\documentclass[11pt,a4paper]{article} \pagestyle{plain} \usepackage[dvips]{graphicx} \usepackage{amsfonts} \usepackage{latexsym} \addtolength{\hoffset}{-1.5cm} \addtolength{\voffset}{-1.5cm} \addtolength{\textheight}{+2cm} \addtolength{\textwidth}{+2.5cm} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABBREVIAZIONI E MACRO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NUMERAZIONE DEI TEOREMI \newtheorem{teor}{Theorem}[section] \newtheorem{prop}[teor]{Proposition} \newtheorem{corol}[teor]{Corollary} \newtheorem{lemma}[teor]{Lemma} \newtheorem{defin}[teor]{Definition} \newtheorem{rem}[teor]{Remark} \newtheorem{example}[teor]{Example} \renewcommand{\theequation}{\thesection.\arabic{equation}} \makeatletter \@addtoreset{equation}{section} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABBREVIAZIONI MATEMATICHE \def\interior{\mathaccent'27} \renewcommand{\epsilon}{\varepsilon} \newcommand{\beeq}{\begin{equation}} \newcommand{\eneq}{\end{equation}} \newcommand{\bear}{\begin{array}} \newcommand{\enar}{\end{array}} \newcommand{\bema}{\begin{displaymath}} \newcommand{\enma}{\end{displaymath}} \newcommand{\beca}{\bepa \bear{ll}} \newcommand{\enca}{\enar \enpa} \newcommand{\beea}{\begin{eqnarray}} \newcommand{\enea}{\end{eqnarray}} \newcommand{\dist}{\displaystyle} \newcommand{\bepa}{\left\{ } \newcommand{\enpa}{\right.} \newcommand{\om}{\Omega} \newcommand{\gam}{\Gamma} \newcommand{\sob}[2]{W^{#1,#2}(\om)} \newcommand{\lp}[1]{L^{#1}(\om)} \newcommand{\boh}{ \mbox{\textbf{??}} } \newcommand{\R}{\mathbf{R}} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\N}{\mathbf{N}} \newcommand{\dx}{\, dx} \newcommand{\dy}{\, dy} \newcommand{\eps}{\varepsilon} \newcommand{\epsj}{\eps_j} \newcommand{\e}{_{\eps}} \newcommand{\ej}{_{\eps_j}} \newcommand{\nona}{\nonumber} \newcommand{\weak}{\rightharpoonup} \newcommand{\trattoa}{\rule[3pt]{3.8pt}{.5pt}} \newcommand{\trattob}{\rule[3pt]{7.5pt}{1pt}} \newcommand{\mint}{\int \!\!\!\! \trattoa} \newcommand{\mmint}{\int \!\!\!\!\!\! \trattob} \newcommand{\bigA}{\mbox{\rm {\textsf{\large A}}}} \newcommand{\A}{\mbox{\rm {\textsf{\scriptsize A}}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SPECIFICHE \newcommand{\M}{\mathbf{M}} \newcommand{\W}{W} \newcommand{\Orth}{\mathbf{Orth}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABBREVIAZIONI DI TESTO \newcommand{\noin}{\noindent} \newcommand{\spa}[1]{ \vspace{#1mm} } \def \qed{ \hfill \rule{6pt}{6pt} \medskip} \newcommand{\proof}{\noindent \textbf{Proof. }} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LABELS \newcommand{\lab}[1]{ \label{#1} } \newcommand{\name}[1]{ \label{#1} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % SECTIONS % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\bf {\Large Linearized elasticity as $\Gamma\mbox{-limit}$ of finite elasticity}} \vspace{1cm} {\small G. Dal Maso \vspace{2pt} M. Negri \vspace{6pt} SISSA - International School for Advanced Studies - Via Beirut 2 - 34013 Trieste - Italy \vspace{11pt} D. Percivale \vspace{6pt} Universit\`a degli studi di Genova - Dipartimento di Metodi e Modelli Matematici \mbox{Piazzale Kennedy - 16129 Genova - Italy} } \end{center} \vspace{1cm} \noin {\small {\bf Abstract.} Linearized elastic energies are derived from rescaled non-linear energies by means of $\Gamma\mbox{-convergence}$. For Dirichlet and mixed boundary value problems in a Lipschitz domain $\om$, the convergence of minimizers takes place in the weak topology of $H^1(\om,\R^n)$ and in the strong topology of $W^{1,q}(\om,\R^n)$ for $1\leq q <2$. \vspace{2pt} \noin {\bf Key words:} linearized elasticity, $\Gamma\mbox{-convergence}$. } \vspace{1cm} \section{Introduction} \name{intro} The stored energy of a hyperelastic material can be written in terms of the deformation gradient $\nabla v$ as \beeq \lab{enelgen} \int_{\om} W(x, \nabla v) \dx\,, \eneq where $\om\subset\R^n$ is the reference configuration, and the energy density $W(x,F)$ is a function defined for $x\in\om$ and $F$ in the space $\M^{n \times n}$ of ${n {\times} n}$ matrices. The stress tensor corresponding to the deformation gradient $\nabla v$ is then given by $T(x,\nabla v)=\partial_F W(x ,\nabla v)$. By frame indifference we can express $W(x, \nabla v)$ in terms of the right Cauchy-Green strain tensor $C(v):= \nabla v^T \nabla v$ or, equivalently, in terms of the Green-St.Venant tensor $\frac{1}{2}(C(v) -I)$, where $I$ is the identity matrix. Thus we can write $W(x,\nabla v)=V(x,\frac{1}{2}({C(v) -I}))$ for a suitable function $V(x,E)$ defined for $x\in\om$ and $E$ in the space $\M^{n \times n}_{sym}$ of symmetric ${n {\times} n}$ matrices. We prefer to express these quantities in terms of the displacement $u$, defined by $u(x):=v(x)-x$. As $\nabla v = I+\nabla u$ the Green-St.Venant tensor $\frac{1}{2}( C(v) -I)$ can be written as $E(u):=e(u) + \frac{1}{2} C(u)$, where $e(u):=\frac{1}{2}( \nabla u^T + \nabla u)$ is the symmetric part of the displacement gradient. We assume that the reference configuration is stress free, i.e., $T(x,I)=0$, and thus $\partial_FW(x,I)=\partial_EV(x,0)=0$. As $W(x,\cdot)$ and $V(x,\cdot)$ are defined up to an additive constant, it is not restrictive to assume also that $W(x,0)=V(x,0)=0$. Since the displacement $u=0$ is an equilibrium configuration when no external loads are applied, it is natural to expect small displacements for small external loads. It is then convenient to rescale the variables and to write the load as $\eps \ell $ and the displacement as $\eps u$ for a suitable (adimensional) small parameter $\eps>0$. Thus we have $v(x)=x+\eps u(x)$, and the equilibrium configurations are stationary points of the total energy \beeq \lab{energyf} \int_{\om} W(x, I+\eps \nabla u) \dx - \eps^2\int_\om \ell u\dx \,. \eneq As \beeq W(x, I+\eps \nabla u)= V(x,\eps e(u) + {\textstyle\frac{1}{2}} \eps^2 C(u))\,, \eneq if $\nabla u$ is bounded we have, by Taylor expansion, \beeq W(x, I+\eps \nabla u)= \eps^2 {\textstyle\frac{1}{2}} \partial_E^2 V(x,0) [ e(u) , e(u) ] + o( \eps^2) \, , \eneq where $\partial_E^2 V(x,\cdot)$ denotes the second derivative of $V(x,\cdot)$ on $\M^{n \times n}_{sym}$, and $o( \eps^2)$ is uniform with respect to~$x$. The tensor $\mbox{\textsf{A}}(x):= \partial_E^2 V(x,0)$ is called the elasticity tensor, and the linearized elastic energy is then defined as \bema \frac{1}{2}\int_{\om} \mbox{\textsf{A}}(x) [ e(u) , e(u) ] \dx \, . \enma The previous discussion shows that, if we rescale the total energy given by (\ref{energyf}), we obtain \beeq \label{pointlim} \lim_{\eps\to0} \frac{1}{\eps^2}\Big( \int_{\om} W(x, I+\eps \nabla u) \dx - \eps^2\int_\om \ell u\dx \Big) = \frac{1}{2} \int_{\om} \mbox{\textsf{A}}(x) [ e(u) , e(u) ] \dx - \int_\om \ell u\dx \eneq for every Lipschitz function $u$. This equality is usually considered as the main justification of linearized elasticity. Note that this argument does not prove that the minimizers $u_\eps$ of (\ref{energyf}), satisfying suitable boundary conditions, actually converge to the minimizer of the corresponding limit problem \bema \frac{1}{2} \int_{\om} \mbox{\textsf{A}}(x) [ e(u) , e(u) ] \dx - \int_\om \ell u\dx\,. \enma Indeed we shall see that this is not always true (see Example \ref{esempio}). In this paper, given a load $ \ell \in L^2(\om,\R^n)$, a boundary value $g\in W^{1,\infty}(\om,\R^n)$, and a closed subset $\partial \om_D$ of $\partial \om$ with ${\cal H}^{n-1} (\partial \om_D)>0$, we consider the minimum problems \beeq \lab{mineps} \min_{u \in H^1_{g,\partial \om_D} } \Big\{ \int_{\om} W(x, I+\eps \nabla u) \dx - \eps^2\int_\om \ell u\dx \Big\}\,, \eneq where $H^1_{g,\partial \om_D}$ denotes the closure in $H^1(\om,\R^n)$ of the space of functions $u \in W^{1,\infty}(\om,\R^n)$ such that $u=g$ on $\partial \om_D$. Suppose that, for every $\eps>0$, there exists a solution $u_\eps$ of (\ref{mineps}) which satisfies the orientation preserving condition $\det(I+\eps\nabla u_\eps)>0$. Under some natural hypotheses on the function $V$, we prove that $u_\eps$ converges weakly in $H^1(\om,\R^n)$ to the (unique) minimizer $u_0$ of the problem \bema \min_{u \in H^1_{g,\partial \om_D}} \Big\{ \frac{1}{2} \int_{\om} \mbox{\textsf{A}}(x) [ e(u) , e(u) ] \dx - \int_\om \ell u\dx \Big\}\,. \enma Moreover we prove the convergence of the rescaled energies, i.e., \beeq \label{minenerg} \lim_{\eps\to 0} \frac{1}{\eps^2} \Big\{ \int_{\om} W(x, I+\eps \nabla u_\eps) \dx - \eps^2\int_\om \ell u_\eps\dx \Big\} = \frac{1}{2} \int_{\om} \mbox{\textsf{A}}(x) [ e(u_0) , e(u_0) ] \dx - \int_\om \ell u_0\dx \,. \eneq More generally, the same results hold if $\det(I+\eps\nabla u_\eps)>0$ and \bema \int_{\om} W(x, I+\eps \nabla u_\eps) \dx - \eps^2\int_\om \ell u_\eps\dx= {\cal J}_\eps + o(\eps^2)\,, \enma where ${\cal J}_\eps$ is the (possibly not attained) infimum of problem (\ref{mineps}). This provides a full variational justification of linearized elasticity. These results are proved under the following additional hypotheses on $V$: \begin{itemize} \item[(a)] $\displaystyle \inf_{|E|\ge\rho}\,\, \inf_{x\in\om} \,V(x,E)>0$ for every $\rho>0$; \item[(b)] there exist $\alpha>0$ and $\rho>0$ such that $\displaystyle \inf_{x\in\om} V(x,E)\ge\alpha |E|^2$ for every $|E|\le\rho$; \item[(c)] $\displaystyle \liminf_{|E|\to +\infty}\, \frac{1}{|E|} \, \inf_{x\in\om} V(x,E)>0$. \end{itemize} These conditions say that $0$ is the unique minimizer of $V(x,\cdot)$ (with a uniform estimate with respect to~$x$) and that $V(x,\cdot)$ grows more than quadratically near the origin and more than linearly at infinity. If (c) is replaced by the slightly stronger condition \begin{itemize} \item[(c')] $\displaystyle \liminf_{|E|\to +\infty}\, \frac{1}{|E|^p} \, \inf_{x\in\om} V(x,E)>0$ \end{itemize} for some exponent $p>1$, then we prove also that $u_\eps$ converges to $u_0$ strongly in $W^{1,q}(\om,\R^n)$ for every $q<2$. The proof is obtained in two steps. First we show that the sequence $u_\eps$ is compact in the weak topology of $H^1(\om,\R^n)$, using a recent lemma proved by Friesecke, James and M\"uller \cite{FJM}. Then we prove that the functionals \bema {\cal F}\e(u) = \frac{1}{\eps^2} \int_{\om} V(x,\eps e(u) + {\textstyle \frac{1}{2}} \eps^2 C(u)) \dx \enma $\Gamma\mbox{-converge}$ to the functional \bema {\cal F}(u)= \frac{1}{2} \int_{\om}\mbox{\textsf{A}}(x) [e(u) , e(u) ] \dx \, . \enma These two facts lead to the weak convergence of the solutions in $H^1(\om,\R^n)$ and to the convergence of the rescaled energies expressed by (\ref{minenerg}). The strong convergence in $W^{1,q}(\om,\R^n)$ for $q<2$ is obtained from (\ref{minenerg}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The main results} \name{convres} Let the reference configuration be an open, bounded, connected domain $\om \subset \R^n$, for $n \geq 2$, having Lipschitz boundary. Let $|F|^2 = \sum_{i,j} |F_{ij}|^2$ be the norm in the space $\M^{n \times n}$ and let $SO(n)$ be the subset of rotations (orthogonal matrices with positive determinant). We will assume that the material is hyperelastic, i.e., there exists a stored energy density $\W : \om \times \M^{n \times n} \to [0, +\infty ]$ such that for a.e. $x \in \om$ we have \beeq \lab{incom} W(x,F)=+\infty \quad \mbox{if $\det F \leq 0$} \eneq (orientation preserving condition), and such that for a.e. $x \in \om$ \beeq \lab{finite} \W(x,F) < +\infty \eneq for $F$ in a neighborhood $U$ of the identity $I$ independent of $x$ (so that small deformation of the reference configuration have finite energy). By frame indifference the stored energy density can be written as \beeq \lab{WhatZ} \W(x,F)=V(x,\textstyle{\frac{1}{2}}( F^T F - I) ) \, . \eneq where $F^T$ denotes the transpose of the matrix $F$. We suppose that $V : \om \times \M^{n \times n}_{sym} \to \R$ is ${\cal L}^n \times {\cal B}^n \mbox{-measurable}$ (where ${\cal L}^n$ and ${\cal B}^n$ are the $\sigma\mbox{-algebras}$ of Lebesgue measurable and Borel measurable subsets of $\R^n$) and that, for some $\delta>0$, the function $B \to V(x,B)$ is of class $C^2$ for $|B| < \delta$ and for a.e. $x \in \om$. Moreover we will assume that the reference configuration has zero energy and is stress free, which means that for a.e. $x \in \om$ \beeq \lab{Z1} V(x,0)=0 \qquad \partial_E V(x,0)=0 \, . \eneq Finally we require the coercivity assumptions (a), (b), (c) and for a.e. $x \in \om$ the upper bound \beeq \lab{Velliptic} |\partial_E^2 V(x,E) [T,T]| \leq 2 \gamma |T|^2 \quad \mbox{for $|E| < \delta$ and $T \in \M^{n \times n}_{sym}$ } \, , \eneq for some constant $\gamma >0$ independent of $x$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {}From (\ref{Z1}) it is easy to deduce by Taylor expansion that for a.e. $x \in \om$ \beeq \lab{12'} V(x,E)=\frac{1}{2} \partial_E^2 V(x, t E) [E , E] \eneq for some $t \in (0,1)$ depending on $x$, hence \beeq \lab{Vupbound} | V(x,E)| \leq \gamma |E|^2 \quad \forall E \in \M^{n \times n}_{sym} \mbox{ with $|E| < \delta$ . } \eneq Let $\bigA(x):=\partial_E^2 V(x,0)$. {}From (\ref{12'}) and (b) it follows that for a.e. $x \in \om$ \beeq \lab{Z3} \bigA(x)[E,E]= \partial_E^2 V (x, 0) \, [E , E] \geq 2 \alpha |E|^2 \quad \forall E \in \M^{n \times n}_{sym} \, . \eneq Finally for every $x \in \om$ and $F \in \M^{n \times n}$ let $ F_{sym} = (F+F^T)/2 $ and \beeq \lab{Weps} W_{\eps} (x,F) := \frac{1}{\eps^2} \, \W(x,I+\eps F) = \frac{1}{\eps^2} \, V(x,\eps \, F_{sym} + \textstyle{\frac{1}{2}}\eps^2 F^T F) \, . \eneq It is easy to see that for a.e. $x \in \om$ \beea \label{pointconv} \lim_{\eps \to 0} W\e(x,F) & = & \frac{1}{2} \partial_F^2 W(x,I) [ F , F ] \nona \\ & = & \frac{1}{2} \partial_E^2 V(x,0) [ F_{sym} , F_{sym} ] = \frac12 \bigA(x) [F_{sym},F_{sym}] \, . \enea We consider the functional ${\cal F}_{\eps} : H^1(\om,\R^n) \to [0,+\infty]$ defined as \beeq \lab{Feps} {\cal F}_{\epsilon}(u)= \int_{\om} W_{\eps}(x,\nabla u) \, dx \, , \eneq and the functional ${\cal F} : H^1(\om,\R^n) \to [0,+\infty) $ given by \beeq \lab{Flim} {\cal F} (u) = \frac{1}{2} \int_{\om} \bigA(x) \, [ e(u) , e(u) ] \dx \, . \eneq Let $\partial \om_D$ a closed subset of $\partial \om$ with ${\cal H}^{n-1}(\partial \om_D)>0$ and let $g \in W^{1,\infty}(\om,\R^n)$. Let $H^1_{g,\partial \om_D}$ be the closure in $H^1(\om,\R^n)$ of the space of functions $u \in W^{1,\infty}(\om,\R^n)$ such that $u=g$ on $\partial \om_D$. By strong (resp. weak) topology in $H^1_{g,\partial \om_D}$ we mean the restriction of the strong (resp. weak) topology of $H^1(\om,\R^n)$. Let ${\cal L}: H^1(\om,\R^n) \to \R$ be a continuous linear operator, representing the work of the (rescaled) loads. We define the functionals ${\cal G}\e , {\cal G} : H^1_{g,\om_D} \to [0,+\infty]$ as ${\cal G}_{\eps}(u)={\cal F}_{\eps}(u)- {\cal L}(u)$ and ${\cal G}(u)={\cal F}(u)-{\cal L}(u)$. The main convergence results, proved in Section \ref{strconv}, are the following. \begin{teor} \lab{convmin} Assume that $V: \om \times \M^{n \times n}_{sym} \to [0,+\infty]$ satisfies conditions (a), (b), (c), (\ref{incom}), (\ref{Z1}), and (\ref{Velliptic}). %\beeq \lab{inftomin} % \inf_{u \in H^1_{g,\om_D}} {\cal G}\e(u) % \longrightarrow % \min_{u \in H^1_{g,\om_D}} {\cal G} (u) \, . %\eneq If $u\e$ satisfies \beeq \lab{o1} {\cal G} (u\e) = \inf_{u \in H^1_{g,\partial \om_D}} {\cal G}\e (u) + o(1) \, \eneq then $u\e$ converges weakly to the (unique) solution $u_0$ of \bema \min_{u \in H^1_{g,\partial \om_D}} {\cal G} (u) \, . \enma \end{teor} \begin{teor} \lab{strongconvmin} Under the hypotheses of the previous theorem, if condition (c') is satisfied then $u\e$ converges to $u_0$ strongly in $W^{1,q} (\om,\R^n)$ for $1\leq q <2$. \end{teor} The proof follows basically from the following results, contained in Section \ref{compact} and \ref{gammaconvergence} respectively. \begin{prop} \name{minima} If $\epsj \to 0$ and $u_{\eps_j} \in H^1_{g,\partial \om_D}$ is a sequence such that \bema {\cal G}_{\eps_j}(u_{\eps_j}) \leq C < +\infty \, , \enma then $u_{\eps_j}$ is equibounded in $H^1 (\om,\R^n)$. \end{prop} \begin{prop} \name{convload} Let $\epsj \to 0$. The functionals ${\cal G}_{\epsj}$ $\Gamma\mbox{-converge}$ to ${\cal G}$ in the weak topology of $H^1_{g,\partial \om_D}$. \end{prop} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \section{Compactness} \name{compact} {}From conditions (a), (b) and (c) it follows easily that there exists a non-decreasing, continuous function $\phi(t)$, of the form \bema \phi(t) = \beca \alpha t^2 & \mbox{ for $0\leq t \leq c$} \\ \alpha c^2 & \mbox{ for $c \leq t \leq d$} \\ (\alpha c^2 d^{-1}) t & \mbox{ for $d \leq t$ ,} \enca \enma such that $\phi(|E|) \leq V(x,E)$ for a.e.\ $x \in \om$ and every $E \in \M^{n \times n}_{sym}$. For a positive $\beta$ let $\psi(t)$ be the function defined as \beeq \lab{psi} \psi(t) = \bepa \bear{ll} \alpha t^2 & \mbox{ for $0\leq t \leq \beta$} \, , \\ (2 \alpha \beta) t - (\alpha \beta^2) & \mbox{ for $t \geq \beta$} \, . \enar \enpa \eneq It is easy to check that $\psi(t)$ is increasing, $C^1$, and convex. Moreover, since \bema \lim_{\beta \to 0} 2 \alpha \beta = 0 \, , \enma for $\beta$ sufficiently small we have $\psi(t) \leq \phi(t)$ and then \beeq \lab{Z2} V(x,E) \geq \psi( |E| ) \eneq for a.e. $x \in \om$ and every $E \in \M^{n \times n}_{sym}$. \begin{lemma} \name{rotation} Let $\eps >0$ and $u_{\eps} \in H^1(\om,\R^n)$. Denote the rescaled deformation $x+\eps u\e(x)$ by $v\e(x)$. Then there exists a function $R\e : \om \to SO(n)$ such that \beeq \lab{rotineq} \int_{\om} | \nabla v\e -R\e |^2 \dx \leq C \eps^2 {\cal F}\e(u\e) \,, \eneq where $C$ depends only on the function $\psi$ (in particular it does not depend on $\eps$ or $v\e$). \end{lemma} \proof We may assume ${\cal F}_\eps(u_\eps)<+\infty$, so that $\det \nabla v_\eps >0$ a.e.\ in $\om$ by (\ref{incom}). Considering that \beeq \lab{Wbound} {\cal F}_{\eps}(u_{\eps}) = \int_{\om} W_{\eps}(x,\nabla u_{\eps}) \dx = \frac{1}{\eps^2} \int_{\om} V(x, \textstyle{\frac{1}{2}} (\nabla v_{\eps}^T \nabla v_{\eps}-I) ) \dx \eneq and using (\ref{Z2}) we get \beeq \lab{23'} \int_{\om} \psi({\textstyle\frac{1}{2}} |\nabla v_{\eps}^T \nabla v_{\eps} -I|) \dx \leq \int_{\om} V(x, \textstyle{\frac{1}{2}} ( \nabla v_{\eps}^T \nabla v_{\eps}-I ) ) \leq \eps^2 {\cal F}_{\eps}(u_{\eps}) \, . \eneq As $\det \nabla v\e >0$ a.e.\ in $\om$ by polar decomposition (see for instance \cite{Ciarl}) for a.e.\ $x \in \om$ there exists a rotation $R\e$ and a symmetric positive definite matrix $U_{\eps}$ such that $\nabla v_{\eps} = R_{\eps} U_{\eps}$. In particular $\nabla v_{\eps}^T \nabla v_{\eps}=U_{\eps}^2$, hence \beeq \lab{impequal} |\nabla v_{\eps}^T \nabla v_{\eps}-I|=|U_{\eps}^2 - I| \, . \eneq Since $U_{\eps}$ is symmetric and positive definite, using an orthonormal basis in which $U\e$ is diagonal, we can prove that \bema |U\e -I| \leq |U\e^2 -I| \, . \enma Thus, by the definition of $\psi$, it follows that for $\textstyle{\frac{1}{2}}|U_{\eps}^2-I| \leq \beta$ \bema \textstyle{\frac{\alpha}{4}} |U_{\eps} -I|^2 = \psi(\textstyle{\frac{1}{2}}|U\e^2 -I|) \, . \enma Moreover for a suitable constant $c_1$, depending on $\beta$, \bema c_1 |U\e -I|^2 \leq |U\e^2 -I| \qquad \mbox{for } \textstyle{\frac{1}{2}}|U_{\eps}^2-I| \geq \beta \, . \enma Indeed, using again the diagonal form, we can write \beea \lefteqn{ \sum_{i=1}^n (\lambda_i -1)^2 \leq \sum_{i=1}^n \lambda_i^2 + n = } \nona \\ & & = \sum_{i=1}^n (\lambda_i^2 -1) + \frac{2n}{\beta} \beta \leq \left( 1+ \frac{n}{\beta} \right) \sum_{i=1}^n | \lambda_i^2 -1 | \, . \nona \enea Moreover there is a constant $c_2$ such that $ 2 c_2 t \leq \psi(t) $ for $t \geq \beta$, hence for $\textstyle{\frac{1}{2}}|U_{\eps}^2-I| \geq \beta$ \bema c_1 c_2 |U\e -I|^2 \leq c_2 |U_{\eps}^2 -I| \leq \psi(\textstyle{\frac{1}{2}}|U\e^2 -I|) \, . \enma By this inequality and by (\ref{23'}) and (\ref{impequal}) there exists a constant $c_3$, depending only on $\psi$, such that \bema \int_{\om} |U_{\eps} -I|^2 \dx \leq c_3 \eps^2 {\cal F}\e(u\e) \, . \enma Finally considering that for a.e. $x \in \om$ we have $\nabla v_{\eps} = R_{\eps} U_{\eps}$ we can write \bema \int_{\om} |\nabla v_{\eps} - R_{\eps}|^2 \dx = \int_{\om} |U_{\eps} -I|^2 \dx \leq c_3 \eps^2 {\cal F}\e(u\e) \, . \enma which is the required estimate. \qed The following Lemma (for which we refer to \cite{FJM}) will be crucial in our proof. \begin{lemma} \lab{FJMlemma} Let $\Omega\subset \R^n$ be an open bounded set with Lipschitz boundary. There exists a constant $C$ such that for every $v \in H^1(\Omega, \R^n)$ there exists a constant rotation $R \in SO(n)$ such that \beeq \int_{\om} | \nabla v(x) - R |^2 \dx \leq C \int_{\om} {\rm dist}(\nabla v(x), SO(n) )^2 \dx \, , \eneq where ${\rm dist}( F, SO(n) )$ denotes the distance from the matrix $F$ to the set $SO(n)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LEMMINO Moreover we will need the following result. \begin{lemma} \lab{geometry} Let $S \subset \R^n$ be a bounded ${\cal H}^n\mbox{-measurable}$ set with $0 < {\cal H}^m(S) < +\infty$, for some $m>0$. Then \bema |F|_{S} : = \Big( \min_{\zeta \in \R^n} \int_{S} | F x - \zeta|^2 d{\cal H}^{m}(x) \Big)^{\frac{1}{2}} \enma is a seminorm on $\M^{n \times n}$. Let $S_0$ be the set of points $x \in S$ such that ${\cal H}^m(S \cap B_\rho(x)) > 0$, and let $\mathrm{aff}(S_0)$ be the smallest affine space containing $S_0$. Let $\mathbf{K} \subset \M^{n \times n}$ be a closed cone such that for every $F \in \mathbf{K}$ with $F \neq 0$ \beeq \lab{kerdim} \mathrm{dim}(\mathrm{ker}(F)) < \mathrm{dim}(\mathrm{aff}(S_0)) \, . \eneq Then there exists a constant $C>0$ such that \beeq \lab{equiv} C |F| \leq |F|_{S} \eneq for every $F \in \mathbf{K}$. \end{lemma} \proof It is not difficult to check that $|F|_{S}$ is a seminorm and the minimum is attained for $\zeta = \mint{}_S F x \, d{\cal H}^m$. We will prove (\ref{equiv}) by contradiction. Suppose that for every integer $k$ it is possible to find a matrix $F_k \in {\mathbf K}$ with $|F_k|=1$ such that \beeq \lab{zero} \frac1k = \frac1k |F_k|^2 > \ \int_{S} |F_k x - \zeta_k|^2 d{\cal H}^{m} \geq 0 \, , \eneq with $\zeta_k := \mint{}_S F_k x \, d{\cal H}^m$. It is not restrictive to assume that $F_k$ converges to $F \in {\mathbf K}$, with $|F|=1$. Then by (\ref{zero}) and by continuity it follows that \bema \int_{S} |F x - \zeta |^2 d{\cal H}^{m} = 0 \, . \enma for $\zeta = \mint{}_S Fx \, d{\cal H}^m$. Then $Fx = \zeta$ for ${\cal H}^m\mbox{-a.e.} \, x \in S$ and hence for every $x \in S_0$. By continuity and linearity $Fx = \zeta$ for every $x \in \mathrm{aff}(S_0)$. Then $\mathrm{dim}(\mathrm{ker}(F)) \geq \mathrm{dim}(\mathrm{aff}(S_0))$ and thus, by (\ref{kerdim}), $F=0$. This is clearly impossible because $|F|=1$. \qed Now we are ready to prove the following compactness result. \begin{prop} \lab{comp1} Let $u\e$ be a sequence in $H^1_{g,\partial \om_D}$. Then \beeq \lab{compineq} \int_{\om} | \nabla u\e |^2 \dx \leq C {\cal F}\e(u\e) + C \int_{\partial \om_D} |g|^2 \, d{\cal H}^{n-1} \eneq where $C$ depends only on $\psi$, $\om$, and $\partial \om_D$. \end{prop} \proof By Lemma \ref{rotation} we have \bema \int_{\om} {\rm dist}(\nabla v\e(x), SO(n) )^2 \dx \leq C \eps^2 {\cal F}\e(u\e) \, \enma and by Lemma \ref{FJMlemma} there exists a constant rotation $R\e$ such that \beeq \lab{est1} \int_{\om} | \nabla v\e(x) -R\e |^2 \dx \leq C \eps^2 {\cal F}\e(u\e) \, . \eneq If $\zeta\e = \mint_{\om} (v\e(x) - R\e x) \dx$, then by the Poincar\'e inequality \bema \| v\e(x) - R\e x - \zeta\e \|_{H^1(\om,\R^n)}^2 \leq C \int_{\om } |\nabla v\e(x) - R\e |^2 \dx \leq C \eps^2 {\cal F}\e(u\e) \, . \enma Moreover by the continuity of the traces \bema \int_{\partial \om_D} | v\e(x) - R\e x- \zeta\e |^2 d{\cal H}^{n-1} \leq C \| v\e(x) - R\e x -\zeta\e \|_{H^1(\om,\R^n)}^2 \leq C \eps^2 {\cal F}\e(u\e) \, . \enma Considering that on $\partial \Omega_D$ we have $v_{\epsilon}(x) = x + \epsilon g (x)$ we can write \beeq \lab{30'} \int_{\partial \Omega_D} | x - R_{\epsilon} x - \zeta \e |^2 d{\cal H}^{n-1} \leq C \epsilon^2 {\cal F}\e(u\e) + C \epsilon^2 \int_{\partial \Omega_D} |g|^2 \, d{\cal H}^{n-1}\, . \eneq Let $\mathbf{K}$ be the closed cone generated by $SO(n)-I$, which is the union of the cone generated by $SO(n)-I$ and of the space of antisymmetric matrices. Therefore, $\mathrm{dim}(\mathrm{ker}(F)) < n-1$ if $F \in \mathbf{K}$ and $F \neq 0$. Let $S:=\partial \om_D$. As $S$ is contained in the Lipschitz manifold $\partial \om$ and ${\cal H}^{n-1}(S) >0$, we have ${\cal H}^{n-1}(S_0) >0$. This implies that $\mathrm{dim}(\mathrm{aff}(S_0)) \geq n-1$ and thus condition (\ref{kerdim}) is satisfied. Using Lemma \ref{geometry} and the previous inequality we obtain \bema | I - R_{\epsilon} |^2 \leq C | I - R\e|_{S}^2 = C \int_{\partial \Omega_D} | x - R_{\epsilon} x - \zeta\e |^2 d{\cal H}^{n-1} \enma and thus by (\ref{30'}) \beeq \lab{est2} \int_{\Omega} | I - R_{\epsilon} |^2 \dx \leq C \eps^2 {\cal F}\e(u\e) + C \epsilon^2 \int_{\partial \Omega_D} |g|^2 \, d{\cal H}^{n-1} \, . \eneq By (\ref{est1}) and (\ref{est2}) we have easily \bema \int_{\Omega} | \nabla v_{\epsilon} - I |^2 \dx \leq C \eps^2 {\cal F}\e(u\e) + C \epsilon^2 \int_{\partial \Omega_D} |g|^2 \, d{\cal H}^{n-1} \, . \enma Substituting $\nabla v\e = I +\eps \nabla u\e$ in the previous inequality we get (\ref{compineq}). \qed \noin \textbf{Proof of Proposition \ref{minima}.} Using Proposition \ref{comp1} we have \bema \int_{\om} | \nabla u\ej|^2 \dx \leq C {\cal F}\ej(u\ej) + C \int_{\partial \Omega_D} |g|^2 \, d{\cal H}^{n-1} \, . \enma Hence we can write \bema \int_{\om} | \nabla u\ej|^2 \dx \leq C \big( {\cal G}\ej(u\ej) + {\cal L}(u\ej) + 1 \big) \, , \enma and by the Poincar\'e and the Holder inequality it follows that \bema \| u\ej \|^2_{H^1(\om,\R^n)} \leq C + C \| u\ej \|_{H^1(\om,\R^n)} \, , \enma which gives the boundedness of $u\ej$ in $H^1(\om,\R^n)$. \qed Finally we remark that for $n=2$ and for a sequence $u\ej \in H^1_0 (\om,\R^2)$ we can prove the compactness result in a more elementary way without using Lemma \ref{FJMlemma}. Indeed for every $\eps_j$ let $R\ej \colon \om \to SO(2)$ be given by Lemma \ref{rotation}. Define $M\ej= (R\ej - I)/\epsj$. Then, substituting $\nabla v\ej = I +\eps_j \nabla u\ej$ in (\ref{rotineq}), we get \bema \int_{\om} |\nabla u_{\eps_j} - M_{\eps_j}|^2 \dx \leq C_1 {\cal F}\ej(u\ej) \, . \enma Note that $M\ej$ has the form \bema M\ej = \left( \bear{cc} a\ej & -b\ej \\ b\ej & a\ej \enar \right) \enma for some real functions $a\ej$ and $b\ej$. Denote the components of $u$ by $u^i$. By a linear combination we obtain \bema \int_{\om} \big| \nabla _1 u^1\ej - \nabla _2 u^2\ej \big|^2 \dx = \int_{\om} \big| ( \nabla_1 u^1\ej - a\ej ) - ( \nabla_2 u^2\ej - a\ej ) \big|^2 \dx \leq C_2 {\cal F}\ej(u\ej) \, , \enma \bema \int_{\om} \big| \nabla_2 u^1\ej + \nabla_1 u^2\ej \big|^2 \dx \int_{\om} \big| ( \nabla_2 u^1\ej + b\ej ) + ( \nabla_1 u^2\ej - b\ej ) \big|^2 \dx \leq C_3 {\cal F}\ej(u\ej) \, . \enma Moreover, being $n=2$, we can write \bema \int_{\om} | \nabla u\ej|^2 \dx = \int_{\om} | \nabla_1 u\ej^1 - \nabla_2 u\ej^2 | ^2 \dx + \int_{\om} | \nabla_2 u\ej^1 + \nabla_1 u\ej^2 |^2 \dx + 2 \int_{\om} \det \nabla u\ej \dx \, . \enma As $u\ej \in H^1_0 (\om,\R^2)$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\bema % \int_{\om} \det \nabla w \dx = C \int_{\partial \om} % \langle ( {\rm cof} \nabla w) \nu \, , \, w \rangle % d{\cal H}^{1} = 0 %\enma %for $w \in C^{\infty}_0(\om,\R^2)$ (see e.g. \cite{Ciarl}), %by a density argument it's easy to see that %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% we have (see e.g. \cite{Ciarl}) \bema \int_{\om} \det \nabla u\ej \dx = 0 \, . \enma Then by the previous inequalities we get \bema \int_{\om} | \nabla u\ej |^2 \dx = \int_{\om} | \nabla_1 u\ej^1 - \nabla_2 u\ej^2 |^2 \dx + \int_{\om} | \nabla_2 u\ej^1 + \nabla_1 u\ej^2 |^2 \dx \leq C {\cal F}\ej(u\ej) \enma and thus $u\ej$ is bounded in $H^1_0 (\om,\R^2)$. \vspace{6pt} The following example shows that, if other potential wells are present, with the same value of the energy, we might lose compactness of solutions. \begin{example} \lab{esempio} Let $\om=(-1,1) \times (-1,1)$, $\ell=1$ and $w \in H^1_0 (\om,\R^2)$ defined as $w^1(x_1 , x_2)=-\max \{ |x_1| \, , |x_2|\} +1$ and $w^2(x_1 , x_2)=0$. Let $\eps_j \to 0$, $w\ej(x)=w(x) / \eps_j$ and $v\ej(x)=x+\epsj w\ej(x)$. Then $\nabla v\ej=I+\eps_j \nabla w\ej=I+\nabla w$ does not depend on $\eps_j$ and takes only four values, denoted by $F_1 , \dots , F_4$. Let $E_i = \frac12 (F_i^T F_i -I)$, for $i=1,\dots,4$. Let $V$ be the function satisfying conditions (b) and (c) and such that $V(x,0)=V(x,E_i)=0$ for $i=1,\dots,4$. Then \beea \inf \{ {\cal G}\ej (u) \, : \, u \in H^1_0(\om) \} & \leq & \frac{1}{\eps_j^2} \int_{\om} V(x, {\textstyle \frac{1}{2} } (\nabla v\ej^T \nabla v\ej -I)) \, dx - \int_{\om} w\ej \dx \nona \\ & = & - \frac{1}{\epsj} \|w\|_{L^1(\om,\R^n)} \nona \, . \enea If $u\ej$ is a sequence satisfying (\ref{o1}) then \bema - \frac{1}{\epsj} \|w\|_{L^1(\om,\R^n)} + o(1) \geq {\cal G}\ej(u\ej) \geq - \, \| u\ej \|_{L^1(\om,\R^n)} \, , \enma hence $\| u\ej \|_{L^1(\om,\R^2)}$ diverges. \end{example} \section{$\Gamma\mbox{-convergence}$ } \name{gammaconvergence} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NOTATION For $x \in \om$ and $E \in \M^{n \times n}_{sym}$ let $|E|_{\A(x)}$ be the norm defined by \beeq \lab{Anorm} |E|_{\A(x)} = \Big\{ \frac12 \bigA(x) [E,E] \Big\} ^{\frac12} = \Big\{ \frac12 \partial^2_E V(x,0) [E,E] \Big\} ^{\frac12} \, . \nona \eneq Note that by (\ref{Velliptic}) and (\ref{Z3}) we have \beeq \lab{Anormequiv} \alpha |E|^2 \leq |E|^2_{\A(x)} \leq \gamma |E|^2 \, . \eneq If $\Phi : \om \to \M^{n \times n}_{sym}$ is a measurable map, the function $x \mapsto |\Phi(x)|_{\A(x)}$ is denoted by $|\Phi|_{\A}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EQUICOERCIVITY Let us fix a sequence $\epsj \to 0$. By Proposition \ref{minima} the functionals ${\cal G}_{\eps_j}$ are equicoercive in $H^1_{g,\partial \om_D}$ and by Proposition 8.10 in \cite{DalM} we can characterize the $\Gamma\mbox{-limit}$ in the weak topology of $H^1_{g,\partial \om_D}$ in terms of weakly converging sequences. In particular we have \bema {\cal G}'(u): = \Gamma\mbox{-}\liminf_{\epsj \to 0} {\cal G}\e(u) = \inf \{ \liminf_{j \to +\infty} {\cal G}_{\eps_j} ( u_{j} ) \, : \, \mbox{for $u_{j} \rightharpoonup u$ in $H^1_{g,\partial \om_D}$} \} \, , \enma \bema {\cal G}''(u): = \Gamma\mbox{-}\limsup_{\epsj \to 0} {\cal G}\e(u) = \inf \{ \limsup_{j \to +\infty} {\cal G}_{\eps_j} ( u_{j} ) \, : \, \mbox{for $u_{j} \rightharpoonup u$ in $H^1_{g,\partial \om_D}$} \} \, . \enma We will prove that for every function $u \in H^1_{g,\partial \om_D} $ we have ${\cal G}''(u) \leq {\cal G}(u) \leq {\cal G}'(u)$, from which Proposition \ref{convload} follows. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LIMSUP \begin{prop} \name{glimsup} For every $u \in H^1_{g,\partial \om_D}$ we have ${\cal G}''(u) \leq {\cal G}(u)$. \end{prop} \proof Consider first the case $u \in W^{1,\infty}(\om,\R^n)$. By (\ref{pointconv}) it follows that for a.e. $x \in \om$ \bema \lim_{\eps_j \to 0} W_{\eps_j} (x, \nabla u ) = \frac{1}{2} \bigA(x) \, [ e(u) , e(u) ] \, . \enma Using the upper bound (\ref{Vupbound}) we deduce that $V_{\eps_j}(x,\nabla u)$ is equi-bounded in $L^{\infty}(\om)$. Then taking the sequence $u_{\eps_j} = u$, by dominated convergence it follows that \beeq \lab{lsup1} \limsup_{\eps_j \to 0} {\cal G}_{\eps_j} ( u_{\eps_j} ) = \lim_{\eps_j \to 0} \int_{\om} V_{\eps_j} (x, \nabla u) \dx - {\cal L}(u) = \frac{1}{2} \int_{\om} \bigA(x) \, [ e(u) , e(u) ] \dx - {\cal L}(u) \, . \eneq If $u \notin W^{1,\infty} (\om,\R^n)$ by the definition of $H^1_{g,\partial \om_D}$ there exists a sequence $u_k$ in $W^{1,\infty}(\om,\R^n)$, which satisfy the boundary condition $u_k=g$ on $\partial \om_D$ and converge to $u$ strongly in $H^1 (\om,\R^n)$. Since, by (\ref{lsup1}), ${\cal G}''(u_k) \leq {\cal G}(u_k)$, the lower semicontinuity of the $\Gamma\mbox{-limsup}$ and the continuity of ${\cal G}$ respect to strong convergence imply that \bema {\cal G}''(u) \leq \liminf_{k \to \infty} {\cal G}''(u_k) \leq \liminf_{k \to \infty} {\cal G}(u_k) = {\cal G}(u) \enma and the proof is concluded. \qed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LEMMA \begin{lemma} \name{convexapprox} Let $\eps_j \to 0$ be a decreasing sequence. For every $k \in \N$ there exist an increasing sequence of Caratheodory functions $V^k_j : \om \times \M^{n \times n}_{sym} \to [0, +\infty)$ and a measurable function $\mu^k:\om \to (0,+\infty)$ such that $V^k_j(x,\cdot)$ is convex for a.e. $x \in \om$ and satisfies \beeq \lab{riscal} V^k_j (x,E) \leq V(x,\eps_j E) / \eps_j^2 \qquad \forall E \in \M^{n \times n}_{sym} \, , \eneq \beeq \lab{identica} V_j^k (x,E) = \Big( 1-\frac{1}{k} \Big) \, |E|^2_{\A(x)} \qquad \mbox{for $|E|_{\A(x)} \leq \mu^k(x) / \eps_j$ }. \eneq \end{lemma} \noin \textbf{Proof.} By Taylor's formula, from (\ref{Z1}) and (\ref{Anormequiv}) it follows that for a.e.\ $x\in \om$ and every $k \in \N$ there exists $r^k(x)>0$ such that \beeq \lab{less1} \Big( 1-\frac{1}{k} \Big) |E|_{\A(x)}^2 \leq V(x,E) \quad \mbox{for $ |E|_{\A(x)} \leq r^k(x) $ .} \eneq Let us consider the function $h^k : \om \times \M^{n \times n}_{sym} \longrightarrow \R$ defined by \bema h^k (x,E) = \bepa \bear{ll} (1-\textstyle{\frac1k}) |E|_{\A(x)}^2 & \mbox{for $ |E|_{\A(x)} \leq r^k(x)$\, ,} \\ & \\ \psi(\gamma^{-\frac12} |E|_{\A(x)}) & \mbox{for $ |E|_{\A(x)} > r^k(x)$\, ,} \enar \enpa \enma which is less than or equal to $V(x,E)$ by (\ref{less1}), (\ref{Anormequiv}), and (\ref{Z2}). For a suitable choice of $\mu^k(x)>0$ the function \bema \phi^k (x,t) = \bepa \bear{ll} (1-\textstyle{\frac1k}) t^2 & \mbox{for $0 \leq t \leq \mu^k(x)$\,,} \\ & \\ 2 (1-\textstyle{\frac1k}) \mu^k(x) t - (1-\textstyle{\frac1k}) (\mu^k(x))^2 & \mbox{for $ t \geq \mu^k(x)$\,,} \enar \enpa \enma is convex in $t$ and satisfies $\phi^k (x,|E|_{\A(x)}) \leq h^k(x,E) \leq V(x,E)$. To conclude the proof it is enough to define $V_j^k (x,E) := \phi^k (x,\eps_j|E|_{\A(x)} )/ \eps_j^2$. {}From the special form of $\phi^k (x,\cdot)$ it is easy to see that $V_j^k (x,\cdot)$ is increasing with respect to $j$ and that (\ref{identica}) holds, while (\ref{riscal}) follows from the inequality $\phi^k(x, |E|_{\A(x)}) \leq V(x,E)$. \qed \begin{lemma} \name{gli-incr} Let $g_j : \om \times \R^m \to [0, +\infty)$ be Caratheodory functions such that $g_j(x,\cdot)$ is convex. Let $g_j(x,\xi)$ be increasing in $j$ and pointwise converging to a function $g(x,\xi)$. If $w_j$ converges weakly to $w$ in $L^1(\om, \R^m)$, then \beeq \lab{gli-incr-ineq} \int_{\om} g(x,w) \dx \leq \liminf_{j \to +\infty} \int_{\om} g_j(x,w_j) \dx \, . \eneq \end{lemma} \proof As $g_i(x,w_j) \leq g_j(x,w_j)$ for $j \geq i$, by the lower semicontinuity of the functional $\int_{\om} g_i(x,v) \dx$ we have \bema \int_{\om} g_i(x,w) \dx \leq \liminf_{j \to +\infty} \int_{\om} g_i(x,w_j) \dx \leq \liminf_{j \to +\infty} \int_{\om} g_j(x,w_j) \dx \, , \enma which proves (\ref{gli-incr-ineq}) for $i \to \infty$. \qed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GAMMALIMINF \begin{prop} \name{liminf} For every $u \in H^1_{g,\partial \om_D}$ and every sequence $u_{j} \in H^1_{g,\partial \om_D} $ weakly converging to $u$, we have the $\Gamma\mbox{-liminf}$ inequality \beeq \lab{gliineq} \frac{1}{2} \int_{\om} \bigA(x) \, [ e(u) , e(u) ] \dx \leq \liminf_{\eps_j \to 0} {\cal G}_{\eps_j} (u_{j}) \, , \eneq from which it follows that ${\cal G}(u) \leq {\cal G}'(u)$. \end{prop} \proof For every $k \in \N$ let $V^k_j(x,E)$ be the sequence given by Lemma \ref{convexapprox}. Note that by (\ref{identica}) for every $E \in \M^{n \times n}_{sym}$ we have \beeq \lab{Vjconv} \lim_{j \to +\infty} V^k_j (x, E) = \Big( 1-\frac{1}{k} \Big) \, |E|^2_{\A(x)} \, . \eneq Then inequality (\ref{riscal}) gives \beea W\ej (x,\nabla u\ej ) & = & \frac{1}{\eps_j^2} V (x,\eps_j e(u\ej) + \textstyle{\frac12}\epsj^2 \nabla u\ej^T \nabla u\ej ) \nonumber \\ & \geq & V_j^k (x, e(u\ej) + \textstyle{\frac12}\epsj \nabla u\ej^T \nabla u\ej ) \, .\nonumber \enea Since $\nabla u_{\eps_j} \rightharpoonup \nabla u$ in $L^2(\om,\M^{n \times n})$ we have that $\eps_j \nabla u_{\eps_j}^T \nabla u_{\eps_j} \longrightarrow 0$ strongly in $L^1(\om,\M^{n \times n})$, hence $e(u\ej) + \frac12 \eps_j \nabla u_{\eps_j}^T \nabla u_{\eps_j} \rightharpoonup e(u)$ weakly in $L^1(\om,\M^{n \times n})$. Then by Lemma \ref{gli-incr} and (\ref{Vjconv}) for every $k \in \N$ we have \beea \liminf_{\eps_j \to 0} \int_{\om} W_{\eps_j} (x,\nabla u_{\eps_j}) \dx & \geq & \liminf_{j \to +\infty} \int_{\om} V_j^k (x, e(u\ej) + \textstyle{\frac12} \epsj \nabla u_{\eps_j}^T \nabla u_{\eps_j}) \dx \nonumber \\ & \geq & \frac{1}{2} \int_{\om} \Big(1-\frac1k \Big) \bigA(x) \, [ e(u), e(u) ] \dx \, . \nonumber \\ \enea Taking the supremum as $k \to \infty$ and considering the weak continuity of ${\cal L}$ we deduce inequality (\ref{gliineq}). \qed \section{Convergence of minimizers} \lab{strconv} We are now in a position to prove Theorem \ref{convmin}. \vspace{6pt} \noin \textbf{Proof of Theorem \ref{convmin}.} It is enough to prove the statement for evey sequence $\epsj \to 0$. Since $ {\cal G}_{\eps_j} (g) \leq C < +\infty$, we have ${\cal G}_{\eps_j}(u_{\eps_j}) \leq C < +\infty$, then by Proposition \ref{minima} $u\ej$ is equibounded in $H^1(\om,\R^n)$. Thus there exists a subsequence $u_{\eps_k}$ converging weakly to some limit $w \in H^1_{g,\partial \om_D}$. By $\Gamma\mbox{-convergence}$ we know that $w$ must be the minimizer $u_0$ of the limit functional ${\cal G}$ (see, e.g., \cite{DalM}, Corollary 7.17) Finally, as the limit $w$ depends neither on the subsequence $w_{\eps_k}$ nor on the sequence $\epsj$, the whole sequence $w_{\eps}$ converges weakly to $u$ in $H^1_{g,\partial \om_D}$. \qed In the sequel we will assume that $V(x,E)$ satisfies conditions (a), (b), and (c'). It is not restrictive to assume that $11$, by a diagonal argument there exists a sequence $j_m\to\infty$ such that for every $k$ \beeq \lab{weak1} V^k_{j_m} (x,\Phi_{j_m})^{\frac{1}{p}} \rightharpoonup w^k \qquad \mbox{weakly in $L^p(\om)$} \, , \eneq for a suitable function $w^k \in L^p(\om)$. Moreover by the weak convergence of $u_j$ it follows that $\Phi_j$ converges weakly to $e(u)$ in $L^1(\om,\M^{n\times n})$. Since the functions $V^k_j (x,\xi)^{\frac{1}{p}}$ are convex in $\xi$, then by Lemma \ref{gli-incr} and by (\ref{identica}), for every Borel set $B \subset \om$ we have \bema \Big( 1 - \frac{1}{k} \Big)^{\frac{1}{p}} \int_B | e(u) |^{\frac{2}{p}}_{\A} \dx \leq \liminf_{m \to \infty} \int_B V^k_{j_m} (x,\Phi_{j_m})^{\frac{1}{p}} \dx = \int_B w^k \dx \, . \enma Thus \beeq \lab{uscpunt} w^k \geq \Big( 1 - \frac{1}{k} \Big)^{\frac{1}{p}} | e(u) |^{\frac{2}{p}}_{\A} \qquad\mbox{a.e.\ in }\om \, . \eneq Moreover, by the weak lower semicontinuity of the norm, from (\ref{convA}), (\ref{lower}), and (\ref{weak1}) it follows that \beeq \lab{lsc||} \int_{\om} (w^k)^p \dx \leq \int_{\om} | e(u) |^2_{\A} \dx \, . \eneq Being $p>1$, there exists $w \in L^p(\om)$ and a subsequence of $w^k$ which converges weakly to $w$ in $L^p(\om)$. Then passing to the limit in (\ref{uscpunt}) we get \bema \big( w \big)^p \geq | e(u) |^2_{\A} %\frac{1}{2} \A(x) [ e(u(x)) , e(u(x)) ] \enma for a.e. $x\in \om$, and by (\ref{lsc||}) we have \bema \int_{\om} w^p \dx \leq \int_{\om} | e(u) |^2_{\A} \dx \,. %\A(x) [ e(u(x)) , e(u(x)) ] \dx \,. \enma These inequalities imply that $w = | e(u) |^{\frac2p}_{\A} $. Being the limit independent of the subsequence, we have proved for whole sequence $w^k$ that \beeq \lab{weak2} w^k \rightharpoonup | e(u) |^{\frac2p}_{\A} \qquad \mbox{weakly in $L^p(\om)$}. \eneq Let $\mu^k(x)$ be the functions defined in Lemma \ref{convexapprox2}. As $\mu^k >0$ a.e.\ in $\om$, there exists a decreasing sequence of constants $\eta_k$ such that \beeq \lab{meas} \mathrm{meas} \big( \{ x \in \om : \mu^k(x) < \eta^k \} \big) \leq \frac1k \, . \eneq Considering that $V^k_{j_m} (x,\Phi_{j_m})^{\frac{1}{p}}$ is bounded in $L^p(\om)$ uniformly with respect to $m$ and $k$, and hence we can use a metric equivalent to the weak topology, by (\ref{weak1}) and (\ref{weak2}) %that $w^k \rightharpoonup w$ and %$V^k_j (x,\Phi_{\eps_n})^{\frac{1}{p}} \rightharpoonup w^k$ %we can use a metric equivalent to the weak topology and thus we can extract a subsequence $i_k$ of $j_m$ such that, writing for simplicity $\eps_k$ instead of $\eps_{i_k}$, we have \beeq \lab{7'} \frac{\eta^k}{\eps_k} > k \qquad \mbox{and} \qquad V^k_{i_k} (x,\Phi_{i_k})^{\frac{1}{p}} \rightharpoonup | e(u) |^{\frac2p}_{\A} \quad \mbox{weakly in $L^p(\om)$} \, . \eneq Then by (\ref{convA}) and (\ref{lower}) we have \bema \limsup_{k \to \infty} \int_{\om} V^k_{i_k} (x,\Phi_{i_k}) \leq \int_{\om} | e(u) |_{\A}^2 \dx \, . \enma By the uniform convexity of the $L^p(\om)$ space this implies that \bema V^k_{i_k} (x,\Phi_{i_k})^{\frac{1}{p}} \longrightarrow | e(u) |^{\frac{2}{p}}_{\A} \qquad \mbox{strongly in $L^p(\om)$}\,. \enma Then we have \bema V^k_{i_k} (x,\Phi_{i_k}) \longrightarrow | e(u) |^2_{\A} \enma strongly in $L^1(\om)$ and a.e.\ in $\om$. Now we can prove that $|e(u_{i_k})|_{\A}$ converges in measure to $|e(u)|_{\A}$. Indeed, for every $\delta >0$ the set $ \big\{ \big| \, |e(u_{i_k})|_{\A} - |e(u)|_{\A} \, \big| > \delta \big\}$ is contained in \beeq \lab{C} \Big\{ \left| |e(u_{i_k})|_{\A} - |e(u_{i_k}) + \textstyle{\frac{1}{2}} \eps_k C(u_{i_k}) |_{\A} \right| > \textstyle{\frac{\delta}{2}} \Big\} \cup \Big\{ \left| |e(u_{i_k}) + \textstyle{\frac{1}{2}} \eps_k C(u_{i_k})|_{\A} - |e(u)|_{\A} \right| > \textstyle{\frac{\delta}{2}} \Big\} \, . \eneq The first set is contained in $ \big\{ | \textstyle{\frac{1}{2}} \eps_k C(u_{i_k}) |_{\A} > \frac{\delta}{2} \big\}$, whose measure tends to zero since $ \eps_k C(u_{i_k}) \longrightarrow 0$ in $L^1(\om,\M^{n \times n})$. Note that for $x \in \{ \mu^k(x) > \eta^k \}$ if \bema |e(u_{i_k}) + \textstyle{\frac{1}{2}} \eps_k C(u_{i_k}) | _{\A} < k \enma then by (\ref{identica}) and (\ref{7'}) we have \bema V^k_{i_k} (x,\Phi_{i_k}) = \frac{k-1}{k} |e(u_{i_k}) + {\textstyle\frac12} \eps_k C(u_{i_k})|^2_{\A} \, . \enma Then the second set in (\ref{C}) is contained in \bema \{ \mu^k(x) < \eta^k \} \cup \big\{ |e(u_{i_k}) + \textstyle{\frac{1}{2}} \eps_k C(u_{i_k}) | _{\A} > k \big\} \cup \big\{ \big| ( \, \textstyle{\frac{k}{k-1}} V^k_{i_k} (x,\Phi_{i_k}) \, )^\frac{1}{2} - | e(u) |_{\A} \big| > \textstyle{\frac{\delta}{2}} \big\} \, . \enma The measure of all these sets tends to zero as $k \to +\infty$. The first one by (\ref{meas}), the second one since $|e(u_{i_k}) + \textstyle{\frac{1}{2}} \eps_k C(u_{i_k})|_{\A}$ is equibounded in $L^1(\om)$, and the third one because $(\, \textstyle{\frac{k}{k-1}} V^k_{i_k} (x,\Phi_{u_k})\, )^\frac{1}{2} \longrightarrow | e(u) |_{\A}$ pointwise. This concludes the proof of the convergence in measure of $|e(u_{i_k})|_{\A}$ to $|e(u)|_{\A}$. Then by Lemma \ref{Olek} it follows that $e(u_{i_k})$ converges in measure to $e(u)$. As $e(u_{i_k})$ is bounded in $L^2(\om,\M^{n \times n})$, we deduce that $e(u_{i_k})$ converges strongly to $e(u)$ in $L^q(\om,\M^{n \times n})$ for $1 \leq q <2$. Since the limit does not depend on the subsequence we have that $e(u_{j})$ converges strongly to $e(u)$ in $L^q(\om,\M^{n \times n})$. By the Korn inequality (see e.g. \cite{Tem}) there exists a constant $C_q$ such that \bema \int_{\om} | \nabla (u - u_j) |^q \dx \leq C_q \int_{\om} | e(u-u_j) |^q \dx + C_q \int_{\om} | u -u_j |^q \dx \, . \enma As $e(u_{j})$ converges strongly to $e(u)$ in $L^q(\om,\M^{n \times n})$ and $u_j$ converges strongly to $u$ in $L^q(\om,\R^{n})$ by the Rellich theorem, we deduce that $u_j$ converges to $u$ in the strong topology of $W^{1,q}(\om,\R^n)$. \qed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% COROLLARIO \textbf{Proof of Theorem \ref{strongconvmin}.} Let $\eps_j \to 0$. By Proposition \ref{minima} $u\ej$ converges weakly to $u$ in $H^1 (\om,\R^n)$ and by $\Gamma\mbox{-convergence}$ we have ${\cal G}(u_{\eps_j}) \longrightarrow {\cal G}(u)$ (see e.g. \cite{DalM} Corollary 7.17). By weak continuity we have ${\cal L}(u\ej) \longrightarrow {\cal L}(u)$, so that (\ref{convA}) holds. The conclusion follows from Proposition \ref{Lpstrong}. \qed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BIBLIO \medskip {\bf Acknowledgements. }This work is part of the European Research Training Network ``Homogenization and Multiple Scales'' under contract HPRN-2000-00109. \begin{thebibliography}{99} \bibitem{Ad} R. A. Adams: \emph{Sobolev spaces}, Academic Press, New York (1975) \bibitem{Ciarl} P. G. Ciarlet: \emph{Three dimensional elasticity}, North Holland, Amsterdam (1988) \bibitem{DalM} G. Dal Maso: \emph{An Introduction to $\Gamma$-$convergence$}, Birkh\"auser, Boston (1993) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\bibitem{EvGer} L. C. Evans, R. F. Gariepy: \emph{Measure Theory and Fine %Properties of Functions}, CRC Press, Boca Raton (1992) %\bibitem{GilbTrud} D. Gilbarg, N. 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