.} A a Carath\'{e}odory function $f:\mathbb{R}^N\times M^N\to\mathbb{R}$ satisfies condition $(C_p)$ if there exists $\alpha_p,\beta_p,c\in\mathbb{R}^N$, such that for $x\in \mathbb{R}^N$ and all $(A,A')\in (M^N)^2$, we have \begin{equation} \label{Cp} \begin{gathered} \alpha_p|A|^p\leq f(x,A)\leq \beta_p(1+|A|^p)\\ |f(x,A)-f(x,A')|\leq c|A-A'|(1+|A|^{p-1}+|A'|^{p-1}). \end{gathered} \end{equation} As we have already mentioned, our method will be based on the notion of $\Gamma$-convergence. Let $(X,\tau)$ be a metrisable topological space, and for every $n\in \mathbb{N}$ let $F_{n}, F:X\to\mathbb{\overline{R}}$ be functions defined on X. For every $x\in X$, the $\Gamma(\tau)$-liminf $F_n$ (respectively, $\Gamma(\tau)$-limsup $F_n$ ) are defined as: \begin{gather*} \Gamma(\tau)-\liminf F_n(x)= \inf\{\liminf F_n(x_n): x_n\stackrel{\tau}{\to}x\} \\ \Gamma(\tau)-\liminf F_n(x)=\inf\{\limsup F_n(x_n):x_n\stackrel{\tau}{\to}x\} \end{gather*} If the two expressions are equal to $F(x)$, then we say that the sequence $(F_n)$ $\Gamma(\tau)$-converges to $F$ on $X$ and we write $F=\Gamma(\tau)$-lim$F_{n}$. An other way to define F=$\Gamma$-lim$F_{n}$ is the following:\\ $(\forall x\in X)(\exists x_{0,n}\in X)\mbox{ such that }x_{0,n}\stackrel{\tau}{\to}x \mbox{ and }\limsup_{n\to+\infty}F_{n}(x_{0,n}) \leq F(x)$ \\ $(\forall x\in X)(\forall x_{n}\in X)\mbox{ such that }x_{n}\stackrel{\tau}{\to}x,\;\liminf_{n\to+\infty}F_{n}(x_{n})\geq F(x)$ The $\Gamma$-convergence method is made precise in item (1) below. \begin{proposition} \label{prop2.1} Suppose that $(F_n)_n$ $\Gamma$-converges to $F$. \noindent(1) \cite[Theorem 2.11]{a2}. Let $x_n\in X$ be such that $F_n(x_n)\leq\inf\{ F_n(x): x\in X \}+\varepsilon_n$, where $\varepsilon_n>0$, $\varepsilon_n\to0$. We assume furthermore that $\{x_n, n\in\mathbb{N}\}$ is $\tau$-relatively compact, then any cluster point $\overline{x}$ of $\{x_n, n\in\mathbb{N}\}$ is a minimizer of $F$ and $$ \liminf_{n\to+\infty}\{F_n(x): x\in X\}=F(\overline{x}). $$ (2) \cite[Theorem 2.15]{a2}. If $L:X\to\mathbb{R}$ is continuous, then $(F_n+L)_n$ $\Gamma$-converges to F+L. \end{proposition} For details about $\Gamma$-convergence, we refer the reader to \cite{a2,d3}. 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 $$ f(A)\leq\frac{1}{\mathop{\rm meas} D}\int_D f(A+\nabla \varphi)dx $$ where D is a bounded domain of $\mathbb{R}^N$, $A\in M^N$ and $\varphi\in W^{1,\,\infty}_0(D,\mathbb{R}^N)$. If f is not quasiconvex, his quasiconvex envelope is given as $$ Qf=\sup\{ g\leq f:\mbox{ g is quasiconvex }\} $$ If f is locally bounded, then the definition of $Qf$ can be expressed as \cite[Page 201]{d1} $$ Qf(A)=inf\{ \frac{1}{\mathop{\rm meas D}}\int_D f(A+\nabla\varphi)dx:\varphi\in W^{1,\,\infty}_0(D,\mathbb{R}^N) \} $$ The following proposition establish sufficiency of quasiconvexity to obtain weak lower semicontinuity in $W^{1,p}$ \begin{proposition} \label{prop2.2} Let $\mathcal{O}$ be an open bounded subset of $\mathbb{R}^N$ and $f:\mathcal{O}\times M^N\to\mathbb{R}$ a continuous quasiconvex function satisfying condition \eqref{Cp}, for $p\geq1$. Then, the functional $F:u\to\int_{\mathcal{O}}f(x,\nabla u(x)))\,dx$ is weakly lower semicontinuous on $W^{1,p}(\mathcal{O},\mathbb{R}^N)$. \end{proposition} For the proof of the above proposition, see \cite[Theorem 2.4 and Remark iv]{d1}. To describe a global subadditive theorem, we consider $\mathcal{B}_b(\mathbb{R}^d)$ the family of Borel bounded subsets of $\mathbb{R}^d$ and $\delta$ Euclidean distance in $\mathbb{R}^d$. for every $A\in\mathcal{B}_b(\mathbb{R}^d)$, $\rho(A)=\sup\{ r\geq 0:\exists \overline{B}_r(x)\subset A \}$, where $\overline{B}_r(x)=\{ y\in\mathbb{R}^d:\delta(x,y)\leq r \}$. A sequence $(B_n)_{n\in\mathbb{N}}\subset\mathcal{B}_b(\mathbb{R}^d)$ is called regular if there exist an increasing sequence of intervals $(I_n)_n\subset\mathbb{Z}^d$ and a constant C independent of n such that $B_n\subset I_n$ and $\mathop{\rm meas}(I_n)\leq C\mathop{\rm meas}(B_n)$, $\forall n$. The global subadditive theorem is essentially based on subadditive $\mathbb{Z}^d$-periodic functions . A function $S: A\in\mathcal{B}_b(\mathbb{R}^d) \to S_A\in \mathbb{R}$ is called subadditive $\mathbb{Z}^d$-periodic if it satisfy the following conditions: \begin{itemize} \item[(i)] For all $A, B\in\mathcal{B}_b(\mathbb{R}^d)$ such that $A\cap B=\emptyset$, $S_{A\cup B}\leq S_A+S_B$. \item[(ii)] For all $A\in\mathcal{B}_b(\mathbb{R}^d)$, all $z\in \mathbb{Z}^d$, $S_{A+z}=S_A$. \end{itemize} Now, we shall see the global subadditive theorem, firstly used in the setting of the calculus of variation by Dal Maso and Modica \cite{d4}, and generalized to sequences indexed by convex sets by Licht and Michaille \cite{l1} \begin{theorem} \label{thm2.3} Let $S$ be a subadditive $\mathbb{Z}^d$-periodic function such that $$ \gamma(S) = \inf\{ \frac{S_I}{\mathop{\rm meas} I}: I=[a,b[, a,b\in\mathbb{Z}^d\mbox{ and }a_i< b_i \; \forall 1\leq i\leq d \} > -\infty $$ In addition, we suppose that $S$ satisfies the dominant property: There exists $C(S)$, for every Borel convex subset $A\subset[0,1[^d,\;|S_A|\leq C(S)$. Let $(A_n)_n$ be a regular sequence of Borel convex subsets of $\mathcal{B}_b(\mathbb{R}^d)$ with $ \lim_{n\to+\infty}\rho(A_n)=+\infty$. Then $\lim_{n\to+\infty}\frac{S_{A_n}}{\mathop{\rm meas} A_n}$ exists and is equal to $$ \lim_{n\to+\infty}\frac{S_{A_n}}{\mathop{\rm meas} A_n} = \inf_{m\in\mathbb{N}^*}\{ \frac{S_{[0,m[^d}}{m^d} \} = \gamma(S) $$ \end{theorem} For the proof of the above theorem see \cite[page 24]{l2}. \section{Statement of the problem} Let $\mathcal{O}$ be a domain of $\mathbb{R}^N$ with Lipschitz boundary, divided in two parts $\mathcal{O}^\pm$ by the plane $\{x_N=0\}$. The common interface is noted $S$. The structure under study contains two adherents filling $\mathcal{O}_\varepsilon=\mathcal{O}_\varepsilon^+\cup \mathcal{O}_\varepsilon^-$, glued perfectly with an adhesive occupying $B_\varepsilon=\{x=(\widetilde{x},x_N)\in \mathcal{O}:\pm x_N\leq \varepsilon\,\gamma^\pm(\frac{\widetilde{x}}{\varepsilon})\} =\mathcal{O}\setminus\overline {\mathcal{O}}_\varepsilon$, along the common surfaces $ S^\pm_\varepsilon=\{x\in\mathcal{O}:\pm x_N=\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon})\} $, where $\varepsilon$ is a small parameter intended to tend toward 0, $\gamma^\pm:\mathbb{R}^{N-1}\longrightarrow\mathbb{R}^+$ are two $\mathcal{C}^1$ $\widetilde{Y}$-periodic functions, $\widetilde{Y}=]0,1[^{N-1}$. The maximum (respectively Minimum) of $\gamma^\pm$ on $\widetilde{Y}$ is noted $\gamma^\pm_M$(respectively $\gamma^\pm_m$). Surface forces are applied on a portion $\Gamma_1$ of $\partial\mathcal{O}$ with surface measure supposed to be positive, and the structure is clamped on his complementary $\Gamma_0$. The illustration of the domain is shown in Figure \ref{fig1}. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1.2mm} \begin{picture}(82,30)(3,5) \put(25,20){\qbezier(-15,0)(-14,14)(0,15) \qbezier(0,15)(14,14)(15,0) \qbezier(15,0)(14,-14)(0,-15) \qbezier(0,-15)(-14,-14)(-15,0)} \multiput(10.4,18)(0.4,0){74}{\line(0,1){4}} \put(3,19){$B_\varepsilon$} \put(6.5,19){$\rightarrow$} \put(19,28){$\mathcal{O}_\varepsilon^+$} \put(19,10){$\mathcal{O}_\varepsilon^-$} \put(23,23){$S_\varepsilon^+$} \put(23,15){$S_\varepsilon^-$} \put(32,27){\qbezier(-2.83,-2.83)(-4.38,-1)(-1.77,1.77 ) \qbezier(-1.77,1.77)(1,4.38)(2.83,2.83) \qbezier(2.83,2.83)(4.38,1)(1.77,-1.77 ) \qbezier(1.77,-1.77)(-1,-4.38)(-2.83,-2.83) \put(-1,-1){$\Gamma_1$}} \put(65,20){\qbezier(-15,0)(-14,14)(0,15) \qbezier(0,15)(14,14)(15,0) \qbezier(15,0)(14,-14)(0,-15) \qbezier(0,-15)(-14,-14)(-15,0)} \put(50,20){\line(1,0){30}} \put(80.3,19){$\leftarrow$} \put(83.5,19){$S$} \put(59,28){$\mathcal{O}^+$} \put(59,10){$\mathcal{O}^-$} \put(72,27){\qbezier(-2.83,-2.83)(-4.38,-1)(-1.77,1.77 ) \qbezier(-1.77,1.77)(1,4.38)(2.83,2.83) \qbezier(2.83,2.83)(4.38,1)(1.77,-1.77 ) \qbezier(1.77,-1.77)(-1,-4.38)(-2.83,-2.83) \put(-1,-1){$\Gamma_1$}} \end{picture} \end{center} \caption{Initial problem (left). Limit problem (right)\label{fig1}} \end{figure} Our study is focused on the minimization problem $(\mathcal{P}_r)$: Find $u\in V_\varepsilon$ such that: \begin{equation} \label{Pr} I_r(u)=\min_{v\in V_\varepsilon} I_r(v)=\min_{v\in V_\varepsilon}{F_r(v)-L(v)} \end{equation} where \begin{itemize} \item $r=(\varepsilon,\mu,\eta)$, 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 $\nabla$. \item $V_\varepsilon=\{v\in W^{1,q}(\mathcal{O}_\varepsilon)\times W^{1,p}(B_\varepsilon):\nabla_s \,v\in L^{p_s}(B_\varepsilon,M^N)\mbox{ and }\nabla_d\,v\in L^{p_d}(B_\varepsilon,M^N), [v]_{S^\pm_\varepsilon}=0, v=0\mbox{ sur }\Gamma_0\}$, $\nabla_s$ and $\nabla_d$ are respectively spherical and deviational components of $\nabla$. $p_s$, $p_d$ and q are constants with $10$ such that for all $\varepsilon\leq\varepsilon_0:(\mathop{\rm supp}f\cup\Gamma_1) \cap B_\varepsilon=\emptyset$. \end{itemize} To lighten notation, we shall often use {\rm const} to designate different constants (independent of $r$) in a same proof. \begin{remark} \label{rmk3.1} (1) If we consider the following norm on $V_\varepsilon$, $$ \|v\|_{V_\varepsilon}=\|v\|_{W^{1,q}(\mathcal{O}_\varepsilon,\mathbb{R}^N)}+ \|\nabla_sv\|_{L^{p_s}(B_\varepsilon,M^N)}+ \|\nabla_dv\|_{L^{p_d}(B_\varepsilon,M^N)}. $$ Then $V_\varepsilon$ will be a reflexive Banach space (because $1-\infty$ (see Theorem \ref{thm2.3} for $\gamma(S)$ definition). Let $\widetilde{A}$ be a convex open bounded subset of $\mathbb{R}^{N-1}$, and $\widetilde{A}_n=\frac{1}{\varepsilon_n}\widetilde{A}$. $(\widetilde{A}_n)_n$ is a regular sequence. Indeed, since $\widetilde{A}$ is a bounded subset, we can find a cube $\widetilde{I}\subset\mathbb{Z}^{N-1}$ such that $\widetilde{A}\subset\widetilde{I}$ and $\alpha$ small enough so that $\alpha\widetilde{I}\subset\widetilde{A}$. If we take $\widetilde{I}_n=\frac{1}{\varepsilon_n}\widetilde{I}$ we obtain regularity. Now, let $\overline{B}_m(x)=\{y\in\mathbb{R}^{N-1}:\delta(x,y)\leq m\} $ where $m\geq0$, $\delta$ euclidian distance in $\mathbb{R}^{N-1}$ and $t\geq0$. Since $t\overline{B}_m(x)=\overline{B}_{tm}(tx)$, then for $A\subset\mathbb{R}^{N-1}$ we have $\rho(tA)=t\rho(A)$ where $\rho(A)=\sup\{m\geq0:\exists\overline{B}_m(x)\subset A\}$. Thus $$ \rho(\widetilde{A}_n)=\rho(\frac{1}{\varepsilon_n}\widetilde{A}) =\frac{1}{\varepsilon}\rho(\widetilde{A})\stackrel{n}{\to}+\infty $$ Conditions of Theorem \ref{thm2.3} are then satisfied for $\widetilde{A}_n$, which prove lemma. \end{proof} \begin{lemma} \label{lem4.5} If a sequence $(v_r)_r\subset X$ satisfies $F_r(v_r)\leq c$, then \begin{gather} \mu\int_{B_\varepsilon}|\nabla_s v_r|^{p_s} dx\leq C_1 \label{e4.3}\\ \eta\int_{B_\varepsilon}|\nabla_d v_r|^{p_d}dx\leq C_2 \label{e4.4}\\ \int_{\mathcal{O}_\varepsilon}|\nabla v_r|^q dx\leq C_3 \label{e4.5} \end{gather} \end{lemma} The proof of the above lemma is a straightforward consequence of (H1). Now, we consider the following regularity condition. \begin{itemize} \item[(H4)] $l_s$ and $l_d\in [0,+\infty[$, $u\in V'$, $v_r\to u$ in $(X,\tau)$ and $\liminf F_r(v_r)<+\infty$. \end{itemize} where $V'=\{v\in V:v^\pm=v_{/\mathcal{O}^\pm}\in \mathcal{C}^1(\mathcal{O}^\pm,\mathbb{R}^N)\}$. We define on $V$, the application \begin{equation} \label{e4.6} R_\varepsilon u(x)=\frac{u(|x_N|)-u(-|x_N|)}{2}\;\Psi_\varepsilon(x) +\frac{u(|x_N|)+u(-|x_N|)}{2}, \end{equation} where $\Psi_\varepsilon(x)=\Psi^\gamma(\frac{x}{\varepsilon})$. We take $\theta=\nabla\Psi^\gamma\in L^\infty(\mathbb{R}^N)$. We also consider \begin{equation} \label{e4.7} \begin{gathered} t^\pm(\varepsilon)=(\int_{\mathbb{R}^{N-1}}|\;(v_r-u)(\widetilde{x},\pm\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon}))\;|^pd\widetilde{x})^\frac{1}{p}\\ B'\varepsilon=\{x\in \mathcal{O}:\pm x_N\leq(1+t^\pm(\varepsilon))\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon})\}\\ \varphi_\varepsilon(\widetilde{x},\pm x_N)=1-\Psi(\frac{\frac{|x_N|}{\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon})}-1}{t^\pm(\varepsilon)}) \end{gathered} \end{equation} Let $\alpha>0$ such that $\alpha\to0$. Let $(S_i)_{i\in I(\alpha)}$ be a family of open bounded disconnected cubes of $\mathbb{R}^{N-1}$ with diameter $\alpha$ so that $\mathop{\rm meas}(\mathbb{R}^{N-1}\setminus\cup_{i\in I(\alpha)} S_i)=0$, and ${B_\varepsilon'}_{, i}=B_\varepsilon'\cap(S_i\times\mathbb{R})$. we denote by $(\lambda,w)$ pair $(\mu,s)$ or $(\eta,d)$ and $b=b_w$. \begin{lemma} \label{lem4.6} With condition (H4), for $\omega_r=\varphi_\varepsilon(v_r-R_\varepsilon u)$ we have \begin{align*} &\liminf_{r\to0}\lambda\int_{B'_{\varepsilon,i}}b(\frac{x}{\varepsilon}, \frac{1}{2\varepsilon}([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w+\nabla_w\omega_r)dx\\ & \geq\liminf_{r\to0}\lambda\int_{B'_{\varepsilon,i}}b(\frac{x}{\varepsilon}, \frac{1}{2\varepsilon}([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w)dx -o(\alpha) \end{align*} \end{lemma} \begin{proof} We have ${B_\varepsilon'}_{, i}=B_\varepsilon'\cap(S_i\times\mathbb{R})$, then \begin{equation} \label{e4.8} \mathop{\rm meas}(B'_{\varepsilon,i}) \leq\alpha^{N-1}\varepsilon(1+t^\pm(\varepsilon))\gamma^\pm_M \quad(\pm\mbox{ in sense of maximum })\\ \end{equation} If we take $p=\min(p_s, p_d)$ \begin{align*} t^\pm(\varepsilon)^p&=\int_{\mathbb{R}^{N-1}}|\;(v_r-u)(\widetilde{x}, \pm\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon}))\;|^pd\widetilde{x}\\ &\leq{\rm const}(\int_{\mathbb{R}^{N-1}}|\;v_r(\widetilde{x}, \pm\varepsilon\gamma^\pm(\frac{\widetilde{x}}{\varepsilon}))\;-u(\widetilde{x},0)|^pd \widetilde{x}\\ &\quad+\int_{\mathbb{R}^{N-1}}|\;u(\widetilde{x},\pm\varepsilon\gamma^\pm (\frac{\widetilde{x}}{\varepsilon}))\;-u(\widetilde{x},0)|^pd\widetilde{x}). \end{align*} Since $v_r\rightharpoonup u$ in $W^{1,q}_{\rm loc}(\mathcal{O}\setminus S)$, $v_r\to u$ in $L^q_{\rm loc}(\mathcal{O}\setminus S)$. According to Proposition \ref{prop4.1}, embedding $L^q\hookrightarrow L^p$ ($p\leq q$) and regularity of $u$, $ t^\pm(\varepsilon)\stackrel{r}{\to}0$. Applying this result on \eqref{e4.8}, \begin{equation} \label{e4.9} \mathop{\rm meas}(B'_{\varepsilon,i}) \leq {\rm const}\,\alpha^{N-1}\varepsilon . \end{equation} Since $b$ satisfies condition \eqref{Cw} and using \eqref{e4.9}, \begin{align*} \lambda\int_{B'_{\varepsilon,i}}b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon} ([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w)dx &\leq {\rm const}\lambda(1+\frac{1}{(2\varepsilon)^p}) \mathop{\rm meas}(B'_{\varepsilon,i})\\ &\leq {\rm const}(\varepsilon\lambda+\frac{\lambda}{(2\varepsilon)^{p-1}})\,\alpha^{N-1} \end{align*} Thus $$ \liminf_{r\to0}\lambda\int_{B'_{\varepsilon,i}}b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon}([u] (a_i)\otimes\theta(\frac{x}{\varepsilon}))_w)dx \leq {\rm const} \,l\,\alpha^{N-1}=o(\alpha) $$ Since $b\geq0$, \begin{align*} &\liminf_{r\to0}\lambda\int_{B'_{\varepsilon,i}} b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon}([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w +\nabla_w\omega_r)dx\\ & \geq0\geq\liminf_{r\to0}\lambda\int_{B'_{\varepsilon,i}} b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon}([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w)dx\\ -o(\alpha) \end{align*} \end{proof} Now we are ready to establish Proposition \ref{prop4.3} \begin{proposition} \label{prop4.7} For every sequence $(v_r)_r\subset X$ and every $u\in X$ such that $v_r\stackrel{\tau}{\to}u$ in $X$, we have $$ F(u)\leq\liminf_{r\to0}F_r(v_r). $$ \end{proposition} \begin{proof} Let $(v_r)_r$ be a sequence in $X$ and $u\in X$ so that $v_r\stackrel{\tau}{\to}u$ in $X$. If $\liminf_{r\to0}F_r(v_r)=+\infty$, then proposition is proved. If not, by Proposition \ref{prop4.1} $u\in V$. \noindent (i) \textbf{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 \cite[Lemmas 4.4, 4.5, 4.6, 4.9]{l1} and by application of Lemma \ref{lem4.6}, we have for $\omega_r=\varphi_\varepsilon(v_r-R_{\varepsilon}u)$, \begin{align*} &\liminf_{r\to0}\lambda\int_{B_\varepsilon}b(\frac{x}{\varepsilon},\nabla_w v_r)dx\\ &=\liminf_{r\to0}\lambda\int_{{B_\varepsilon'}}b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon} ((u(|x_N|)-u(-|x_N|))\otimes \theta(\frac{x}{\varepsilon})\;)_w+\nabla_w\omega_r)dx\\ &\geq \liminf_{r\to0}\lambda\sum_{i\in I(\alpha)}\int_{{B_\varepsilon'}_{,i}}b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon} ([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w +\nabla_w\omega_r)dx-o(\alpha)\\ &\geq \liminf_{r\to0}\lambda\sum_{i\in I(\alpha)}\int_{{B_\varepsilon'}_{,i}}b(\frac{x}{\varepsilon},\frac{1}{2\varepsilon} ([u](a_i)\otimes\theta(\frac{x}{\varepsilon}))_w)-o(\alpha)\\ &\geq l_w \sum_{i\in I(\alpha)}\mathop{\rm meas}(S_i) (b^{\infty,w})^{\rm hom}([u](a_i))-o(\alpha). \end{align*} As $\alpha\to0$, we obtain \begin{equation} \label{e4.10} \liminf_{r\to0}\lambda\int_{B_\varepsilon}b(\frac{x}{\varepsilon},\nabla_w v_r)dx \geq l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x}. \end{equation} By the characterization of quasiconvex envelope (see section 2), we have $$ Qh(x,\nabla v_r(x))=\inf\{\frac{1}{\mathop{\rm meas}\,D}\int_D h(x,\nabla v_r(x)+\nabla\varphi(y)):\varphi\in W^{1,\infty}_0(D)\} $$ where $D$ is a bounded domain of $\mathbb{R}^N$. If we take $\varphi=0$, then $$ Qh(x,\nabla v_r(x))\leq\frac{1}{\mathop{\rm meas}\,D}\int_D h(x,\nabla v_r(x))dy =h(x,\nabla v_r(x)) $$ Let $\delta$ be a fixed real less than 1. For a given $\varepsilon$ small enough, $\mathcal{O}_\delta\subset\mathcal{O}_\varepsilon$. Thus $$ \int_{\mathcal{O}_\delta}Qh(x,\nabla v_r(x))dx \leq \int_{\mathcal{O}_\varepsilon}h(x,\nabla v_r(x))dx $$ Since the sequence $(F_r(v_r))_r$ is bounded, and according to \eqref{e4.5}, the fact that $\mathcal{O}_\delta\subset\mathcal{O}_\varepsilon$, then $v_r\rightharpoonup u$ in $W^{1,q}(\mathcal{O}_\delta,\mathbb{R}^N)$. $Qh$ being quasiconvex, by Proposition \ref{prop2.2} the functional $I(v)=\int_{\mathcal{O}_\delta}Qh(x,\nabla v(x))dx$ is then weakly lower semicontinuous on $W^{1,q}(\mathcal{O}_\delta,\mathbb{R}^N)$. Thus $$ \liminf_{r\to0}\int_{\mathcal{O}_\varepsilon}h(x,\nabla v_r(x))dx \geq \int_{\mathcal{O}_\delta}Qh(x,\nabla u(x))dx $$ tending $\delta$ toward 0 \begin{equation} \label{e4.11} \liminf_{r\to0}\int_{\mathcal{O}_\varepsilon}h(x,\nabla v_r(x))dx \geq \int_{\mathcal{O}}Qh(x,\nabla u(x))dx \end{equation} According to \eqref{e4.10} and \eqref{e4.11} \begin{equation} \label{e4.12} \liminf_{r\to0}F_r(v_r) \geq F(u), \mbox{ for $u$ regular} \end{equation} If u is not regular, we consider a regular vector valued function $u_\delta$ so that $\|u-u_\delta\|_{W^{1,q}(\mathcal{O}\setminus S,\mathbb{R}^N)}\leq \delta$ and we take $ v_{\delta,r}=v_r-R_\varepsilon u+R_\varepsilon u_\delta$. Now, let us verify that $R_\varepsilon u\stackrel{\tau}{\to}u$. Since $R_\varepsilon u=u$ on $\mathcal{O}_\varepsilon$ (see \eqref{e4.6}) and $\psi_\varepsilon\leq 1$, it follows that for $p=\min(p_s,p_d)$, \begin{equation} \label{e4.13} \begin{aligned} \int_\mathcal{O}|R_\varepsilon u-u|^pdx & = \int_{B_\varepsilon}|R_\varepsilon u-u|^pdx\\ &\leq {\rm const}\{\int_{B_\varepsilon}|R_\varepsilon u|^pdx+\int_{B_\varepsilon}|u|^pdx\}\\ &\leq {\rm const} \int_{B_\varepsilon}|u|^pdx\\ &\stackrel{\varepsilon}{\to} 0 \end{aligned} \end{equation} So we have the result for $u$ and $u_\delta$, thus $v_{\delta,r}\stackrel{\tau}{\to}u_\delta$. Using \eqref{e4.12}, \begin{equation} \label{e4.14} \liminf_{r\to0}F_r(v_{\delta,r}) \geq F(u_\delta) \end{equation} According to conditions \eqref{Cw} and \eqref{Cw} with $w$ replaced by $q$, we have \begin{equation} \label{e4.15} \begin{aligned} F_r(v_r) & = F_r(v_{\delta,r}+R_\varepsilon u-R_\varepsilon u_\delta)\\ & \geq F_r(v_{\delta,r}) - {\rm const} \, \{ \int_{\mathcal{O}_\varepsilon}|\nabla\,(u_\delta-u)|(1+|\nabla v_r|^{q-1}+|\nabla v_{\delta,r}|^{q-1}\,)\\ &\quad +\sum_{(\lambda,w)}\lambda\int_{B_\varepsilon}|\nabla_w\,R_\varepsilon(u_\delta-u)|(1+|\nabla_w v_r|^{p_w-1}+|\nabla_w v_{\delta,r}|^{p_w-1}\,)\} \end{aligned} \end{equation} Let $v_\delta=u_\delta-u$, $v^{\varepsilon}_{\delta}=R_\varepsilon(v_\delta)$ and \begin{gather*} A_1 =\int_{\mathcal{O}_\varepsilon}|\nabla\,v_\delta|(1+|\nabla v_r|^{q-1}+|\nabla v_{\delta,r}|^{q-1})\\ A_2 = \sum_{(\lambda,w)}\lambda\int_{B_\varepsilon}|\nabla_w v^{\varepsilon}_{\delta}| (1+|\nabla_w v_r|^{p_w-1}+|\nabla_w v_{\delta,r}|^{p_w-1}\,)dx\,. \end{gather*} By Holder inequality \begin{align*} A_1 & \leq (\int_{\mathcal{O}_\varepsilon}|\nabla v_\delta|^q)^{\frac{1}{q}} (\int_{\mathcal{O}_\varepsilon}(1+|\nabla v_r|^{q-1}+|\nabla v_{\delta,r}|^{q-1})^{q'}dx)^{1/q'} \\ & \leq {\rm const} \|v_\delta\|_{W^{1,q}(\mathcal{O}\setminus S)} .(\int_{\mathcal{O}_\varepsilon}1+|\nabla v_r|^{q}+|\nabla u|^{q}+|\nabla u_\delta|^q dx)^{1/q'} \end{align*} ($q'$ is the conjugate exponent of $q$). We have \begin{align*} \int_{\mathcal{O}_\varepsilon}|\nabla u_\delta|^qdx & \leq \|u_\delta\|^q_{W^{1,q}(\mathcal{O}\setminus S)}\\ & \leq {\rm const} \, (\|v_\delta|^q_{|W^{1,q}(\mathcal{O}\setminus S)} +\|u\|^q_{W^{1,q}(\mathcal{O}\setminus S)})\\ & \leq {\rm const} \, ( 1 + \|u\|^q_{W^{1,q}(\mathcal{O}\setminus S)} ). \end{align*} Using this result and \eqref{e4.5} \begin{equation} \label{e4.16} A_1 \leq {\rm const} \, \|v_\delta\|_{W^{1,q}(\mathcal{O}\setminus S)} ( 1 + \|u\|^q_{W^{1,q}(\mathcal{O}\setminus S)})^{1/q'}. \end{equation} On the other hand, by Holder inequality, \[ A_2 \leq {\rm const} \sum_{(\lambda,w)}\lambda ( \int_{B_\varepsilon}|\nabla v^{\varepsilon}_{\delta}|^{p_w} )^\frac{1}{p_w} ( \int_{B_\varepsilon}1+|\nabla_w v_r|^{p_w}+|\nabla v^{\varepsilon}_{\delta}|^{p_w}dx)^{\frac{p_w-1}{p_w}}. \] We have \begin{align*} \int_{B_\varepsilon}|\nabla v^{\varepsilon}_{\delta}|^{p_w}dx & = \int_{B_\varepsilon}|\nabla R_\varepsilon v_\delta|^{p_w}dx\\ & = \int_{B_\varepsilon}|\frac{1}{2\varepsilon}(v_\delta|x_N|-v_\delta(-|x_N|))\otimes\theta(\frac{x}{\varepsilon})\\ &\quad +\frac{1}{2}(\nabla v_\delta|x_N|-\nabla v_\delta(-|x_N|))\psi_\varepsilon(x)|^{p_w}dx\,. \end{align*} Using $\theta\in L^\infty(\mathbb{R}^N)$, $\psi_\varepsilon \leq 1$ and a change of variable, \begin{equation} \label{e4.17} \int_{B_\varepsilon}|\nabla u^{\varepsilon}_{\delta}|^{p_w}dx \leq {\rm const} (\frac{1}{(2\varepsilon)^{p_w}}\int_{B_\varepsilon}|v_\delta|^{p_w} +\int_{B_\varepsilon}|\nabla v_\delta|^{p_w}dx). \end{equation} Since $v_\delta\in V$, by \cite[Lemma 3.1]{l1}, \begin{align*} \frac{\lambda}{(2\varepsilon)^{p_w}}\int_{B_\varepsilon}|v_\delta|^{p_w}dx & \leq {\rm const} ( \lambda\int_{B_\varepsilon}|\nabla v_\delta|^{p_w}dx+\frac{\lambda}{\varepsilon^{p_w-1}}\|v_\delta\|^{p_w}_{W^{1,q}(\mathcal{O}\setminus S)} )\\ & \leq {\rm const} ( o(r)+\frac{\lambda}{\varepsilon^{p_w-1}} \|v_\delta\|^{p_w}_{W^{1,q}(\mathcal{O}\setminus S)} ). \end{align*} By \eqref{e4.17}, we have $$ \lambda\int_{B_\varepsilon}|\nabla u^{\varepsilon}_{\delta}|^{p_w} \leq {\rm const} ( o(r)+\frac{\lambda}{\varepsilon^{p_w-1}}\| v_\delta\|^{p_w}_{W^{1,q}(\mathcal{O}\setminus S)} ). $$ Using this result, \eqref{e4.3} and \eqref{e4.4}, \begin{equation} \label{e4.18} \begin{aligned} &A_2\\ &\leq {\rm const} \sum_{(\lambda,w)} (o(r) +\frac{\lambda}{\varepsilon^{p_w-1}}\|v_\delta\|^{p_w}_{W^{1,q}(\mathcal{O} \setminus S)} )^{\frac{1}{p_w}} ( 1+o(r)+\frac{\lambda}{\varepsilon^{p_w-1}}\|v_\delta\|^{p_w}_{W^{1,q} (\mathcal{O}\setminus S)} )^{\frac{p_w-1}{p_w}} \end{aligned} \end{equation} Applying \eqref{e4.14}, \eqref{e4.15}, \eqref{e4.16} and \eqref{e4.18}, we obtain \begin{equation} \label{e4.19} \liminf_{r\to0}F_r(v_r) \geq F(u_\delta)-C(u)\|v_\delta\|_{W^{1,q}(\mathcal{O}\setminus S)} \end{equation} where $C(u)$ is a constant depending on $u$. On the other hand, since $b^{\infty,w}$ and $h$ satisfies respectively conditions \eqref{Cw} and \eqref{Cw} with $w$ replaced by $q$, $(b^{\infty,w})^{\rm hom}$ and $Qh$ are lipshitz functions (the proof is an adaptation of the proof of \cite[Proposition 2.1]{m1}). Then \begin{align*} F(u_\delta) & \geq F(u)- {\rm const} \{ \int_{\mathcal{O}}|\nabla v_\delta| (1+|\nabla u|^{q-1}+|\nabla u_\delta|^{q-1} )\\ &\quad +\int_S |[v_\delta]| ( 1+|[u]|^{p_w-1}+|[u_\delta]|^{p_w-1} )d\widetilde{x}\}. \end{align*} Using the fact that $p_w\leq q$, Holder inequality, continuity of the jump, the compact embeding $W^{1,q}(\mathcal{O}\setminus S)\hookrightarrow L^q(S)$ and that $\|u_\delta\|_{W^{1,q}(\mathcal{O}\setminus S)} \leq \|u\|_{W^{1,q}(\mathcal{O}\setminus S)}+1$, we have $$ F(u_\delta) \geq F(u) - C(u)\|v_\delta\|_{W^{1,q}(\mathcal{O}\setminus S)}. $$ We then use this result and \eqref{e4.19}, and we let $\delta$ approach 0. Thus $$ \liminf_{r\to0}F_r(v_r) \geq F(u) $$ (ii) \textbf{Case $l_s=+\infty$ and $l_d<+\infty$:} We have $[u_N]=0$. Indeed, let $\sigma\in \mathcal{D(\mathcal{O},M^N)}$. By Green formula and Proposition \ref{prop4.1} \begin{align*} \int_{B_\varepsilon}\sigma:\nabla v_r\,dx & = \int_\mathcal{O}\sigma:\nabla v_r\,dx-\int_\mathcal{O}\sigma:(\mathcal{X}_{\mathcal{O}_\varepsilon}\nabla v_r)\,dx\\ & = -\int_\mathcal{O}div\sigma:v_r\,dx-\int_\mathcal{O}\sigma:(\mathcal{X}_{\mathcal{O}_\varepsilon}\nabla v_r)\,dx\\ & \stackrel{r}{\to} \int_S\sigma n.[u]\,d\widetilde{x}, \end{align*} where $n$ is the unit vector normal exterior to $\mathcal{O}^+$ . If we take $\sigma=\phi.I_N$, where $I_N$ is the unit matrix of $\mathbb{R}^N$ and $\phi\in \mathcal{D}(\mathcal{O})$, we have \begin{equation} \label{e4.20} \lim_{r}\int_{B_\varepsilon} \phi\,div v_r\,dx = \int_S\phi.[u_N]\,d\widetilde{x} \end{equation} According to \eqref{e4.5} and that $l_s=+\infty$ \begin{align*} \big|\int_{B_\varepsilon} \phi\,div v_r\,dx\big| & \leq \|\phi\|_{L^{p'_s}(B_\varepsilon)}\|div v_r\|_{L^{p_s}(B_\varepsilon)}\\ & \leq {\rm const} (\frac{\varepsilon^{p_s-1}}{\mu})^{\frac{1}{p_s}} \stackrel{r}{\to} 0 \end{align*} where $p'_s$ is the conjugate exponent of $p_s$. By \eqref{e4.20}, we obtain $[u_N]=0$. Thus, $u\in V_{0,N}$. And we have $(b^{\infty, s})^{\rm hom}[u]=0$. Indeed let us take $(b^{\infty, s})^{\rm hom}(a)=(b^{\infty, s})^{\rm hom}(a,\gamma)$. According to \cite[Proposition 3.8]{l1}, we have $$ (b^{\infty, s})^{\rm hom}([u]) \leq (b^{\infty, s})^{\rm hom}([u], \gamma_m). $$ Let $k\in \mathbb{N}$ and $\varphi=\Psi^{\gamma_m}.[u]$, where for a given $y\in B_k$, $\Psi^{\gamma_m}(y)=sign(y_N)\Psi(\frac{|y_N|}{\gamma^\pm_m}) =\pm\frac{y_N}{{\gamma^\pm}_m}$. Definition of $(b^{\infty, s})^{\rm hom}$ implies $$ (b^{\infty, s})^{\rm hom}([u]) \leq \frac{1}{k^{N-1}} \int_{B_k}b^{\infty,s}(y, \;\nabla_s\varphi)dy. $$ Since $[u_N]=0$, $\nabla_s\varphi(y)=(\nabla\Psi^{\gamma_m}(y)\otimes[u])_s =\pm\frac{1}{{\gamma^\pm}_m}(e_N\otimes[u])_s=0$. By the fact that $b_s(x,0)=0$, we deduce that $(b^{\infty, s})^{\rm hom}([u])=0$ and we conclude using result of case (i). \noindent(iii) \textbf{case $l_d=+\infty$:} In this case, $[u]=0$. Indeed, let $\sigma\in \mathcal{D}(\mathcal{O},M^N)$. We have $$ \lim_r\int_{B_\varepsilon}\sigma:\nabla_d v_r dx=\int_S\sigma_d n.[u]d\widetilde{x} $$ for all $\sigma\in \mathcal{D}(\mathcal{O},M^N)$. Thus $u\in V_0$, and $\varphi=\Psi^{\gamma_m}.[u]=0$. Consequently $$ (b^{\infty,w})^{\rm hom}([u])\leq\frac{1}{k^{N-1}}\int_{B_k}b^{\infty,w}(y, \nabla_w\varphi)dy=0 $$ for $w=s$ and $w=d$. The result is then proved. \end{proof} \begin{proposition} \label{prop4.8} If $u\in X$, then there exist a sequence $(v_r)_r\subset X$ such that $v_r\stackrel{\tau}{\to}u$ and $$ \limsup_{r\to0} F_r(v_r) \leq F(u) $$ \end{proposition} \begin{proof} (i) \textbf{Case $l_s$ and $l_d$ are finite:} Let $u\in X$. If $u\not\in V$, $F(u)=+\infty$, 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 $\mathbb{R}^{N-1}$ with diameter $\alpha$ so that $\mathop{\rm meas}(\mathbb{R}^{N-1}\setminus\cup_{i\in I(\alpha)} S_i)=0$, and $v_r=R_\varepsilon u$ \eqref{e4.6}. By \eqref{e4.13}, $v_r\stackrel{\tau}{\to}u$. Let $a\in\mathbb{R}^N$, for $(\lambda,w)=(\mu,s)$ or $(\eta,d)$ we have \begin{equation} \sum_{i\in I(\alpha)}l_w \mathop{\rm meas}(S_i)(b^{\infty,w})^{\rm hom} ([u](a)) \geq \lim_{r\to0} \lambda \int_{B_\varepsilon}b(\frac{x}{\varepsilon},\nabla_w v_r) dx-o(\alpha). \end{equation} Indeed, let $u_{\varepsilon,i}$ be an $\varepsilon$-minimizer of $S_{\frac{1}{\varepsilon}S_i}(a)$ defined by $$ S_{\frac{1}{\varepsilon}S_i}(a)=\inf \{\int_{\frac{1}{\varepsilon}B_{\varepsilon,i}}b^{\infty,w}(y,\nabla_w \varphi)dy:\varphi\in\Psi^\gamma a +W^{1,p_w}_0(\frac{1}{\varepsilon}B_{\varepsilon,i}) \}, $$ where $B_{\varepsilon,i}=B_\varepsilon\cap(S_i\times\mathbb{R})$. Let $\theta=\nabla\Psi^\gamma\in L^\infty(\mathbb{R}^N)$. Using lemma \ref{lem4.4} and the change of variable $x=\varepsilon y$, we have \begin{equation} \label{e4.22} \begin{aligned} &l_w \mathop{\rm meas}(S_i)(b^{\infty,w})^{\rm hom}([u](a))\\ & = \lim_{r\to0}\varepsilon^{N-1}\frac{\lambda}{2(2\varepsilon)^{p_w-1}}\int_{\frac{1}{\varepsilon}B_{\varepsilon,i}}b^{\infty,w}(y,([u](a)\otimes\theta(x))_w+\nabla_w u_{\varepsilon,i})dy\\ & = \lim_{r\to0}\frac{\lambda}{(2\varepsilon)^{p_w}}\int_{B_{\varepsilon,i}} b^{\infty,w}(\frac{x}{\varepsilon},([u](a)\otimes\theta(\frac{x}{\varepsilon}))_w +(\nabla_w u_{\varepsilon,i})(\frac{x}{\varepsilon}))dx \end{aligned} \end{equation} According to (H2) and the inequalities $\mathop{\rm meas}(B_{\varepsilon,i})\leq\gamma^\pm_M\alpha^{N-1}\varepsilon$ and $$ \int_{B_{\varepsilon,i}}|(\nabla_w u_{\varepsilon,i})(\frac{x}{\varepsilon})|^{p_w}dx\leq {\rm const} \,\varepsilon(\alpha^{N-1}+\varepsilon^N) $$ (this last result is obtained using the $u_{\varepsilon,i}$ definition and condition \eqref{Cw} satisfied by $b^{\infty,w}$), \eqref{e4.22} becomes \begin{equation} \label{e4.23} \begin{aligned} &l_w \mathop{\rm meas}(S_i)(b^{\infty,w})^{\rm hom}([u](a)) \\ & = \lim_{r\to0}\lambda\int_{B_{\varepsilon,i}}b(\frac{x}{\varepsilon}, \frac{1}{2\varepsilon}([u](a)\otimes\theta(\frac{x}{\varepsilon}))_w +\frac{1}{2\varepsilon}(\nabla_w u_{\varepsilon,i})(\frac{x}{\varepsilon}))dx\,. \end{aligned} \end{equation} By Holder inequality, condition \eqref{Cw} and the result $|\nabla R_{\varepsilon} u| \leq {\rm const} (1+\frac{1}{\varepsilon})$, we deduce \begin{align*} & l_w \mathop{\rm meas}(S_i)(b^{\infty,w})^{\rm hom}([u](a)) \\ & \geq \lim_{r\to0}\lambda\int_{B_{\varepsilon,i}}b(\frac{x}{\varepsilon},\nabla_w R_\varepsilon u+\frac{1}{2\varepsilon}(\nabla_w u_{\varepsilon,i})(\frac{x}{\varepsilon}))dx -o(\alpha)\\ & \geq \lim_{r\to0}\lambda\int_{B_{\varepsilon,i}}b(\frac{x}{\varepsilon},\nabla_w R_\varepsilon u)dx-o(\alpha) \end{align*} Summing over $I(\alpha)$ and tending $\alpha$ towards 0, we deduce that $$ l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x} \geq \lim_{r\to0}\lambda\int_{B_\varepsilon}b(\frac{x}{\varepsilon},\nabla_w v_r)dx\,. $$ Since $v_r=u$ on $\mathcal{O}_\varepsilon$, \begin{align*} &\lim_{r\to0} \{ \int_{\mathcal{O}_\varepsilon}h(x,\nabla v_r)dx+\sum_{\lambda,w}\lambda\int_{B_\varepsilon}b(\frac{x}{\varepsilon},\nabla_w v_r)dx \}\\ &\leq \int_{\mathcal{O}}h(x,\nabla u)dx+\sum_w l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x} \end{align*} Thus \begin{align*} G(u)&=\inf\{ \limsup_r F_r(v_r):v_r\stackrel{\tau}{\to}u \}\\ &\leq \int_{\mathcal{O}}h(x,\nabla u)dx+\sum_w l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x} \end{align*} If we take the weak lower semicontinuous envelope on $W^{1,q}(\mathcal{O}\setminus S)$ denoted $\Gamma_\tau$ for the two members, we obtain $$ \Gamma_\tau G(u) \leq \int_{\mathcal{O}}Qh(x,\nabla u)dx+\sum_w l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x} $$ (we use the integral representation of quasiconvex envelope for the first integral term and compact embeeding $W^{1,q}(\mathcal{O}\setminus S)\hookrightarrow L^{p_w}(S)$ for the second, noticing that function $(b^{\infty,w})^{\rm hom}$ is convex \cite[Proposition 2.6]{l3}. Since $G$ is the $\Gamma$-limsup of $F_r$, it will be $\tau$-lower semicontinuous \cite[Theorem 2.1]{a2}; thus \[ G(u)=\Gamma_{\tau} G(u) \leq \int_{\mathcal{O}}Qh(x,\nabla u)dx+\sum_w l_w \int_S(b^{\infty,w})^{\rm hom}([u])d\widetilde{x} \leq F(u). \] 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 \ref{prop4.7}. \noindent(ii) \textbf{Case $l_s=+\infty$ and $l_d<+\infty$:} If $u\not\in V_{0,N}$, $F(u)=+\infty$ and we take for example $v_r=u$. If not, $u\in V_{0,N}\subset V$ and it suffice to apply results of case (i) noticing that $(b^{\infty,s})^{\rm hom}([u])=0$. \noindent (iii) \textbf{Case $l_d=+\infty$:} It is deduced from the fact that $(b^{\infty,w})^{\rm hom}([u])=0$ for $w= s$ and $w=d$. \end{proof} The proof of Proposition \ref{prop4.3} is a direct consequence of Propositions \ref{prop4.7} and \ref{prop4.8}. Recall the functional $I_r=F_r-L$ is defined on the space $(X,\tau)$ and take $I = F-L$. Let $$ W= \begin{cases} V &\mbox{ if $l_s$ and $l_d$ are finite}\\ V_{0,N} &\mbox{ if $l_s=+\infty$ and $l_d$ is finite} \\ V_0 &\mbox{ if $l_d=+\infty$} \end{cases} $$ \begin{corollary} \label{coro4.9} Let $(\overline{u}_r)_r$ be a \eqref{Pr}-minimizing sequence. Thus $(\overline{u}_r)_r$ is relatively compact in $(X,\tau)$. Moreover, for every cluster point $\overline{u}$ and a subsequence, we have $$ \lim_{r\to0}I_r(\overline{u}_r)=I(\overline{u})=\inf\{I(v):v\in W \}. $$ \end{corollary} The proof of this corollary is a straightforward application of Remark \ref{rmk4.2}, propositions \ref{prop2.1} and \ref{prop4.3}. \begin{thebibliography}{00} \bibitem{a1} E. Acerbi, G. Buttazzo and D. Percivale, {\em Thin inclusions in linear elasticity: a variational approach}, J. Reine Angew. Math., 386, P. 99-115 (1988). \bibitem{a2} H. Attouch, {\em Variational Convergence for Functions and Operators}. 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