\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $1 0$, 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 $10$ 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 \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} 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 $10$ 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 $$\label{intfdmuintermsofmux} \int_{\Omega \times \mathbb{R}^d }f d\mu = \int_{\Omega }\int_{\mathbb{R}^d }f(x,A)d\mu _x (A) dx;$$ \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 $$\label{elemYmconverge} \mathcal{E}_{V_{n_k }} \rightharpoonup \mu \ {\rm weakly * \ in } \ C_{0}(\Omega \times \mathbb{R} ^d )'.$$ A necessary and sufficient condition for $\mu$ to be a Young measure is that $$\label{muisaYm} \lim_{R\to \infty }\sup_{k \in \mathbb{N}}\mathcal{L}^{N} \left( \{ x \in \Omega : |V_{n_k }(x)| \geq R \} \right) = 0,$$ 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 $$\label{primainegalitate} \mathcal{W}(v,c) \geq \inf_{\nu \in Y_{v,c}^{p}}\int_{\Omega \times \mathbb{M}^{3\times 3}}Wd\nu ,$$ 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 \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} 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 $$\label{Wcaligraphic(v,b)is a limit} \begin{gathered} v_n \rightharpoonup v \quad \text{weakly in } W^{1,p}(\Omega;\mathbb{R} ^3), \\ \frac{1}{\varepsilon_n}\nabla_3 v_n \rightharpoonup c \quad \text{weakly in } L^p(\Omega;\mathbb{R}^3), \\ \lim_{n \to \infty}\int_{\Omega }W\left(x, \left(\nabla_\alpha v_{n } \Big| \frac{1}{\varepsilon_{n} }\nabla_3 v_{n} \right)(x) \right) dx = \mathcal{W}(v,c). \end{gathered}$$ 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. \begin{thebibliography}{00} \bibitem{AmbrosioFuscosiPallara} Ambrosio, L., Fusco, N., Pallara, D., \emph{Functions of Bounded Variation and Free Discontinuity Problems}, Oxford University Press, Oxford, 2000. \bibitem{AnzBalPer:1994} Anzelotti, E., Baldo, S., Percivale, D., Dimensional reduction in variational problems, asymptotic developments in $\Gamma$-convergence, and thin structures in elasticity. \emph{Asymptotic Anal.} {\bf 9} (1994), 61-100. \bibitem{Bal:1984} Balder, E. J., A general approach to lower semicontinuity and lower closure in optimal control theory. \emph{ SIAM J. Control Opt.} {\bf 22} (1984), 570-598. \bibitem{Ball:1989} Ball, J.M., \emph{A version of the fundamental theorem for Young mesures,} in PDE's and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., Lecture Notes in Phys. 344, Springer-Verlag, Berlin (1989), 207-215. \bibitem{BerliocchiandLasri:1973} Berliocchi, H., Lasry, J.-M., Int\'{e}grands normales et mesures param\'{e}tr\'{e}es en calcul des variations. \emph{Bull. Soc. Math. France.} {\bf 101} (1973), 129-184. \bibitem{BhattaFonsecaFrancfort:1999} Bhattacharya, K., Fonseca, I., Francfort, G., An asymptotic study of the debonding of thin films. \emph{Arch. Rat. Mech. Anal.} {\bf 161} (2002), 205-229. \bibitem{BhattaJames:1999} Bhattacharya, K., James, R.D., A theory of thin films of martensitic materials with applications to microactuators. \emph{J. Mech. Phys. Solids} {\bf 47} (1999), 531-576. \bibitem{BoceaFonsecaFebr2002} Bocea, M., Fonseca, I., Equi-integrability results for 3D-2D dimension reduction problems. \emph{ESAIM: Control, Optimisation and Calculus of Variations} {\bf 7} (2002), 443-470. \bibitem{BoceaFonseca2004} Bocea, M., Fonseca, I., A Young measure approach to a nonlinear membrane model involving the bending moment. \emph{Royal Society of Edinburgh Proceedings A} {\bf 134} No. 5 (2004), 845-883. \bibitem{Bourbaki:1969} Bourbaki, N., Integration, Chap. IX, \'{E}l\'{e}ments de Math\'{e}matique, Hermann, Paris, 1969. \bibitem{BraidesFonsecaFrancfort:2000} Braides, A., Fonseca, I., Francfort, G., 3D-2D asymptotic analysis for inhomogeneous thin films. \emph{Indiana Univ. Math. J.} {\bf 49} (2000), 1367-1404. \bibitem{BraidesFonseca:2001} Braides, A., Fonseca, I., Brittle thin films. \emph{Applied Math. and Optimization} {\bf 44} (2001), 299-323. \bibitem{FonsecaFrancfort:2001} Fonseca, I., Francfort, G., On the inadequacy of scaling of linear elasticity for 3D-2D asymptotics in a nonlinear setting. \emph{J. Math. Pures Appl.} {\bf 80} (2001), 547-562. \bibitem{FL:2001} Fonseca, I., Leoni, G., \emph{Modern Methods in the Calculus of Variations with Applications to Nonlinear Continuum Physics.} Springer-Verlag, to appear. \bibitem{FMaquasiconvexity} Fonseca, I., M{\"{u}}ller, S.: $A$-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. {\bf 30} (1999), 1355-1390. \bibitem{FMP} Fonseca, I., M\"{u}ller, S., Pedregal, P., Analysis of concentration and oscillation effects generated by gradients. \emph{ SIAM J. Math. Anal. } {\bf 29} (1998), 736-756. \bibitem{FoxRaoultandSimo:1993} Fox, D.D., Raoult, A., Simo, J.C., A justification of nonlinear properly invariant plate theories. \emph{Arch. Rat. Mech. Anal.} {\bf 124} (1993), 157-199. \bibitem{Hungerbuhler:1997} Hungerb\"{u}ler, N., A Refinement of Ball's Theorem on Young Measures. \emph{New York J. Math.} {\bf 3} (1997), 48-53. \bibitem{KinderlehrerPedregal:1991} Kinderlehrer, D., Pedregal, P., Characterizations of Young mesures generated by gradients. \emph{Arch. Rat. Mech. Anal.} {\bf 115} (1991), 329-365. \bibitem{KinderlehrerPedregal:1994} Kinderlehrer, D., Pedregal, P., Gradient Young mesures generated by sequences in Sobolev spaces. \emph{J. Geom. Anal.} {\bf 4} (1994), 59-90. \bibitem{Kr:1994} Kristensen, J., Finite functionals and Young measures generated by gradients of Sobolev functions, \emph{Mat. Report 1994-34,} Mathematical Institute, Technical University of Denmark, Lyngby, Denmark, 1994. \bibitem{Kr:1999} Kristensen, J., Lower semicontinuity in spaces of weakly differentiable functions. \emph{Math. Ann.} {\bf 313} (1999), 653-710. \bibitem{LeDretRaoult:1995} Le Dret, H., Raoult, A., The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. \emph{J. Math. Pures Appl.} {\bf 74} (1995), 549-578. \bibitem{LeDretRaoult:2000} Le Dret, H., Raoult, A., Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. \emph{Arch. Rat. Mech. Anal.} {\bf 154} (2000), 101-134. \bibitem{Pedregal'sbook:1997} Pedregal, P., \emph{Parametrized mesures and Variational Principles,} Birkh\"{a}user, Boston, 1997. \bibitem{Tartarheriotwatt:1979} Tartar, L., \emph{Compensated compactness and applications to partial differential equations,} in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, R. Knops, ed., Vol. IV, Pitman Res. Notes Math., 39, Longman, Harlow, U.K. (1979), 136-212. \bibitem{Tartarconslaws:1983} Tartar, L., \emph{The compensated compactness method applied to systems of conservation laws, } in Systems of Nonlinear Partial Differential Equations, J. M. Ball, ed., D. Riebel, Dordrecht, 1983 \bibitem{Tartaroscillations:1984} Tartar, L., \emph{\'{E}tude des oscillations dans les \'{e}quations aux d\'{e}riv\'{e}es partielles nonlin\'{e}aires,} in Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Phys. 195, Springer-Verlag, Berlin, New York (1984), 384-412. \bibitem{Young:1937} Young, L.C., Generalized curves and the existence of an attained absolute minimum in the calculus of variations. \emph{Comptes Rendus de la Soci\'{e}t\'{e} des Sciences et des Lettres de Varsovie, classe III. } {\bf 30} (1937), 212-234. \bibitem{Young:1969} Young, L.C., \emph{Lectures on the calculus of variations and optimal control theory.} Saunders, 1969 (reprinted by Chelsea 1980). \end{thebibliography} \end{document}