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\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 $1<p<2$.
Let $\alpha$ be the constant appearing in (b). {}From (a), (b), and (c') 
it follows that there exists 
a nondecreasing, 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^{-p}) t^p   & \mbox{ for $d \leq t$ ,} \enca
	\qquad \qquad \mbox{for $0<c<d$} \, ,
%	(for some constants $0<c<d$) 
\enma
such that $\phi(|E|) \leq V(x,E)$ for a.e.\ $x \in \om$.
Consider the function $\psi_p(t)$ defined as 
\beeq \lab{psi2}
	\psi_p(t) = \beca
	\alpha t^2 	& \mbox{ for $0 \leq t \leq \mu$ ,} \\
	a (t-b)^p   & \mbox{ for $\mu \leq t$ ,} \enca
\eneq
for $a=\alpha \, p^{-p} \, 2^p \, \mu^{2-p}$ and 
$b=(1-\frac{p}{2}) \mu$.
It is not difficult to check that $\psi_p(t)$ is increasing, $C^1$, 
and convex. As $1<p<2$, we have 
\beeq \lab{limmu}
	\lim_{\mu \to 0} \alpha \, p^{-p} \, 2^p \, \mu^{2-p} = 0 \, , 
\eneq
thus for $\mu$ sufficiently small $\psi_p(t) \leq \phi(t)$ for every $t\geq 0$
and then $\psi_p(|E|) \leq V(x,E)$ for a.e.\ $x \in \om$ and every 
$E \in \M^{n \times n}_{sym}$.

\begin{lemma} \name{convexapprox2} Let $\eps_j \to 0$. 
For every $k \in \N$ there exists 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 for a.e. $x \in \om$ the function 
$V^k_j(x,\cdot)^{\frac1p}$ is convex and (\ref{riscal})
and (\ref{identica}) hold. 
\end{lemma}

\noin \textbf{Proof.} We follow the proof of Lemma \ref{convexapprox},
with $\psi$ replaced by $\psi_p$, and
we consider the functions
\bema
	\phi_p^k (x,t) = \bepa \bear{ll}
	(1-\textstyle{\frac1k}) t^2 
	& \mbox{for $0 \leq t \leq \mu^k(x)$ ,} \\
	& \\
	a(x) (t - b(x))^p 
	& \mbox{for $ t \geq \mu^k(x)$ .} \enar \enpa
\enma
Note that $\phi_p^k (x,t)^{\frac1p}$ is convex 
for $a(x)=(1-\textstyle{\frac1k}) 2^{p} p^{-p} (\mu^k(x))^{2-p}$ and 
$b(x)=(1-\frac{p}{2}) \mu^k(x)$. 
By (\ref{limmu}) for $\mu^k(x)$ sufficiently small we have that
\bema
	\phi_p^k (x,|E|_{\A(x)}) \leq V(x,E)
\enma
for a.e.\ $x \in \om$ and every $E \in \M^{n \times n}_{sym}$. 
Then the sequence defined by 
$V^k_j (x,E):=\phi_p^k(x,\eps_j |E|_{\A(x)})/ \eps_j^2$ 
satisfies (\ref{riscal}) and (\ref{identica}), is increasing with respect 
to $j$, and $V^k_j(x,\cdot)^{\frac1p}$ is convex for a.e.\ $x \in \om$. \qed 


\begin{lemma} \lab{Olek} Let $\Phi_n \rightharpoonup \Phi$  weakly in
$L^1(\om,\M^{n \times n})$ such that
$|\Phi_n|_{\A}$ converges
to $|\Phi|_{\A}$  in measure. Then $\Phi_n$ converges
to $\Phi$ in measure.
\end{lemma}


\proof By passing to a subsequence and to a suitable measurable
subdomain, it is not restrictive to suppose
that $\Phi(x) \neq 0$ for every $x \in \om$ and
that $|\Phi_n|_{\A}$ converges to $|\Phi|_{\A}$ pointwise.


By (\ref{Anormequiv}) and by weak convergence we have
\beeq \lab{wconvy}
	\int_{\om} \langle \frac{\Phi}{|\Phi|_{\A}} \, 
	, \, \Phi_n - \Phi \rangle_{\A} \dx
	\ \longrightarrow 0 \, ,
\eneq
where $\langle \cdot , \cdot \rangle_{\A}$ is the scalar product
associated with the norm $| \cdot |_{\A}$, i.e., 
$$ \langle \Psi_1 , \Psi_2 \rangle_{\A}=\frac12 \mbox{\textsf{A}}(x)  
[\Psi_1(x) , \Psi_2(x) ]\,.$$
Moreover by the Schwarz inequality
\bema
	\int_{\om} \left(
	\langle \frac{\Phi}{|\Phi|_{\A}} \, , \, \Phi_n - \Phi \rangle_{\A}
	\right)^+ \dx \leq
	\int_{\om} \left(
	|\Phi_n|_{\A} - |\Phi|_{\A} \right)^+  \dx \, .
\enma
As $|\Phi_n|_{\A}$ is equiintegrable and converges
to $|\Phi|_{\A}$ in measure, it converges also in $L^1 (\om)$.
Thus
\bema
	\int_{\om} \left(
	\langle \frac{\Phi}{|\Phi|_{\A}} \, , \, \Phi_n - \Phi \rangle_{\A}
	\right)^+ \dx \
	\ \longrightarrow 0 \, .
\enma
Then by (\ref{wconvy})
$\langle \frac{\Phi}{|\Phi|_{\A}} \, , \, \Phi_n -\Phi \rangle_{\A} \longrightarrow 0$
in $L^1 (\om)$ and, up to a subsequence, it converges for a.e.\
$x \in \om$, hence $ \langle \frac{\Phi}{|\Phi|_{\A}} \,
	, \, \Phi_n \rangle_{\A}\longrightarrow |\Phi|_{\A}$ pointwise a.e.\ in
	$\om$.

Considering the identity
\bema
	|\Phi_n -\Phi|^2_{\A} =
	\langle \frac{\Phi}{|\Phi|_{\A}} \, , \, \Phi_n -\Phi \rangle_{\A}^2 
	+ |\Phi_n|^2_{\A} -  \langle \frac{\Phi}{|\Phi|_{\A}} \,
	, \, \Phi_n \rangle_{\A} ^2\,,
\enma
we deduce that $|\Phi_n -\Phi|^2_{\A} \longrightarrow 0$ pointwise for a.e.
$x \in \om$.

Since for every subsequence of $\Phi_n$ we can find a further
subsequence converging pointwise to $\Phi$, it follows that
$\Phi_n$ converges to $\Phi$ in measure. \qed


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LPSTRONG 


\begin{prop} \lab{Lpstrong} Let $\eps_j\to0^+$ and let
     $u_j \rightharpoonup u$ weakly in
$H^1(\om,\R^n)$ such that
\beeq \lab{convA}
	\frac{1}{\epsj^2}
	\int_{\om} V ( x, \epsj e(u_j) +
	{\textstyle \frac{1}{2}\epsj^2 }
	C(u_j) ) \dx \longrightarrow
	\int_{\om} | e(u) |_{\A}^2 \dx \, .
\eneq
Then $u_j \longrightarrow u$ strongly in $W^{1,q}(\om,\R^{n})$
for $1 \leq q < 2$.
\end{prop}

\proof For every $k$ let $V^k_j(x,E)$ be the sequence given by Lemma
\ref{convexapprox2}. Denote
$e(u_j)+{\textstyle \frac{1}{2} }\epsj C(u_j)$ by $\Phi_j$. By (\ref{riscal})
for every $k$ and every $j$ we have
\beeq \lab{lower}
	V^k_j(x,\Phi_j) \leq \frac{1}{\epsj^2} V ( x, \epsj e(u_j) +
	{\textstyle \frac{1}{2}\epsj^2 } C(u_j) ) \, .
\eneq
By (\ref{convA}) it follows that, for every $k$,
$V^k_j (x,\Phi_j)^{\frac{1}{p}}$ is bounded in $L^p(\om)$ uniformly with respect 
to $j$ and $k$. 
Being $p>1$, 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.


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\end{document}






