\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small Sixth Mississippi State Conference
on Differential Equations and Computational Simulations, {\em
Electronic Journal of Differential Equations},
Conference 15 (2007), pp. 41--50.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{41}
\title[\hfilneg EJDE-2007/Conf/15\hfil Young measure minimizers]
{Young measure minimizers in the asymptotic analysis of thin
films}

\author[M. Bocea\hfil EJDE/Conf/15 \hfilneg]
{Marian Bocea}  

\dedicatory{Dedicated to Klaus Schmitt on the occasion of his 65th
birthday}

\address{Marian Bocea \newline
Department of Mathematics,
300 Minard Hall, North Dakota State University, Fargo,
ND 58105-5075, USA}
\email{marian.bocea@ndsu.edu}

\thanks{Published February 28, 2007.}
\subjclass[2000]{49J45, 74B20, 74G10, 74K15, 74K35}
\keywords{Equi-integrability; concentrations;
  oscillations; relaxation; \hfill\break\indent  Young measure}

\begin{abstract}
 An integral representation for a relaxed functional
 arising in the membrane theory is obtained in terms of Young
 measures generated by sequences
 $\{( \nabla_{\alpha}u_{\varepsilon_n } \big| \frac{1}{\varepsilon_n }
 \nabla _3 u_{\varepsilon_n})\} $
 of scaled gradients.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

Let $\omega \subset \mathbb{R}^2 $ be an open bounded domain with
Lipschitz boundary, and consider a thin three dimensional domain
$\Omega _{\varepsilon}:= \omega \times (-\varepsilon, \varepsilon)$ filled with an
elastic material with elastic energy density  $W_\varepsilon$ and subject
to dead loading body forces  of densities $f_\varepsilon\in L^{p'
}(\Omega _\varepsilon, \mathbb{R}^3) $ where $1<p<+\infty,$ and $p' $
stands for the conjugate exponent of $p$, i.e. $ \frac{1}{p} +
\frac{1}{p' } = 1$. We assume that for fixed $\varepsilon> 0$, in
order to reach equilibrium, $u_\varepsilon$ seeks to minimize
\[
E_\varepsilon(u):= \int_{\Omega _\varepsilon}W_\varepsilon(x, \nabla u (x))\ dx -
\int_{\Omega _\varepsilon}f_\varepsilon(x) \cdot u(x)\ dx,
\]
among all kinematically admissible fields $u$.

To study the effective behavior of a very thin film, we consider a
sequence $\lbrace \varepsilon_n \rbrace $ of positive real numbers
(half-thickness) converging to zero and we recast energy functionals
over varying domains $\Omega _{\varepsilon_n}$ into functionals with
a fixed domain of integration $\Omega := \omega \times (-1, 1)$ by
means of a reformulation of the problem through a
$\frac{1}{\varepsilon_n}$-dilation in the transverse direction $x_3
$. With $x = (x_1, x_2, x_3) $, set
\begin{gather*}
W^{(\varepsilon_n )} (x, \cdot )  := W_{\varepsilon_n }(x_1 , x_2
, \varepsilon_n x_3 ; \cdot ),\\
f^{(\varepsilon_n )} (x)  := f_{\varepsilon_n }(x_1 , x_2
, \varepsilon_n x_3 ),\\
v_n (x)  := u_{\varepsilon_n }(x_1 , x_2 , \varepsilon_n x_3 ).
\end{gather*}
After an appropriate rescaling, $v_n $ seeks to minimize
\[
E^{(\varepsilon_n )} (v):= \int_{\Omega }W^{(\varepsilon_n )}
\left(x, \left( \nabla _{\alpha }v \Big| \frac{1}{\varepsilon_n
}\nabla _3 v\right) (x)\right) dx - \int_{\Omega }f^{(\varepsilon_n
)}(x) \cdot v(x) \ dx,
\]
among all kinematically admissible fields $v = (v_1, v_2, v_3)$ on
$\Omega $, where $\nabla _{\alpha }v$ stands for the $3\times 2$
matrix of partial derivatives $\frac{\partial v_i }{\partial
x_\alpha }, \, i\in \{1, 2, 3\} , \alpha \in \{1, 2\}, \ \nabla_3 v$ is
the three-dimensional vector of partial derivatives
$\frac{\partial v_i }{\partial x_3}, \, i\in \{1, 2, 3\},$ and $
\left( A|a\right) $ denotes a $3\times 3$ matrix whose first two
columns are those of the $3\times 2$ matrix $A$ and the last
column is the vector $a \in \mathbb{R}^3$. We assume that the
rescaled energy density $W^{(\varepsilon_n )}$ does not explicitly depend
on $\varepsilon_n$. Precisely, $W^{(\varepsilon_n )} = W $ where $W: \Omega \times
\mathbb{M}^{3\times 3}\to \mathbb{R}$ is a Carath\'{e}odory
integrand (see Definition
\ref{defofnormalandCaratheodoryintegrands}) satisfying for some
$1<p<+\infty $ the \emph{p}-growth and coercivity condition
\begin{equation}\label{pgrowthandcoercforW}
\frac{1}{C}|A|^p - C \leq W(x, A) \leq C( 1 + |A|^p)
\end{equation}
for $\mathcal{L}^{3}$-a.e $x \in \Omega $ and for all $A \in
\mathbb{M}^{3\times 3}$, where $C>0$ is a real constant and
$\mathbb{M}^{3\times 3}$ denotes the space of real $3\times 3$
matrices endowed with the usual Euclidean norm $|A|:= \sqrt{{\rm
tr}\left( A^{T}A \right) }$. Assuming, moreover, that the rescaled
body force density $f^{(\varepsilon_n )}$ is independent of $n$ (see e.g.
\cite{FoxRaoultandSimo:1993}), the study of the effective energy
of the limiting system is hinged on the understanding of the
asymptotic behavior of the energies
\[
I_{n}(v_n ):=  \int_{\Omega }W\left(x, \left( \nabla _{\alpha }v_n
\Big| \frac{1}{\varepsilon_n }\nabla _3 v_n \right) (x)\right) dx.
\]

An extensive literature in this direction (see \cite
{AnzBalPer:1994, BhattaFonsecaFrancfort:1999, BhattaJames:1999,
BraidesFonsecaFrancfort:2000, BraidesFonseca:2001,
FonsecaFrancfort:2001, FoxRaoultandSimo:1993, LeDretRaoult:1995,
LeDretRaoult:2000}, among others) is usually formulated in the
natural mathematical setting of $\Gamma $-convergence, and this
approach gives rise to the so-called membrane theory. In view of the
a priori bound
\[
\sup _{n\in \mathbb{N}} \int_{\Omega } \Big| \left( \nabla
_{\alpha}v_n \Big| \frac{1}{\varepsilon _n}\nabla
_{3}v_{n}\right)(x)\Big| ^{p}dx < +\infty
\]
for energy bounded sequences, and derived from
(\ref{pgrowthandcoercforW}), in this paper we obtain an integral
representation of the relaxed energy functional $ \mathcal{W}:
W^{1,p}(\omega; \mathbb{R}^3)\times L^{p}(\Omega; \mathbb{R}^3) \to
\mathbb{R}$ defined by
\begin{equation}\label{relaxedfunctional}
\begin{aligned}
\mathcal{W}(v,c)
 := \inf \Big\{& \liminf_{n
\to +\infty }\int_{\Omega }W\left( x, \left( \nabla_\alpha  v_n
\Big| \frac{1}{\varepsilon_n }\nabla_3 v_n \right)(x) \right) dx :
\varepsilon_n \to
0^{+}, \\
& v_n \rightharpoonup v \text{ weakly   in $W^{1,p}(\Omega;\mathbb{R} ^3)$},\\
& \frac{1}{\varepsilon_n }\nabla_3 v_n \rightharpoonup c \text{
weakly  in $L^{p}(\Omega; \mathbb{R}^3)$} \Big\} ,
\end{aligned}
\end{equation}
in terms of scaled gradient $p$-Young measures, which are
essentially Young measures generated by sequences of scaled
gradients $ \big\{ \big( \nabla_\alpha  v_n \big| \frac{1}{\varepsilon_n }\nabla_3 v_n
\big) \big\}$ (see Definitions \ref{defofYmandelemYM} and
\ref{defofaYmgenbyasequence}).

\begin{definition}\label{defofagradpYoungmeasures} \rm
Let $ \Omega := \omega \times (-1,1),$ where $\omega \subset \mathbb{R}
^2 $ is an open domain, and let $1\leq p \leq +\infty $.
A Young measure $\mu $ on $\Omega \times \mathbb{R}^9 $ is called a
\emph{scaled gradient} $p$-Young measure  (scaled gradient
Young measure if $ p = + \infty $) if there exist sequences
$\varepsilon_n \to 0^+ $  and $\{ v_n \} \subset W^{1,p}(\Omega;\mathbb{R} ^3) $ such that
\begin{itemize}
\item[(i)] $\{ v_n \} $ is weakly
(weakly * if $p = + \infty$) convergent in $W^{1,p}(\Omega;\mathbb{R} ^3) $,

\item[(ii)] $\{ \frac{1}{\varepsilon_n }\nabla_3 v_n \} $ is weakly
(weakly * if $p = + \infty $) convergent in
$L^{p}(\Omega; \mathbb{R}^3 ) $,

\item[(iii)] $\mathcal{E}_{\left( \nabla_\alpha  v_n \big|
\frac{1}{\varepsilon_n }\nabla_3 v_n \right)} \rightharpoonup \mu $
weakly * in $ C_{0}(\Omega \times \mathbb{R}^9 )'$.

\end{itemize}
The weak (weak * if $p = + \infty$) limit of $v_n $ in
$W^{1,p}(\Omega;\mathbb{R} ^3) $ is called an \emph{underlying deformation for} $\mu $ while
the weak (weak * if $p = + \infty$) limit of
$\frac{1}{\varepsilon_n }\nabla_3 v_n$ in $L^p(\Omega;\mathbb{R}^3) $ is called a
\emph{Cosserat vector associated to} $\mu $. For $1\leq p < +\infty$,
 we set
\begin{align*}
Y_{v,c}^{p} := \Big\{ & \nu \in Y(\Omega \times \mathbb{R}^9) : \nu
\ \text{is\ a  scaled  gradient  $p$-Young
measure  with} \\
& \text{underlying   deformation  $v$  and  associated  Cosserat
vector  $c$} \Big\} .
\end{align*}
\end{definition}

Our relaxation result is the following.

\begin{theorem}\label{relaxationtheorem}
Let $1<p<+\infty$, $\Omega := \omega \times (-1,1)$, where
$\omega \subset \mathbb{R}^2 $ is an open, bounded Lipschitz domain,
 and let $W: \Omega \times \mathbb{M}^{3\times 3} \to \mathbb{R}$ be a
Carath\'{e}odory integrand satisfying \eqref{pgrowthandcoercforW}.
Let $v \in W^{1,p}(\omega; \mathbb{R}^3) $ and $ c \in L^{p}(\Omega;
\mathbb{R}^3) $ be given, and define the relaxed functional
$\mathcal{W}: W^{1,p}(\omega; \mathbb{R}^3)\times L^p(\Omega;\mathbb{R}^3) \to \mathbb{R}$
by \eqref{relaxedfunctional}. Then
\begin{equation}\label{equalityofinfsinrelaxationthm}
\mathcal{W}(v,c) = \inf_{\nu \in
Y_{v,c}^{p}}\int_{\Omega \times \mathbb{M}^{3\times 3}}Wd\nu .
\end{equation}
Moreover, the infimum on the right hand side is attained, i.e.
there exists a scaled gradient $p$-Young measure $\mu _0 \in
Y_{v,c}^{p}$ such that
\begin{equation}\label{infimumisattainedatmu0}
\inf_{\nu \in Y_{v,c}^{p}}\int_{\Omega \times
\mathbb{M}^{3\times 3}}Wd\nu = \int_{\Omega \times
\mathbb{M}^{3\times 3}}Wd\mu _0 .
\end{equation}
\end{theorem}

A challenging open problem is the identification of algebraic and
analytical conditions on parametrized probability measures both
necessary and sufficient to guarantee that they belong to
$Y_{v,c}^{p}$. The corresponding program for probability measures
generated by gradients bounded in $L^p$ was carried out by
Kinderlehrer and Pedregal (see \cite{KinderlehrerPedregal:1991,
KinderlehrerPedregal:1994}) and subsequently generalized to the
realm of $\mathcal{A}$-{\rm quasiconvexity} by Fonseca and
M\"{u}ller in \cite{FMaquasiconvexity}. In the case of Young
measures generated by sequences of scaled gradients, recent progress
was made by Bocea and Fonseca \cite{BoceaFonseca2004}, where the
slightly broader class of \emph{bending Young measures} has been
completely characterized.


The crucial ingredient needed to prove Theorem
\ref{relaxationtheorem} is the following decomposition result
whose proof may be found in \cite{BoceaFonsecaFebr2002}.

\begin{theorem}\label{epsilondecompositionlemma}
 Let $\Omega:= \omega \times (-1,1) $, where
$\omega \subset \mathbb{R} ^2 $ is an open, bounded Lipschitz
domain, let  $\{ \varepsilon _n \}$ be a sequence of positive real
numbers converging to zero, and consider a sequence $ \lbrace v_n
\rbrace $ bounded in $W^{1,p}(\Omega; \mathbb{R} ^3) \ (1<p<+\infty
) $ and satisfying
\[
\sup_{n\in \mathbb{N}} \int_{\Omega } \Big| \left( \nabla
_{\alpha}v_n \Big| \frac{1}{\varepsilon _n}\nabla
_{3}v_{n}\right)(x)\Big| ^{p}dx < +\infty .
\]
Suppose further that $v_n \rightharpoonup v$ in $W^{1,p}(\Omega;
\mathbb{R} ^3) $ and $\frac{1}{\varepsilon_n}\nabla _{3}v_{n}
\rightharpoonup c$ in $L^{p}(\Omega; \mathbb{R} ^3)$. Then there
exists a subsequence $\lbrace v_{n_k} \rbrace $ of $ \lbrace v_n
\rbrace $ and a sequence $\lbrace w_k \rbrace \subset
W^{1,p}(\Omega; \mathbb{R} ^3) $ such that
\begin{gather}\label{(i)epsilondecompositionlemma}
\lim_{k \to \infty} \mathcal{L}^3 \left( \lbrace x\in \Omega : w_k
(x) \not= v_{n_k}(x) \rbrace \cup \lbrace x\in \Omega : \nabla w_k
(x) \not= \nabla v_{n_k}(x) \rbrace\right) = 0,
\\
\label{(ii)epsilondecompositionlemma} \left\{ \left( \nabla
_{\alpha}w_k \Big| \frac{1}{\varepsilon _{n_k}}\nabla
_{3}w_{k}\right) \right\} \text{  is  $p$-equi-integrable,}
\\
\label{(iii)epsilondecompositionlemma}
w_k \rightharpoonup v \text{ weakly   in  } W^{1,p}(\Omega; \mathbb{R} ^3),
\\
\label{(iv)epsilondecompositionlemma}
\frac{1}{\varepsilon_{n_k }}\nabla _{3}w_{k} \rightharpoonup c \text{  weakly  in
 } L^{p}(\Omega; \mathbb{R} ^3 ).
\end{gather}
\end{theorem}

This theorem allows us to decompose (up
to a subsequence) $\big\{ \big( \nabla _{\alpha }v_n \big|
\frac{1}{\varepsilon_n }\nabla _3 v_n \big) \big\} $ as a sum of a
sequence $\big\{ \big( \nabla _{\alpha }w_n \big| \frac{1}{\varepsilon
_n }\nabla _3 w_n \big) \big\}$ whose $p$-th power is
equi-integrable and a remainder  converging to zero in measure.
We may say that $\big\{ \big( \nabla _{\alpha }w_n \big|
\frac{1}{\varepsilon_n }\nabla _3 w_n \big) \big\}$ carries the
oscillations, while the remainder accounts for the concentration
effects.

In the next section we present the basic facts about Young
measures needed for the proof of our relaxation theorem. The proof
of Theorem \ref{relaxationtheorem} is the subject of Section 3 of
the paper.

\section{Young measures}

The characterization of (oscillatory) limits of nonlinear
quantities in the Calculus of Variations has been successfully
analyzed in several contexts by means of Young measures. Young
measures were originally introduced in Optimal Control Theory by
L.C. Young in connection to nonconvex problems, thus providing the
appropriate framework for the description of generalized
minimizers in the Calculus of Variations
(see \cite{Young:1937,Young:1969}). Later, Tartar introduced the use of Young
measures in the PDE framework
(see \cite{Tartarheriotwatt:1979,Tartarconslaws:1983,
Tartaroscillations:1984}).  For
more details regarding the study of Young measures we refer the
reader to \cite{Bal:1984,Ball:1989,BerliocchiandLasri:1973,
FMP,KinderlehrerPedregal:1991,KinderlehrerPedregal:1994,Kr:1994,
Kr:1999, Pedregal'sbook:1997}, among others.

In this section we recall the definition and the relevant results
about Young measures that we will need in the sequel; we follow
closely the approach of Kristensen (see \cite{Kr:1999}). Let $D$
be an open subset of $\mathbb{R}^l  (l \geq 1), \ C(D) $ be the
space of real-valued continuous functions on $D$ and define
\begin{align*}
C_{0}(D) := \Big\{& \varphi \in C(D) :
\text{for  every $\varepsilon>0$ there  exists a  compact  set $K \subset D$}\\
&  \text{such that $\ |\varphi (x)| \leq \varepsilon$  if
$ x\in D\setminus K$} \Big\}.
\end{align*}
Endowed with the supremum norm, $C_{0}(D) $ is a separable Banach
space. In view of Riesz' Theorem the dual space $C_{0}(D)'$ can be identified with the space of bounded Radon measures on
$D$ with the norm $\|\mu \| := |\mu |(D)$, via the duality pairing
\[
\langle \mu , \varphi \rangle = \int_{D}\varphi (x) \cdot
\frac{d\mu }{d|\mu |}(x)d|\mu |(x),
\]
where $|\mu | $ stands for the total variation of $\mu $ and is a
non-negative, finite Radon measure on $D$.

\begin{definition}\label{defofYmandelemYM} \rm
\begin{itemize}
\item[(i)] A non-negative Radon measure $\mu $ on
$\Omega \times \mathbb{R}^d $ with the property
\[
\mu (B\times \mathbb{R}^d ) = \mathcal{L}^{N}(B)\quad
\text{for  all  Borel  subsets  of }  \Omega ,
\]
is called a \emph{Young measure}. The set of Young measures on
 $\Omega \times \mathbb{R}^d $ is denoted by $Y(\Omega \times \mathbb{R}^d )$.

\item[(ii)] A Young measure $\mu $ for which there exists a
$\mathcal{L}^N$-measurable mapping $V:\Omega \to \mathbb{R}^d $ such
that
\[
\int_{\Omega \times \mathbb{R}^d }f d\mu = \int_{\Omega }
f(x,V(x)) dx, \quad\text{for all } f \in C_{0}(\Omega \times
\mathbb{R}^d ),
\]
is called an {\rm elementary Young measure}. We write
\[
\mu = \mathcal{E}_{V} := \int_{\Omega }\delta _{x}\otimes
\delta _{V(x)}dx,
\]
where $\delta _{x}$ and $\delta _{V(x)}$ are the Dirac measures on
$\Omega $ concentrated at $x$ and on $\mathbb{R}^{d}$ concentrated at
$V(x)$, respectively.

\item[(iii)] A product measure
$\big(\mathcal{L}^{N}\lfloor \Omega \big) \otimes \tilde{\mu } $ on
$\Omega \times \mathbb{R}^d $, where $\tilde{\mu }$  is a probability
measure on $\mathbb{R}^d $, is called a homogeneous Young measure.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk2.2} \rm
The definition of Young measures in Definition
\ref{defofYmandelemYM} (i) follows that of Berliocchi and Lasry
(see \cite{BerliocchiandLasri:1973}). It can be shown (cf.
\cite{Kr:1999}) to be equivalent to the original definition of
L.C. Young \cite{Young:1937} and the ones used in literature
(e.g., \cite{Bal:1984, Ball:1989,Pedregal'sbook:1997}).
\end{remark}

\begin{proposition}\label{bourbaki}
Let $\mu \in Y(\Omega \times \mathbb{R}^d )$.  Then there exists a
mapping $x\mapsto \mu_x $ from $\Omega $ into the set of non-negative,
finite Radon measures on $\mathbb{R}^d $, such that
\begin{itemize}
\item[(i)] $   \mu = \int_{\Omega }\delta
_{x}\otimes \mu _x dx,$ i.e. for any Borel function $f: \Omega
\times \mathbb{R}^d \to [0, +\infty ]$ the function $\displaystyle x
\mapsto \int_{\mathbb{R}^d }f(x,A)d\mu _x (A)$ is
$\mathcal{L}^{N}$-measurable, and
\begin{equation}\label{intfdmuintermsofmux}
\int_{\Omega \times \mathbb{R}^d }f d\mu = \int_{\Omega
}\int_{\mathbb{R}^d }f(x,A)d\mu _x (A) dx;
\end{equation}

\item[(ii)] $\mu _x (\mathbb{R}^d ) = 1$,
 for $\mathcal{L}^N$-a.e. $x \in \Omega$.
\end{itemize}
Moreover, if $x\mapsto \nu _x $ is another such mapping then $\nu_x
= \mu _x$ for $\mathcal{L}^N$-a.e. $x \in \Omega$.
\end{proposition}

\begin{remark} \rm
Proposition \ref{bourbaki} is a special case of a result in
\cite{Bourbaki:1969} (Proposition 13, pp. 39-40). See also
\cite{AmbrosioFuscosiPallara}.
\end{remark}

Consider a sequence $\{ V_n \}$ of measurable mappings of
$\Omega $ into $\mathbb{R}^d $. The corresponding sequence
$\{ \mathcal{E}_{V_n } \} $ of elementary Young measures is bounded in
$C_{0}(\Omega \times \mathbb{R} ^d )'$ and thus, by virtue
of Banach-Alaoglu's Theorem, there exists a subsequence
$\{ V_{n_k } \}$ and a measure
$\mu \in C_{0}(\Omega \times \mathbb{R} ^d )'$ such that
\begin{equation}\label{elemYmconverge}
\mathcal{E}_{V_{n_k }} \rightharpoonup \mu  \ {\rm weakly * \ in }
\ C_{0}(\Omega \times \mathbb{R} ^d )'.
\end{equation}
A necessary and sufficient condition for $\mu $ to be a Young
measure is that
\begin{equation}\label{muisaYm}
\lim_{R\to \infty }\sup_{k \in
\mathbb{N}}\mathcal{L}^{N} \left( \{ x \in \Omega : |V_{n_k }(x)|
\geq R \} \right) = 0,
\end{equation}
or, equivalently (see \cite{Hungerbuhler:1997, Kr:1994}):
There  exists  a  Borel  function
$g: \mathbb{R}^d \to [0, + \infty ]$  such  that
$ \lim_{|A| \to +\infty}g(A) = +\infty$,
 and
\[
 \sup_{k \in \mathbb{N}}\int_{\Omega }g(V_{n_k }(x))dx < +\infty .
\]

\begin{definition}\label{defofaYmgenbyasequence} \rm
If (\ref{elemYmconverge}) and (\ref{muisaYm}) hold, then we say
that the Young measure $\mu $ is generated by the sequence
$\{V_{n_k } \}$.
\end{definition}

\begin{definition}\label{defofnormalandCaratheodoryintegrands} \rm
\begin{itemize}
\item[(i)] A function $f: \Omega \times \mathbb{R}^d \to
\mathbb{R}\cup \{ + \infty \}$ is called a normal integrand if $f$
is Borel measurable and $f(x,\cdot ) : \mathbb{R}^d \to
\mathbb{R}\cup \{ + \infty \} $ is lower semicontinuous for every
$x \in \Omega $.

\item[(ii)] A real-valued function $f: \Omega \times
\mathbb{R}^d \to \mathbb{R} $ is called a Carath\'{e}odory
integrand if both $f$ and $-f$ are normal integrands.
\end{itemize}
\end{definition}

Set $f^- := -\min \{f, 0\} .$ The following result is well-known
(see \cite{Bal:1984,Ball:1989,BerliocchiandLasri:1973,FL:2001,
KinderlehrerPedregal:1994,Kr:1994,Kr:1999,Pedregal'sbook:1997}).

\begin{lemma} \label{propertiesofYm}
 Let $\{ v_n \}$ be a sequence of
measurable mappings from $\Omega $ into $\mathbb{R}^d $ which
generates the Young measure $\mu $.
\begin{itemize}
\item[(i)] If $f: \Omega \times \mathbb{R}^d \to \mathbb{R}\cup
\{ + \infty \}$ is a normal integrand and if  $\{ f^- (\cdot , v_n
)\} $ is equi-integrable then
\[
\int_{\Omega \times \mathbb{R}^d }f d\mu  \leq
\liminf_{n \to \infty }\int_{\Omega }f(x, v_n (x))dx.
\]
Moreover, if $f$ is a Carath\'{e}odory integrand then $\{ f(\cdot
, v_n )\} $ is equi-integrable if and only if
\[
\int_{\Omega \times \mathbb{R}^d }f d\mu  = \lim_{n \to
 \infty }\int_{\Omega }f(x, v_n (x))dx.
\]

\item[(ii)] If $\{ w_n \} $ is a sequence of measurable
mappings from $\Omega $ into $\mathbb{R}^d $ such that $v_n - w_n \to
0 $ in measure then $\{ w_n \} $ also generates $\mu $.
\end{itemize}
\end{lemma}

\section{Proof of Theorem \ref{relaxationtheorem}}
\noindent We will identify $\mathbb{R}^9 $ with the space of real
$3\times 3$ matrices $\mathbb{M}^{3\times 3}$. To prove that
\begin{equation}\label{primainegalitate}
\mathcal{W}(v,c) \geq \inf_{\nu \in
Y_{v,c}^{p}}\int_{\Omega \times \mathbb{M}^{3\times 3}}Wd\nu ,
\end{equation}
let $\varepsilon_n \to 0^+$ and $\{v_n \}\subset W^{1,p}(\Omega;\mathbb{R} ^3) $ be such that
$v_n \rightharpoonup v $ weakly in $W^{1,p}(\Omega;\mathbb{R} ^3)$ and $\frac{1}{\varepsilon
_n}\nabla_3 v_n \rightharpoonup c$ weakly in $L^p(\Omega;\mathbb{R}^3) $. Extract a
subsequence (not relabelled) so that
\begin{equation}\label{liminfisalimatstartofpfofrelaxthm}
\begin{aligned}
& \liminf_{n \to \infty }\int_{\Omega }W\left(x, \left(
\nabla_\alpha  v_n \Big| \frac{1}{\varepsilon_n }\nabla_3 v_n \right)(x) \right) dx\\
& = \lim_{n \to \infty }\int_{\Omega }W\left(x, \left( \nabla_\alpha
v_n \Big| \frac{1}{\varepsilon_n }\nabla_3 v_n \right)(x) \right) dx
< + \infty,
\end{aligned}
\end{equation}
where the last inequality follows by (\ref{pgrowthandcoercforW}). By
Banach-Alaoglu's Theorem, there exists a subsequence $\big\{\big(
\nabla_\alpha  v_{n_j } \big| \frac{1}{\varepsilon_{n_j } }\nabla_3
v_{n_j } \big) \big\}$ of $\big\{ \big( \nabla_\alpha  v_n \big|
\frac{1}{\varepsilon _n }\nabla_3 v_n \big) \big\} $ such that
\[
\mathcal{E}_{\left(\nabla_\alpha  v_{n_j } \big| \frac{1}{\varepsilon_{n_j } }\nabla_3
v_{n_j } \right) } \rightharpoonup \mu  \quad \text{weakly *  in }
C_{0}(\Omega \times \mathbb{R} ^9 )',
\]
for some Radon measure $\mu $ on $\Omega \times \mathbb{R}^9 $. Since
\[
\sup_{j\in \mathbb{N}} \int_{\Omega } \Big| \left( \nabla
_{\alpha}v_{n_j } \Big| \frac{1}{\varepsilon_{n_j }}\nabla
_{3}v_{n_j }\right)(x)\Big| ^{p}dx < +\infty ,
\]
it follows that (\ref{muisaYm}) holds, with $k$ replaced by $j$,
and $V_{n_j }$ by $\left( \nabla _{\alpha}v_{n_j } \big|
\frac{1}{\varepsilon_{n_j }}\nabla _{3}v_{n_j }\right)$. Thus, $\mu $
is a Young measure. It is clear that we actually have $\mu \in
Y_{v,c}^{p}$. By (\ref{pgrowthandcoercforW}), $W(x,A) + C \geq 0 $
for $ \mathcal{L}^3$-a.e. $x\in \Omega $ and all $A \in
\mathbb{M}^{3\times 3}$. Thus, we can apply Lemma
\ref{propertiesofYm} (i) (take $f = W + C$) and we obtain
\begin{equation*}
\begin{aligned}
\int_{\Omega \times \mathbb{M}^{3\times 3}}( W + C ) d\mu & \leq
\liminf_{j \to \infty }\int_{\Omega }\left( W\left(x, \left(
\nabla_\alpha v_{n_j } \Big| \frac{1}{\varepsilon_{n_j } }\nabla_3
v_{n_j } \right)(x) \right) + C\right)
dx\\
& =  \lim_{j \to \infty }\int_{\Omega }W\left(x, \left(
\nabla_\alpha v_{n_j } \Big| \frac{1}{\varepsilon_{n_j } }\nabla_3
v_{n_j }
\right)(x) \right)dx + C\mathcal{L}^3 (\Omega )\\
& =  \lim_{n \to \infty }\int_{\Omega }W\left(x, \left(
\nabla_\alpha v_{n } \Big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n}
\right)(x) \right)dx + C\mathcal{L}^3 (\Omega ),
\end{aligned}
\end{equation*}
where we have used (\ref{liminfisalimatstartofpfofrelaxthm}). By
Proposition \ref{bourbaki} we have
\[
\int_{\Omega \times \mathbb{M}^{3\times 3}} C d\mu =
\int_{\Omega }\int_{\mathbb{M}^{3\times 3}}C d\mu _x dx
= C \int_{\Omega }\mu _x (\mathbb{M}^{3\times 3})dx = C \mathcal{L}^3
(\Omega ),
\]
and in view of the previous equation we obtain
\[
\int_{\Omega \times \mathbb{M}^{3\times 3}}W d\mu \leq \liminf_{n
\to \infty }\int_{\Omega }W\left(x, \left( \nabla_\alpha  v_{n }
\Big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n} \right)(x) \right)dx.
\]
Thus,
\[
\inf_{\nu \in Y_{v,c}^{p}}\int_{\Omega \times \mathbb{M}^{3\times
3}}Wd\nu \leq \liminf_{n \to \infty }\int_{\Omega }W\left(x, \left(
\nabla_\alpha  v_{n } \Big| \frac{1}{\varepsilon _{n} }\nabla_3
v_{n} \right)(x) \right)dx.
\]
By the arbitrariness of $\{ \varepsilon_n  \}$ and $\{ v_n  \}$
satisfying the admissibility conditions we assert
(\ref{primainegalitate}).

Conversely, let $\nu \in Y_{v,c}^{p}$. There exist sequences $\varepsilon
_n \to 0^+$ and $\{v_n \}\subset W^{1,p}(\Omega;\mathbb{R} ^3) $ such that $ v_n
\rightharpoonup v $ weakly in $W^{1,p}(\Omega;\mathbb{R} ^3) , \ \frac{1}{\varepsilon_n}\nabla_3 v_n
\rightharpoonup c$ weakly in $L^p(\Omega;\mathbb{R}^3),$ and the sequence $ \{
( \nabla_\alpha  v_{n } \big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n} )\} $ generates $\nu $.

By Theorem \ref{epsilondecompositionlemma} there exist a subsequence
$\lbrace v_{n_k} \rbrace $ of $ \lbrace v_n \rbrace $ and a sequence
$\lbrace w_k \rbrace \subset W^{1,p}(\Omega; \mathbb{R} ^3) $ such
that
(\ref{(i)epsilondecompositionlemma})-(\ref{(iv)epsilondecompositionlemma})
hold. In view of (\ref{(i)epsilondecompositionlemma}), the sequence
\[
 \left\{ \left( \nabla_\alpha  w_{k } \Big|
\frac{1}{\varepsilon_{n_k } }\nabla_3 w_k \right) - \left(
\nabla_\alpha v_{n_k } \Big| \frac{1}{\varepsilon_{n_k } }\nabla_3
v_{n_k } \right) \right\}
\]
converges to zero in measure and thus, by  Lemma
\ref{propertiesofYm} (ii),  $\{( \nabla_\alpha  w_{k } \big|
\frac{1}{\varepsilon_{n_k } }\nabla_3 w_k )\} $ also generates $
\nu $. Since by (\ref{pgrowthandcoercforW}) and
(\ref{(ii)epsilondecompositionlemma}) the sequence
$ \Big\{ W\Big( \cdot , \big( \nabla_\alpha  w_{k } \big| \frac{1}{\varepsilon_{n_k }}\nabla_3
w_k \big) \Big) \Big\} $ is equi-integrable, we deduce by
(i) of Lemma \ref{propertiesofYm} that
\[
\int_{\Omega \times \mathbb{M}^{3\times 3}}W d\nu = \lim_{k \to
\infty }\int_{\Omega }W\left(x, \left( \nabla_\alpha w_{k } \Big|
\frac{1}{\varepsilon_{n_k } }\nabla_3 w_k  \right)(x) \right) dx
\geq \mathcal{W}(v,c),
\]
where the last inequality follows by
(\ref{(iii)epsilondecompositionlemma}) and
(\ref{(iv)epsilondecompositionlemma}). Passing to the infimum over
all $\nu \in Y_{v,c}^{p}$ we have that
\[
\inf_{\nu \in Y_{v,c}^{p}}\int_{\Omega \times
\mathbb{M}^{3\times 3}}Wd\nu \geq \mathcal{W}(v,c).
\]
Taking into account (\ref{primainegalitate}), we obtain that
(\ref{equalityofinfsinrelaxationthm}) holds. It remains to prove
(\ref{infimumisattainedatmu0}).
 \smallskip

\noindent \textbf{Claim:}  There exist sequences
$\varepsilon_n \to 0^+$ and $\{v_n \}\subset W^{1,p}(\Omega;\mathbb{R} ^3) $ such that
\begin{equation}  \label{Wcaligraphic(v,b)is a limit}
\begin{gathered}
v_n \rightharpoonup v  \quad \text{weakly   in  } W^{1,p}(\Omega;\mathbb{R} ^3), \\
\frac{1}{\varepsilon_n}\nabla_3 v_n \rightharpoonup c \quad \text{weakly  in } L^p(\Omega;\mathbb{R}^3),
\\
\lim_{n \to \infty}\int_{\Omega }W\left(x, \left(\nabla_\alpha v_{n
} \Big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n} \right)(x) \right)
dx = \mathcal{W}(v,c).
\end{gathered}
\end{equation}
Assuming that the claim holds, let $\mu _0$ be the scaled gradient
$p$-Young measure generated by a subsequence (not
relabelled) of $\{ (\nabla_\alpha  v_{n } \big| \frac{1}{\varepsilon_{n}
}\nabla_3 v_{n} ) \}$. Let $\{n_k \} \subset \{ n \}$ and
$\{ w_k \} \subset W^{1,p}(\Omega; \mathbb{R}^3)$ be the sequences
provided by Theorem \ref{epsilondecompositionlemma}. Taking into
account
(\ref{(i)epsilondecompositionlemma})-(\ref{(iv)epsilondecompositionlemma}),
(\ref{Wcaligraphic(v,b)is a limit}), and making use of Lemma
\ref{propertiesofYm}, we have
\begin{align*}
\mathcal{W}(v,c) & \leq \liminf_{k \to \infty }\int_{\Omega
}W\left(x, \left( \nabla_\alpha  w_k \Big| \frac{1}{\varepsilon
_{n_k }}\nabla_3 w_k \right)(x) \right) dx\\
&  = \int_{\Omega \times \mathbb{M}^{3\times 3}}W d\mu_0\\
& \leq \lim_{n \to \infty }\int_{\Omega }W\left(x, \left(
\nabla_\alpha v_{n } \Big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n}
\right)(x) \right)dx\\
& = \mathcal{W}(v,c).
\end{align*}
Thus,
\[
\int_{\Omega \times \mathbb{M}^{3\times 3}}W d\mu _0 =
\mathcal{W}(v,c),
\]
and in view of (\ref{equalityofinfsinrelaxationthm}), we deduce
(\ref{infimumisattainedatmu0}).

\begin{proof}[Proof of Claim:]
 For any $n \in \mathbb{N}$,
let $\{ \varepsilon_{k,n} \} \subset (0, + \infty )$ and
$\{v_{k,n}\} \subset W^{1,p}(\Omega;\mathbb{R} ^3) $ be such that
$\lim_{k\to \infty }\varepsilon_{k,n}= 0$, $v_{k,n}\rightharpoonup v
$ weakly in $W^{1,p}(\Omega;\mathbb{R} ^3),
 \frac{1}{\varepsilon_{k,n}}\nabla_3 v_{k,n} \rightharpoonup c$
weakly in $L^p(\Omega;\mathbb{R}^3) $ as $k \to \infty $ and, in
addition,
\[
\mathcal{W}(v,c)\leq \liminf_{k \to \infty }\int_{\Omega }W\left(x,
\left( \nabla_\alpha  v_{k,n} \Big| \frac{1}{\varepsilon_{k,n}
}\nabla_3 v_{k,n} \right)(x) \right) dx \leq \mathcal{W}(v,c) +
\frac{1}{n}.
\]
Extract an increasing subsequence $\{k(j,n)\} _j $ of $\{ k \} $
so that
\begin{align*}
&\liminf_{k \to \infty }\int_{\Omega }W\left(x, \left( \nabla_\alpha
v_{k,n} \Big| \frac{1}{\varepsilon_{k,n} }\nabla_3 v_{k,n}
\right)(x) \right) dx\\
& = \lim_{j \to \infty }\int_{\Omega }W\left(x, \left( \nabla_\alpha
v_{k(j,n),n} \Big| \frac{1}{\varepsilon_{k(j,n),n} }\nabla_3
v_{k(j,n),n} \right)(x) \right) dx,
\end{align*}
and put $\overline{\varepsilon}_{j,n}:= \varepsilon_{k(j,n),n} $ and
$\overline{v}_{j,n} := v_{k(j,n),n}.$ Thus,
\begin{gather}\label{(1)}
\lim_{j\to \infty }\overline{\varepsilon}_{j,n} = 0,
\\  \label{(2)}
\lim_{n \to \infty}\lim_{j \to \infty}\int_{\Omega }W\left(x, \left(
\nabla_\alpha  \overline{v}_{j,n} \Big|
\frac{1}{\overline{\varepsilon}_{j,n}}\nabla_3 \overline{v}_{j,n}
\right)(x) \right) dx = \mathcal{W}(v,c),
\end{gather}
$\overline{v}_{j,n}\rightharpoonup v$  weakly  in $W^{1,p}(\Omega;\mathbb{R} ^3)$
as $j \to \infty$,
and
\[
\frac{1}{\overline{\varepsilon}_{j,n}}\nabla_3 \overline{v}_{j,n}
\rightharpoonup c \quad\text{ weakly  in $L^p(\Omega;\mathbb{R}^3)$ as
$j \to \infty$}.
\]
Consider a countable family $\{\varphi_i\}_{i \in \mathbb{N}}$
dense in $L^{p'}(\Omega)$. The weak convergence of
$\overline{v}_{j,n}$ to $v$ in $W^{1,p}(\Omega;\mathbb{R} ^3)$ and that of
$\frac{1}{\overline{\varepsilon}_{j,n}}\nabla_3 \overline{v}_{j,n}$ to $c$ in
$L^p(\Omega;\mathbb{R}^3)$ imply that for each $i,n \in \mathbb{N}$ we have
\begin{gather}\label{(3)}
\lim_{j \to \infty}\int_{\Omega}\varphi_i(x)\overline{v}_{j,n}(x)dx =
\int_{\Omega}\varphi_i(x)v(x)dx,
\\ \label{(4)}
\lim_{j \to \infty}\int_{\Omega}\varphi_i(x)
\Big(\frac{1}{\overline{\varepsilon}_{j,n}}\nabla_3
\overline{v}_{j,n}(x)\Big)dx = \int_{\Omega}\varphi_i(x)c(x)dx,
\\  \label{(5)}
\lim_{j \to \infty}\int_{\Omega}\varphi_i(x)\nabla
\overline{v}_{j,n}(x)dx = \int_{\Omega}\varphi_i(x)\nabla
v(x)dx.
\end{gather}
Taking into account (\ref{(1)}), (\ref{(2)}), (\ref{(3)}),
(\ref{(4)}), and (\ref{(5)}), a diagonalization process allows us
to find an increasing subsequence $\{j(n)\}$ of $\{j\}$ such that,
after denoting $\varepsilon_n := \overline{\varepsilon}_{j(n),n}$ and
$v_n := \overline{v}_{j(n),n}$ we have
\begin{gather*}
\varepsilon_n \to 0^+, \quad v_n \rightharpoonup v \quad
\text{weakly  in }W^{1,p}(\Omega;\mathbb{R} ^3), \\
\frac{1}{\varepsilon_n}\nabla_3 v_n \rightharpoonup c \quad\text{weakly  in } L^p(\Omega;\mathbb{R}^3),
\end{gather*}
and (\ref{Wcaligraphic(v,b)is a limit}) holds.
\end{proof}

\subsection*{Acknowledgement} Part of this work has
been written while the author was supported by a Burgess Assistant
Professorship at the University of Utah.

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\end{document}