Regularity of intrinsically convex $H^2$ surfaces and the derivation of homogenized bending shell models
Peter Hornung
FB Mathematik
TU Dresden
01062 Dresden (Germany)
Igor Vel\v ci\'c
University of Zagreb
Faculty of Electrical Engineering and Computing
Unska 3
Zagreb (Croatia)
We prove smoothness of $H^2$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We use this regularity result to rigorously derive homogenized bending models of convex shells from three-dimensional nonlinear elasticity.
1506.02571.tex
\documentclass[12pt, a4paper]{article}

\usepackage{amsfonts, latexsym, amssymb, amsthm, amsmath, mathrsfs, esint, color}
\usepackage[latin1]{inputenc}
\usepackage{enumerate}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage{enumerate}
\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{esint}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{relsize}
\usepackage{hyperref}
\usepackage{comment}

\def\bs{}
\newcommand{\U}{\mathcal U}\newcommand{\FF}{\mathcal F}
\newcommand{\n}{^{(n)}}
\newcommand{\zbar}{\overline{z}}
\newcommand{\p}{\varphi}
\newcommand{\la}{\lambda}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand{\D}{\nabla}
\renewcommand{\L}{\mathcal{L}}
\def\embed{\hookrightarrow}
\def\d{\partial}
\def\Rtilde{\tilde{R}}
\newcommand{\h}{\widehat}
\def\e{\varepsilon}
\def\II{\mathrm{II}}
\def\xbar{\bar{x}}
\def\xtilde{\tilde{x}}
\def\stilde{\tilde{s}}
\def\dt{\tilde{\delta}}
\def\Rt{\tilde{R}}
\def\ut{\tilde{u}}
\def\gb{\bar{\gamma}}
\def\Nt{\tilde{N}}
\def\k{^{(k)}}
\def\hk{\hat{\kappa}}
\def\bk{\bar{\kappa}}
\def\bK{\bar{K}}
\def\vk{\vec{\kappa}}
\def\sgn{\text{ sgn }}
\def\diag{\text{diag }}
\def\weaks{\stackrel{*}{\rightharpoonup}}
\def\gt{\tilde{\gamma}}

\def\down{\downarrow}
\def\weak{\rightharpoonup}
\def\RR{\mathbb{R}}
\renewcommand{\t}{\widetilde}
\newcommand{\R}{\mathbb{R}}
\def\QQ{\mathbb{Q}}
\def\CC{\mathbb{C}}
\def\NN{\mathbb{N}}
\def\HH{\mathcal{H}}
\def\C{\mathcal{C}}
\def\GG{\mathcal{G}}
\def\FF{\mathcal{F}}
\def\MM{\mathcal{M}}
\def\LL{\mathcal{L}}
\def\AA{\mathcal{A}}
\def\BB{\mathcal{B}}
\def\VV{\mathcal{V}}
\def\TT{\mathcal{T}}
\def\SS{\mathbb{S}}
\def\DD{\mathcal{D}}
\def\W{\mathcal{W}}
\def\WW{\mathcal{W}}
\newcommand{\Wtil}{\widetilde{\W}}
\newcommand\calY{\mathcal{Y}}
\newcommand{\Y}{\mathcal Y}


\newcommand{\SM}{\mathcal{S}M}
\newcommand\wto{\rightharpoonup}
\newcommand\tto{\stackrel{2}{\to}}
\newcommand\wtto{\xrightharpoonup{2}} 
\newcommand\otto{\xrightharpoonup{osc}}
\newcommand\stto{\xrightarrow{2}}
\newcommand\hx{x}
\renewcommand\boldsymbol{}

\newcommand{\bbv}{b_V}
\newcommand\ya{y}
\newcommand\ua{u}
\newcommand\phia{\phi} 
\newcommand\ra{r}
\newcommand\wa{w}
\newcommand\na{n}
\newcommand\aaa{a}
\newcommand\ba{b}
\newcommand\ca{c}
\newcommand\da{d}
\newcommand\ga{g}
\newcommand\va{v}
\newcommand\oa{o}
%\newcommand\la{l}
\newcommand\ma{m}
\newcommand\za{z}
\newcommand\ea{e}
\newcommand\qa{q}
\newcommand\ta{t}
\newcommand\fa{f}
\newcommand\pa{p}
%\newcommand\weak{\rightharpoonup}
\renewcommand{\t}{\widetilde}
\renewcommand{\o}{\tilde}




%\newcommand{\p}{\varphi}
\renewcommand{\d}{\partial}
\newcommand{\dd}{\partial}
%\renewcommand{\labelitemi}{--}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\O}{\Omega}
\newcommand\Oo{{O}}



%\usepackage[active]{srcltx}
\DeclareMathOperator{\Defy}{Def_{\mathcal Y}}
\DeclareMathOperator{\Def}{Def}
\DeclareMathOperator{\Hess}{Hess}
\DeclareMathOperator{\Hessy}{Hess_{\mathcal Y}}
%\newcommand{\Y}{\mathcal Y}




\newcommand\I{Id}
%\newcommand\QQ{T^*S\otimes T^*S}
\newcommand\B{B}
\newcommand\Q{Q}
\newcommand\G{G}
\newcommand\V{V}
\newcommand\A{A}
\newcommand\Psib{\Psi}
%\newcommand\R{\mathbb{R}}
\newcommand\M{\mathbb{M}}
\newcommand\N{\mathbb{N}}
\newcommand\Z{\mathbb{Z}}
\renewcommand\L{\mathbb{L}}
\newcommand\taub{\tau}
\newcommand\etab{\eta}
\newcommand\xib{\xi}
\newcommand\thetab{\theta}
\newcommand\F{F}
\newcommand\Ma{M}
\newcommand\sigmab{\sigma}
\newcommand\K{K}
\newcommand\Ra{R}
%\newcommand\W{W}
\renewcommand\AA{{\mathbf S}}
%\newcommand{\HH}{\mathcal{H}}
%\newcommand\D{\nabla}
\newcommand{\super}[1]{^{(#1)}}
\newcommand\zetaa{{\boldsymbol \zeta}}
%\newcommand\otto{\xrightharpoonup{osc}}
%\newcommand\stto{\xrightarrow{2}}
%\newcommand\hx{x}
\renewcommand\boldsymbol{}

\renewcommand{\t}{\widetilde}
\renewcommand{\o}{\tilde}

\newcommand\calA{\mathcal A}
\newcommand\calN{\mathcal N}
\newcommand\calE{\mathcal E}
\newcommand\calG{\mathcal G}
\newcommand\calB{\mathcal B}
\newcommand\calL{\mathcal L}
\newcommand\calI{\mathcal I}
\newcommand\calW{\mathcal{W}}
\newcommand\calQ{\mathcal{Q}}
%\newcommand\calY{\mathcal{Y}}
\newcommand\calF{\mathcal{F}}
\newcommand\ttan{\textrm{tan}}
\newcommand\ts{\textrm{ts}}
\newcommand\Err{\textrm{\textit{Error}}}


\newcommand\unfold{\mathcal T}
\newcommand{\SO}[1]{\operatorname{SO}(#1)}
\newcommand\id{I}
\newcommand\sym{\operatorname{sym}}
\newcommand\skw{\operatorname{skw}}
\newcommand\trace{\operatorname{trace}}
\newcommand\transpose{\textrm{t}}
\newcommand\rot{\mathop{\operatorname{rot}}}
\newcommand\meas{\operatorname{meas}}
\newcommand{\so}[1]{\operatorname{so}(#1)}


%\newcommand\wto{\rightharpoonup}
%\newcommand\tto{\stackrel{2}{\to}}
%\newcommand\wtto{\xrightharpoonup{2}}
%\newcommand\stto{\xrightarrow{2,\gamma}}

\newcommand\esssup{\mathop{\operatorname{ess\,sup}}}
\newcommand\ud{\,\mathrm{d}}
\newcommand\supp{\mathop{\operatorname{supp}}}
\newcommand\dist{\operatorname{dist}}
%\newcommand{\e}{\varepsilon}
\newcommand\iso{\textrm{iso}}
% \newcommand{\ho}{\mathrm{hom}}




\DeclareMathOperator{\osc}{osc}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\divg}{div_g}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\diam}{diam}
%\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\spt}{spt}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\co}{co}
\DeclareMathOperator{\Lip}{Lip}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\ran}{ran}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\clos}{clos}
\DeclareMathOperator{\cof}{cof}


\newcommand{\step}[1]{\noindent \textbf{Step #1.}}

\newcommand\ho{\textrm{hom}}
\newcommand\loc{\textrm{loc}}
\newcommand\ee{\mathbf e}














\def\D{\nabla}
\def\a{\alpha}
\def\b{\beta}

\def\Xint#1{\mathchoice
   {\XXint\displaystyle\textstyle{#1}}%
   {\XXint\textstyle\scriptstyle{#1}}%
   {\XXint\scriptstyle\scriptscriptstyle{#1}}%
   {\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
   \!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$}
     \vcenter{\hbox{$#2#3$}}\kern-.5\wd0}}
\def\ddashint{\Xint=}
\def\dint{\Xint-}





 \setlength{\parindent}{0em}

\newcounter{bei}

\renewcommand{\S}{\mathbb{S}}
%\newcommand{\B}{\mathcal{B}}
\renewcommand{\o}{\overline}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\vect}[2]{\begin{pmatrix} {#1}\\{#2} \end{pmatrix}}
\newcommand{\zwo}[2]{\begin{pmatrix} {#1}\\{#2} \end{pmatrix}}
\newcommand{\al}[1]{\left\langle #1 \right\rangle}
\newcommand{\sect}{\par\noindent\refstepcounter{bei}{\bf\thebei. }}
\newcommand{\Leq}{\stackrel{C}{\leq}}
\renewcommand{\div}{\mathrm{div\ }}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}



\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\graph}{graph}
\DeclareMathOperator{\End}{End}


\begin{document}
\title{Regularity of intrinsically convex $H^2$ surfaces
and the derivation of homogenized bending 
shell models}


\author{
Peter Hornung
\\
FB Mathematik, TU Dresden
\\
01062 Dresden (Germany)
\\
\\
Igor Vel\v ci\'c
\\
University of Zagreb, \\Faculty of Electrical Engineering and Computing
\\
Unska 3, Zagreb (Croatia)
}



\date{}


\maketitle
\begin{abstract}
We prove smoothness of $H^2$ isometric immersions of surfaces endowed with a smooth Riemannian
metric of positive Gauss curvature.
We use this regularity result to rigorously derive homogenized bending models of 
convex shells from three-dimensional nonlinear elasticity. 
\end{abstract}
\vspace{10pt}


\noindent {\bf Keywords:}
isometric immersions, positive Gauss curvature, regularity,
elasticity, dimension reduction, homogenization, shell theory,
two-scale convergence, Gamma convergence.

\section{Introduction}

For a local $C^2$ isometric immersion $u$ of a two-dimensional Riemannian manifold with positive Gauss
curvature into $\R^3$, there is a link between the regularity of the metric and the regularity
of $u$; in particular, if the metric is smooth then so is $u$.
Without a priori assumptions on the regularity of $u$ this link is broken;
as shown by classical results of Nash and Kuiper even $C^1$ regularity is not enough.
\\
In the present paper, we show that square integrability of the second fundamental form of $u$ 
is sufficient for the link to persist. In particular, if the metric is smooth,
then $u$ is smooth in the interior, provided that initially it belongs to the Sobolev space $H^2$.
\\
We note in passing that replacing the assumption of positive Gauss curvature by zero Gauss curvature
also breaks the regularity link. Indeed, local isometric immersions of class $H^2$ of the standard
flat metric in $\R^2$ into $\R^3$ may fail to be $C^2$, see \cite{Hornung-arma1} and \cite{Hornung-arma2} for
examples in this direction.
\\
Our regularity results for metrics with positive Gauss curvature rely upon earlier work by {\v{S}}ver{\'a}k
on the Monge-Amp\`ere equation. Due to the low regularity, 
the passage from the scalar problem to the vectorial problem addressed here is not trivial.
\\
Relaxing $C^2$ regularity to regularity on the Sobolev scale is important for variational
problems: the $H^2$ isometric immersions studied here arise naturally in thin film elasticity.
In the present paper, we use this regularity result to derive homogenized bending 
models for convex shells from three dimensional nonlinear elasticity. 


Much work has been done on shell theories in elasticity. An
overview of the derivation of models for linear and nonlinear shells by the method of formal
asymptotic expansions can be found in \cite{Ciarletshell00}.
In the case of linearly elastic shells, the models thus obtained can also be justified by a rigorous convergence 
result, starting from three dimensional linearized elasticity.

In the last two decades, rigorous justifications of nonlinear models for
rods, curved rods, plates and shells were obtained by means of $\Gamma$-convergence,
starting from three dimensional nonlinear
elasticity. The first papers in that direction are
\cite{AcBuPe,LDRa95, LDRa96} for strings and membranes (plates and
shells, respectively). For plates, the nonlinear bending theory
the F\"oppl-von K\'arm\'an theory were derived in \cite{FJM-02} and \cite{FJM-06}.
For shells, the nonlinear bending theory was derived in \cite{FJMMshells03} and
the von K\'arm\'an theory in \cite{Lewicka1}; see also \cite{LewMorPak11}
for limiting models in an intermediate energy scaling regime between
bending and von K\'arm\'an theories, for the particular case of convex surfaces.
\\
Here we are interested in the derivation of the homogenized bending
theory of shells by means of simultaneous homogenization and dimension reduction.
Our starting point is the energy functional of three dimensional nonlinear elasticity:
the elastic energy stored in a deformation $u\in H^1(S^h, \R^3)$
of a reference shell $S^h \subset \R^3$ of thickness $h > 0$ around a surface $S \subset \R^3$
is given by
\begin{equation}\label{intro:1}
\frac{1}{h^2\,|S^h|}\int_{S^h}W_\e(x,\nabla \bs u(x))\,dx.
\end{equation}
%
Here, $W_\e$ is a non-degenerate stored energy function that oscillates periodically
in $x$, with some period $\e\ll 1$. We are interested in the
effective behavior when both the thickness $h$ and the period $\e$ are
small.
\\
The separate limits $h\to 0$ and $\e\to 0$ are reasonably well understood:
in \cite{FJMMshells03} it is shown that, when $W_{\e}$ does not depend on $\e$, then
the functionals \eqref{intro:1}  $\Gamma$-converge, as
$h\to 0$, to a two-dimensional
bending shell theory. Regarding the limit $\e\to
0$, which is related to homogenization, the first rigorous results relevant in nonlinear
elasticity were obtained by Braides~\cite{Braides-85} and
independently by M\"uller~\cite{Mueller-87}. They proved that, under suitable growth
assumptions on $W_\e$, the energy \eqref{intro:1} $\Gamma$-converges as $\e\to 0$
(for fixed $h$) to the functional obtained by replacing $W_\e$ in
\eqref{intro:1} with the homogenized energy
density given by an infinite-cell homogenization formula.

In this paper we study the asymptotic behavior when both the
thickness $h$ and the period $\e$ tend to zero simultaneously.
Such a combination of  dimensional reduction and homogenization
has been the subject of numerous papers: in the 
paper \cite{Braides-Fonseca-Francfort-00}  the authors study the same effects for nonlinear systems (membrane plate) 
by means of $\Gamma$-convergence, also without periodicity assumptions. In their general approach they also consider 
the possibility of oscillating boundary. The 1d case of rods was addressed in \cite{Neukamm-11};
more recently the plate model in the von K\'arman regime (see \cite{NeuVel-12}) and in the bending
regime (see \cite{Horneuvel12,Vel13}) were analyzed.
In these cases one does not obtain
an infinite-cell homogenization formula as in the membrane case
(see \cite{Braides-Fonseca-Francfort-00}). This is because
the energy is essentially convex for small strain.
Therefore one can use two-scale convergence techniques in all these cases.
\\
However, each case has its own peculiarities. In the von K\'arm\'an theory of plates, one obtains
a limiting quadratic energy density which is continuous in the asymptotic ratio $\gamma$ between $h$ and $\e$, for 
all $\gamma \in [0,\infty]$.
Moreover, the case $\gamma=0$ corresponds to the situation when the dimensional reduction dominates
and the resulting model is just the homogenized plate model in von K\'arm\'an regime.
The situation $\gamma=\infty$ corresponds to the case when homogenization dominates and the
resulting model is the plate model  in the von K\'arm\'an  regime corresponding to the homogenized functional.
The same phenomenology is observed in bending regime for rods in \cite{Neukamm-11}. The limiting model of the 
homogenized plate in von K\'arm\'an regime without periodicity assumption is obtained in \cite{Velcic-13} and the 
model of rod in bending regime in \cite{MarVel-14}.
The nonlinear bending theory of plates is more involved in the periodic case.
In \cite{Horneuvel12} the authors obtained asymptotic models in the case
$\gamma \in (0,\infty]$. In \cite{Vel13} the author obtained the
asymptotic model corresponding to the regime $\gamma = 0$
under the additional assumption that $\e^2 \ll h\ll \e$.
\\
In the critical  case $h \sim \e^2$ one may expect that 
the oscillations needed for the relaxation would have to include
some other frequencies in addition to the one on which the 
material is oscillating. So two-scale analysis might not be sufficient in this case. In 
\cite{NeuOlb-13} the authors derived the homogenized bending plate model by performing
homogenization of $2D$ bending 
plate.

The homogenization of shells in the von K\'arm\'an regime (cf. \cite{Hornungvel12}) 
also displayed a separation of the models for 
the case $\gamma = 0$.
For generic shells, we were able to identify the model for the situations $\e^2 \ll h\ll \e$ 
and, surprisingly, also for the siuation $h \sim \e^2$.
Moreover, for convex shells we were able to completely resolve the case $\gamma = 0$.
The geometric framework developed in \cite{Hornungvel12} will be used in the present paper as well.

As it was already emphasized in \cite{Hornungvel12} the bending theory for shells seems likely to be more involved. 
In this paper we address that case, but only for convex shells. To answer the question of homogenization of bending 
shell we need to answer on the question of additional regularity of isometries on convex surfaces, for which we 
apriori know (by the compactness result) that are $H^2$. The main results of this paper are  Theorem \ref{thm1} and 
Theorem \ref{tm:glavni}. 
We emphasize the fact that this regularity result is also needed for the simpler case of studying the  shells that 
are layered in the thickness direction (this model also incorporated in the analysis here).  For the plate model of 
layered materials in the bending regime, see \cite{Schmidt-07}).


Let us explain the technical relevance of the regularity result Theorem \ref{thm1}
for the homogenization. In the compactness result (Proposition \ref{dvoskalnilimesi})
we are unable to identify --- i.e., to obtain 
any additional information on --- the $L^2$ matrix field $B$ 
which arises as (part of) the weak limit of approximate 
strains.
In the construction of the recovery sequence it is natural to try 
to recover this field by adding to the standard recovery sequence an additional 
term of order $h$; this term is called $w$ in our proof of the upper bound.
Under the convexity assumption, we can prove that with this term we can indeed
recover any $L^2$ matrix field $B$. This completes the proof.
\\
In order to carry out this plan, one must analyze a system of equations 
whose coefficients depend on the limiting deformation, 
which is an $H^2$ isometry. The analysis of this system requires regularity of its coefficients, see Lemma 
\ref{pomocnalema}.
\\
No such regularity result was needed in \cite{FJMMshells03}. Instead, the gradient truncation method used in \cite{FJM-02} was employed. However, such a truncation (Lusin type approximation) cannot be used in
the present paper.
\\
Note that in contrast to the situation encountered here, in the case of plates (see \cite{Schmidt-07, Horneuvel12}) 
not every $L^2$ matrix field $B$ can be recovered 
by adding an additional term. But one can recover any matrix field that is zero 
on all planar parts of the deformation. And this class of 
fields contains the ones relevant for the relaxation.



\subsection*{Notation}
$C^{k,\alpha}$ denotes the space of functions that are $k$-times differentiable and their $k$-th derivatives are 
H\"older continuous with exponent $\alpha$. $W^{k,p}$, $H^k$ denote the standard Sobolev spaces.
For $A \subset \R^n$ we denote by $C^k_0(A)$ the space of $k$-differnetiable functions with continuous derivatives 
that have compact support in $A$. 
We set $Y=[0,1)^2$ and we denote
by $\calY$ the Euclidean space $\R^2$ equipped with the torus
topology, that is for all $z\in\Z^2$ the points $y{+}z$ and $y$ are identified in $\calY$.
We write $C^0(\calY)$ to
denote the space of continuous functions $f:\,\R^2\to\R$ satisfying $f(y+z)=f(y)$ for all
$z\in\Z^2$.
We denote by $C^k(\calY)$ those functions in
$C^k(\R^2)\cap C^0(\calY)$ whose derivatives up to the $k$-th order belong to $C^0(\calY)$.
We denote by $L^2(\calY)$, $H^1(\calY)$ and $H^1(S{\times}\calY)$ the Banach spaces obtained as
the closure of $C^\infty(\calY)$
and $C^\infty(\bar S,C^\infty(\calY))$ with respect to the norm in $L^2(Y)$,
$H^1(Y)$ and $H^1(S{\times} Y)$,
respectively.
By $\dot{L}^2(\calY), \dot{H}^k(\calY)$ etc. we
denote the subspaces of $L^2(\calY), H^k (\calY)$ etc. whose mid-value over $\calY$ is zero.
%
For $A\subset\R^d$ measurable and $X$ a Banach space, $L^2(A,X)$ is understood in the
sense of Bochner. We identify the spaces $L^2(A,L^2(B))$ and
$L^2(A\times B)$ in usual way. Standard basis vectors in $\R^2$ are denoted by $e_i$.
By $\SO 3$ we denote the set of rotational matrices in $\R^{3 \times 3}$, by $\so 3$ the space of skew symmetric 
matrices in $\R^{3 \times 3}$, while by $\R^{3 \times 3}_{\sym}$ the space of symmetric matrices in $\R^{3 \times 
3}$. 
For a matrix $M$ by $\sym M$ we denote the symmetic part of $M$, $\sym M=\frac{1}{2} (M+M^t)$.
For $A,B \subset \R^n$, with $A \Subset B$ we mean that $A$ is compactly contained in $B$. 

\section{Regularity of intrinsically convex $H^2$ surfaces}
The purpose of this chapter is to prove the following result:

\begin{theorem}\label{thm1}
	Let $U\subset\R^2$ be a bounded domain and 
	let $g\in C^{\infty}( U, \R^{2\times 2}_{\sym})$ be a smooth Riemannian metric on $U$. 
	Assume that the Gauss curvature $K$ of $g$ is positive 
	and let $u\in H^2_g(U)$, i.e., $u\in H^2(U, \R^3)$ solves
	$$
	(\D u)^T(\D u) = g \mbox{ almost everywhere on $U$.}
	$$
	Then $u\in C^{\infty}(U)$.
\end{theorem}

We use ideas and a key result from the unpublished (but widely circulated)
manuscript \cite{sverak}. 
For our purposes, its main result is to deduce convexity of $H^2$
solutions $f$ of the Monge-Amp\`ere inequality $\det \D^2 f \geq c > 0$,
cf. Lemma \ref{le-sverak} below.
\\
In \cite{sverak}, this result is combined with a local graphical representation 
to prove smoothness of $C^{1,1}$ isometric immersions of subdomains of the sphere, 
endowed with the standard metric.
\\
Our proof of Theorem \ref{thm1} also uses this idea of representing $u$ locally as a graph of a function $f$.
However, a priori $u$ is not $C^1$. Instead, we show that the 
normal $n_u$ to $u$ is continuous. This is enough for $u$ to be locally a graph.
\\
Finally, a bootstrap argument, using classical facts about two-dimensional Monge-Amp\`ere equations
on one hand and exploiting the link between $u$ and its graphical representation on the other hand, 
implies that $u$ is smooth.



\subsection{Continuity of the normal}

The purpose of this section is to provide a fairly self-contained proof of Proposition \ref{pro1} below.
In doing so, we combine ideas from \cite{BrezisNirenberg1, BrezisNirenberg2} 
and others, and we introduce a suitable notion of topological degree. For the reader's convenience,
we include proofs of its relevant properties.

In what follows, we use the notation $|g| = \det g$
and $\sqrt{g} = (\det g)^{1/2}$. The Christoffel symbols of $g$ are denoted by $\Gamma_{\a\b}^{\gamma}$.
Define $h : B\to\R^{2\times 2}_{\sym}$ by $h = n_u\cdot\D^2 u$.
The Gauss equation is easily seen to remain true for $u\in H^2$. It reads:
\begin{equation}
\label{gk-5}
\d_{\a}\d_{\b} u = h_{\a\b} n_u + \Gamma_{\a\b}^{\gamma}\d_{\gamma} u.
\end{equation}
Since we are dealing with $H^2$ maps, we should verify the validity
of Gauss' Theorema Egregium.


\begin{lemma}\label{egregium}
If $u\in H^2_g(U)$ and $h = n_u\cdot\D^2 u$, then $\det h = K |g|$ almost everywhere on $U$.
\end{lemma}
\begin{proof}
As shown in \cite[Proof of Proposition 6]{FJM-06}, by approximation it is easy to see
that the map $u$ satisfies
\begin{equation}
\label{egr-1}
|\d_1\d_2 u|^2 - \d_1\d_1 u\cdot \d_2\d_2 u = 
\frac{1}{2}\left( \d_2\d_2 g_{11} + \d_1\d_1 g_{22} - 2 \d_1\d_2 g_{12} \right)
\end{equation}
%
almost everywhere on $U$. Denote by $P(x)$ the orthogonal projection
from $\R^3$ onto the subspace spanned by $\d_1 u$ and $\d_2 u$.
Recall that $h = n_u\cdot\D^2 u$. Then we deduce from \eqref{egr-1} that
\begin{equation}\label{egr-2}
\det h = - |P(\d_{1}\d_2 u)|^2 + P(\d_1\d_1 u)\cdot P(\d_2\d_2 u)
+ \frac{1}{2}\left( \d_2\d_2 g_{11} + \d_1\d_1 g_{22} - 2 \d_1\d_2 g_{12} \right).
\end{equation}
But in view of \eqref{gk-5} we have
$$
P(\d_{\a}\d_{\b} u)\cdot P(\d_{\gamma}\d_{\delta} u) = \Gamma_{\a\b}^{\rho}\Gamma^{\sigma}_{\gamma\delta}
g_{\rho\sigma}.
$$
We conclude that the right-hand side of \eqref{egr-2} can be computed from $g$ and its derivatives.
Since $g$ is smooth, the right-hand side of \eqref{egr-2} agrees with $K |g|$.
\end{proof}



\begin{proposition}\label{pro1}
Let $U\subset\R^2$ be a bounded domain and 
let $g\in C^{\infty}( U)$ be a smooth Riemannian metric on $U$. 
Assume that the Gauss curvature $K$ of $g$ is positive 
and let $u\in H^2_g(U)$. Then the normal 
$$
n_u = \frac{\d_1 u\times\d_2 u}{|\d_1 u\times\d_2 u|}
$$ 
to $u$ is continuous on $U$.
\end{proposition}


To prove Proposition \ref{pro1}, let $R > 0$ and let $\p\in H^1(B_{2R}, \S^2)$. 
Then, for $L^1$ almost every $r\in (0, R)$ we have
$
\p|_{\d B_r}\in H^1(\d B_r)$, hence by Sobolev embedding
\begin{equation}
\label{p1-1}
\p|_{\d B_r}\in C^0(\d B_r).
\end{equation}
%
By a classical result of Schoen and Uhlenbeck,
there exist $\p_k\in C^{\infty}(B_{2R}, \S^2)$ converging strongly in $H^1$
to $\p$. After possibly passing to a subsequence, we may assume that
$\p_k\to\p$ in $H^1(\d B_r)$ for almost every $r\in (0, R)$, hence
%
\begin{equation}
\label{p1-2}
\p_k\to \p \mbox{ uniformly on }\d B_r.
\end{equation}
%
In fact, setting $f_k(r) = \int_{\d B_r}|\D\p_k - \D\p|^2$, by the coarea formula we have
$$
\|f_k\|_{L^1(B_R)} = \int_0^R\ dr\ \int_{\d B_r}|\D\p_k - \D\p|^2 = \int_{B_R} |\D\p_k - \D\p|^2\to 0.
$$
Hence there is a subsequence such that $f_k(r)\to 0$ for $L^1$ almost every $r\in (0, R)$.
\\
For the rest of this section we fix $r\in (0, R)$ 
such that \eqref{p1-1} and \eqref{p1-2} are satisfied simultaneously.
Note in particular that $\p(\d B_r)$ is compact, due to \eqref{p1-1}.
Define the degree $Q : \S^2\to\R$ of $\p$ with respect to $B_r$ by setting
\begin{equation}
\label{p1-3}
Q(y) = \int_{B_r}\p^*\eta,
\end{equation}
%
where $\eta$ is any smooth $2$-form on $\S^2$ with $\int_{\S^2}\eta = 1$
which is supported in the connected
component $\Lambda_y$ of $\S^2\setminus \p(\d B_r)$ that contains $y$.
\\
We claim that $d$ is well-defined, i.e., that it is independent of the choice of $\eta$.
We use the following lemma.
%
\begin{lemma}
Let $\Lambda\subset\S^2$ be connected and let 
$\t\eta$ be a smooth $2$-form on $\S^2$ whose support is contained in $\Lambda$
and which is such that $\int_{\S^2}\t\eta = 0$. Then there exists 
a smooth $1$-form $w$ on $\S^2$ with support in $\Lambda$ and such that $\eta = dw$.
\end{lemma}
\begin{proof}
The proof can be found in \cite{schwartz69}.
\end{proof}

In view of the lemma it remains to show that if $w$ is a smooth $1$-form 
supported in $\Lambda_y$ then $\int_{B_r}\p^*(dw) = 0$.
\\
Since $dw$ is a $2$-form and since $\p_k\to\p$ strongly in $H^1(B_r)$, we see that
$$
\p_k^*(dw)\to \p^*(dw) \mbox{ strongly in }L^1(B_r).
$$
Hence 
\begin{equation}
\label{p1-4}
\int_{B_r} d(\p_k^*w) = \int_{B_r} \p_k^*(dw) \to \int_{B_r} \p^*(dw).
\end{equation}
%
Due to \eqref{p1-2}, the compact set $\p_k(\d B_r)$ does not intersect the support of
$w$ for $k$ large enough, because the latter has positive distance from the compact
set $\p(\d B_r)$. Therefore, $\p_k^* w$ has compact support in $B_r$. Hence,
by Stokes' theorem, the left-hand side of \eqref{p1-4} is zero. This concludes the proof
showing that $Q$ is well-defined by \eqref{p1-3}.
\\
Recall that the essential range of $\p|_{B_r}$ is the smallest closed set
$F$ such that $\p(x)\in F$ for almost every $x\in B_r$; as shown in \cite{BrezisNirenberg1}
it is well-defined.
More or less directly from the definition of $Q$, we see the following:

\begin{lemma}\label{le2}
Let $\p\in H^1(U, \S^2)$ and define $Q$ as in \eqref{p1-3}.
Then the following are true:
%
\begin{enumerate}[(i)]
\item \label{p1-6}
$Q$ is constant on every connected component of $\S^2\setminus\p(\d B_r)$;
\item \label{p1-7}
If $Q(y)\neq 0$ then $\Lambda_y$ is contained in the essential range $F$ of
$\p|_{B_r}$.
%
\item \label{p1-7b}
$Q$ takes integer values.
\end{enumerate}
\end{lemma}
\begin{proof}
To prove \eqref{p1-7}, assume that $\Lambda_y$ is not contained in $F$. Then
there exists an open set $\Lambda\subset\Lambda_y\setminus F$
and a normalized smooth $2$-form $\eta$ supported on $\Lambda$.
So $\p^*\eta = 0$ almost everywhere on $B_r$. Hence we would have $Q(y) = 0$.
\\
To prove \eqref{p1-7b} just note that the last convergence in \eqref{p1-4}
is also true for any other $2$-form; in particular for the form $\eta$ in \eqref{p1-3}.
But for smooth $\p$, the right-hand side of \eqref{p1-3}, which 
is known to attain only integer values.
\end{proof}


\begin{lemma}\label{le3}
If $\p = n_u$, then we have $Q \geq 0$ on $\S^2\setminus n_u(\d B_r)$.
Moreover, if $Q(y) = 0$ then $\Lambda_y$ does not intersect the essential
range of $n|_{\d B_r}$.
\end{lemma}
\begin{proof}
Denote by $\eta_{\S^2}$ the standard area form on $\S^2$. 
Then $n_u^*\eta_{\S^2} = K d\vol_g$, due to Lemma \ref{egregium}.
Applyig \eqref{p1-3} with $\eta = \rho\eta_{\S^2}$, 
we see that
$$
\left(\int_{B_r} \rho\eta_{\S^2}\right)\ Q(y) = \int_{B_r} (\rho\circ n_u) K\ d\vol_g. 
$$
for every $\rho\in C^{\infty}_0(\Lambda_y)$.
%
Since $K > 0$ on $B_r$, we conclude that if $Q(y) = 0$ then
$\rho\circ n = 0$ almost everywhere on $B_r$. Since $\rho$ was arbitrary,
this implies that $n(x)\in \S^2\setminus \Lambda_y$ for almost every $x\in B_r$.
Since $\S^2\setminus\Lambda_y$ is closed, by minimality of the essential range
we conclude that it must be contained in $\S^2\setminus\Lambda_y$.
\end{proof}

\begin{lemma}\label{le4}
If $n_u = \p$ and if $R > 0$ is small enough, then $Q$ is zero at some point in $\S^2\setminus n_u(\d B_r)$.
\end{lemma}
\begin{proof}
There exists a constant $C$ depending only on $g$ such that
$$
\int_{B_r} K\ d\vol_g \leq CR^2.
$$
We choose $R$ so small that the right-hand side is bounded by $1/4$ times the area of $\S^2$.
\\

Assume for contradiction that $Q\neq 0$ on $\S^2\setminus n_u(\d B_r)$. Then by Lemma \ref{le3}
we know that $Q$ is positive and so by Lemma \ref{le2} we have $Q\geq 1$ on $\S^2\setminus n_u(\d B_r)$.
\\
Since $n_u\in H^1(\d B_r)$, it maps $\d B_r$ into a set of zero area, cf. 
\cite{Reshetnyak}.
So the area of $\S^2\setminus n_u(\d B_r)$ is that of $\S^2$.
Hence there exist finitely many pairwise disjoint connected components $\Lambda_1, ..., \Lambda_M$ of
$\S^2\setminus n_u(\d B_r)$ and $\psi_i\in C^{\infty}_0(\Lambda_i)$ 
satisfying $\psi_i\in [0, 1]$, such that
\begin{equation}
\label{le4-1}
\sum_{i = 1}^M \HH^2\left( \{\psi_i = 1\} \right) \geq \frac{1}{2}\HH^2(\S^2).
\end{equation}
%
Let $y_i\in \Lambda_i$ and note that $\sum_i\psi_i \leq 1$ pointwise on $\S^2$. Hence, recalling that
$Q(y_i) \geq 1$,
\begin{align*}
\int_{B_r} K\ d\vol_g &\geq \sum_{i = 1}^M \int_{B_r} (\psi_i\circ n_u)\ K\ d\vol_g
\\
&= \sum_{i = 1}^M \int_{B_r} n_u^*(\psi_i\eta_{\S^2})
\\
&= \sum_{i = 1}^M Q(y_i) \int_{\S^2} \psi_i\eta_{\S^2}
\\
&\geq \sum_{i = 1}^M \int_{\S^2} \psi_i\eta_{\S^2}
\\
&\geq \sum_{i = 1}^M \HH^2\left( \{\psi_i = 1\} \right).
\end{align*}
%
In view of \eqref{le4-1} this contradicts our choice of $R$.
\end{proof}


Since $Q$ vanishes at some point, it is in fact zero on a whole connected component $\Lambda$ of
the open set $\S^2\setminus n_u(\d B_r)$. But this means that (after possibly redefinig $n_u$ on a set of measure
zero) $n_u$ does not take values in $\Lambda$.
\\
We assume without loss of generality that $e_3\in \Lambda$ and we denote by $\Psi : \S^2\setminus\{e_3\}\to\R^2$
the stereographic projection. Since $\Lambda\subset \S^2\setminus n_u(\d B_r)$ is open, 
there exists $\rho > 0$ such that
$\S^2\cap B_{2\rho}(e_3)$ does not intersect $n_u(\o B_r)$. And 
$\Psi\in C^{\infty}\left(\R^3\setminus \o{B_{\rho}(e_3)}\right)$.
Hence $\Psi\circ n\in (H^1\cap L^{\infty})(U, \R^2)$. Since $\Psi$
is conformal, we deduce from $K > 0$ 
that the Jacobian of $\Psi\circ n_u$ is positive (or negative) and bounded
away from zero. Hence $\Psi\circ n_u$ is continuous, cf. \cite{Reshetnyak}. (In fact, our arguments
essentially even show that
$\Psi\circ n_u$ is monotone in the Lebesgue sense, which quite directly implies continuity.) 
Hence $n_u$ is continuous as well.


\subsection{Proof of Theorem \ref{thm1}}

Fix a point $x_0\in U$ and let us prove that $u$ is smooth in a neighbourhood of this point.
We assume without loss of generality that $x_0 = 0$ and that $n_u(0) = e_3$. We write
$u = \zwo{\Psi}{V}$ where $V = e_3\cdot u$ and $\Psi : U\to\R^2$ is the in-plane component.

\begin{lemma}\label{le1}
There exists a positive constant $c$ such that $\det\D \Psi \geq c$ near $0$.
\end{lemma}
\begin{proof}
We compute
\begin{align*}
\det\D \Psi &= e_3\cdot \zwo{\d_1 \Psi}{0}\times \zwo{\d_2 \Psi}{0}
\\
&= e_3\cdot\d_1 u\times \d_2 u = \sqrt{g} (n_u\cdot e_3).
\end{align*}
%
By continuity of $n_u$ and since $\sqrt{g}\geq c$, there is a neighbourhood of 
$0$ on which the right-hand side is bounded from below by a positive constant.
\end{proof}

Since $u$ is Lipschitz by hypothesis, we see that $\Psi$ is Lipschitz and, by Lemma \ref{le1},
that its Jacobian is bounded from below by a positive constant. Hence Clarke's
inverse function theorem \cite{Clarke} shows that $\Psi$ is locally bi-Lipschitz on some ball $B$ centered at $0$.
\\
Define $\Phi = \Psi^{-1}$ and define $f = V\circ \Phi$, which is a map
$
\Psi(B)\to\R.
$
Then $u(B) = \graph f|_{\Psi(B)}$. For $z\in \Psi(B)$ we define
$$
G(z) = (z, f(z)),
$$
so that $u = G(\Psi)$. We assume without loss of generality that $\Psi(0) = 0$.
\\



Clearly, $f\in W^{1, \infty}(B)$. Moreover, $f\in H^2$ by the chain rule
for the composition of Sobolev functions with a bi-Lipschitz map $\Phi$.
The usual chain rule applies. 
We claim that $f$ satisfies the prescribed Gauss curvature equation
\begin{equation}
\label{gk-1}
\det\D^2 f = K(\Phi) \cdot (1 + |\D f|^2)^2
\end{equation}
almost everywhere on $\Psi(B)$. In fact, denoting the metric induced by $G$ by
$$
\t g = (\D G)^T (\D G) = I + \D f\otimes\D f,
$$
%
and the normal to $G$ by
$\t n_u = \frac{\d_1 G\times \d_2 G}{|\d_1 G\times \d_2 G|}$, we have
\begin{equation}
\label{gk-2}
\t n_u = \frac{(-\D f, 1)^T}{\left( 1 + |\D f|^2 \right)^{1/2}},
\end{equation}
%
because $\det\t g = 1 + |\D f|^2$.
Moreover, $\d_{\a}\d_{\b} G = (0, 0, \d_{\a}\d_{\b} f)^T$.
Therefore, $\t h = \t n_u \cdot \D^2 G$ satisfies
\begin{equation}
\label{gk-3}
\t h = \frac{\D^2 f}{(1 + |\D f|^2)^{1/2}}.
\end{equation}
%
Taking determinants in \eqref{gk-3}, we see that
\begin{equation}
\label{gk-1b}
\det\D^2 f = (1 + |\D f|^2) \det\t h.
\end{equation}
%
Using the chain rule, it is easy to verify that 
\begin{equation}
\label{gk-4}
h = (\D\Psi)^T\ \t h(\Psi)\ (\D\Psi) \mbox{ almost everywhere.}
\end{equation}
A similar relation applies to $g$ and $\t g$. Therefore, using Lemma \ref{egregium},
we see that \eqref{gk-1b} implies \eqref{gk-1}.
\\

Observe that the right-hand side of \eqref{gk-1} is positive and bounded away from zero and infinity. 
Hence the following lemma implies that $f$ is a (locally) convex function.
%
\begin{lemma}\label{le-sverak}
Suppose $f\in H^2(B)$ satisfies $\det\D^2 f \geq c > 0$. Then $f$ is either locally convex on $B$
or locally concave on $B$.
\end{lemma}
\begin{proof}
This is proven in \cite{sverak}. As observed in \cite{sverak}, the results in \cite{IwaniecSverak} 
(which were conjectured in \cite{sverak})
indeed allow to relax the $W^{2,\infty}$ hypothesis in \cite{sverak} to the
$H^2$ hypothesis used here. We refer to \cite{h-sverak, LewickaMahadevanPakzad}
and the references therein for more on this.
\end{proof}

We may assume that $f$ is convex, and then, e.g. by a result of \cite{Heinz-MZ1959} or
by \cite{Caff1, Caff3}, we may assume that $f$ is strictly convex on $B$. 
One can then deduce from \eqref{gk-1} that $f\in C^1$, but we give a direct proof of this:
%
\begin{lemma}
We have $f\in C^1(\Psi(B))$.
\end{lemma}
\begin{proof}
We know that $n_u : B\to\S^2$ is continuous. And so is $\Phi$.
Since $\t n_u = n_u(\Phi)$, we see that $\t n_u$ is continuous.
Upon scalar multiplication of \eqref{gk-2} with $e_3$, we have
$$
(1 + |\D f|^2)^{-1/2} = n_u(\Phi)\cdot e_3.
$$
Since $n_u(\Phi)\cdot e_3$ is continuous and strictly positive, we conclude that
$(1 + |\D f|^2)^{-1/2}$ is continuous, hence so is $\D f$, by \eqref{gk-2}.
\end{proof}

The following lemma is \cite[Theorem 1']{NikolaevShefel}; cf. also \cite{Caff2}.

\begin{lemma}\label{le-nish}
Let $0 < m < M < \infty$ and let $\t F : B_R\to [m, M]$. If $\p\in C^0(\o B_R)$ is a convex 
Aleksandrov solution of
$$
\det\D^2\p = \t F,
$$
%
then there exists $p\geq 1$ such that $\p\in W_{loc}^{2,p}(B_R)$. Moreover,
$p\to\infty$ as $M/m\to 1$.
\end{lemma}

Since $F$ is continuous, its oscillation is as small as we please on $B_R$,
provided $R$ is small enough. Hence Lemma \ref{le-nish} implies that there
exists $R > 0$ and $p > 2$ such that $f\in W^{2,p}(B_R)$.
\\
Since $f\in W^{2,p}(B_R)$, also $G\in W^{2,p}(B_R)$. And \eqref{gk-3} implies that
$\t h\in L^p(B_R)$. But $\D\Psi\in L^{\infty}$ because $u\in W^{1,\infty}$.
Therefore, \eqref{gk-4} shows that $h\in L^p(\Phi(B_R))$.
\\
Since $\Gamma$ is smooth and $\D u$ is bounded, we deduce from \eqref{gk-5} that $u\in W^{2,p}(\Phi(B_R))$.
In particular, by Morrey-Sobolev embedding, there exists a constant $\delta > 0$ such that
$\Psi\in C^{1,\delta}\left( \Phi(B_R) \right)$ and $f\in C^{1,\delta}(B_R)$. Since
$K$ is, in particular, Lipschitz on $B$, we have $F\in C^{0,\delta}$.
E.g. by the results in \cite{Schulz-MZ1982}, 
we therefore deduce from $\det\D^2 f = F$ that $f\in C^{2,\delta}$.
Hence \eqref{gk-3} shows that $\t h\in C^{0, \delta}$. Hence $h\in C^{0, \delta}$ by \eqref{gk-4}.
Thus \eqref{gk-5} implies $u\in C^{2,\delta}$ for some positive $\delta$ in some neighbourhood
of the origin.

Since on some neighbourhood of the origin 
$u\in C^{2,\delta}$, for every constant unit vector $e$ the function $v = e\cdot u$
is a $C^{2,\delta}$ solution of the Darboux equation
$$
\det\left( \D^2 v + \Gamma\cdot\D v \right) = K |g| \left( 1 - g^{-1} : (\D v\otimes\D v) \right),
$$
where $\Gamma\cdot\D v$ denotes a suitable contraction of indices.
This equation is elliptic with respect to $v$. Since $g^{-1}$, $\Gamma$ and $K$ are smooth,
and since $v\in C^{2, \delta}$, we conclude by standard theory 
that $v\in C^{\infty}$, hence $u$ is smooth in a neighbourhood of the origin.
\begin{remark}\label{regulrem1}
Notice that if we only assume that $g \in C^{2,\alpha}(U, \R^{2 \times 2}_{\sym})$ then we can conclude that $u \in 
C^{2,\delta}(U;\R^3)$, for some $\delta>0$. If $g \in C^{k,\alpha}(U, \R^{2 \times 2}_{\sym})$, for $k \geq 3$ then 
we can conclude that $u \in C^{k+2,\alpha}(U, \R^3)$, by standard elliptic regularity (see, e.g., \cite[Lemma 17.16]
{GT01}).
\end{remark}

\section{Homogenization for shells} 



\subsection{Preliminaries}

Unless additional properties are stated, $\omega\subset\R^2$ is assumed to be a bounded domain with boundary of class 
$C^3$.
We set $I = (-\tfrac{1}{2},\tfrac{1}{2})$ and $\Omega^h = \omega\times(hI)$, and $\Omega = \omega\times I$.
From now on, $S$ denotes a compact connected oriented
surface with boundary which is embedded in $\R^3$.
For convenience we assume that $S$ is parametrized by a single chart: From now on,
$\xi\in C^{4,\alpha}(\overline{\omega}; \R^3)$ denotes an embedding
with $\xi(\omega) = S$. The inverse of $\xi$ is denoted by
$r: S \to \omega$, and we assume it to be $C^{4,\alpha}$ up to the boundary.
\\
As before by $g = (\D\xi)^T(\D\xi)$ we denote the Riemannian metric on $\omega$ induced by $\xi$.
Its Christoffel symbols are denoted by $\Gamma_{\a\b}^{\gamma}$.
In what follows we recall some standard notions and set the notation.
 All notions are discussed in detail
in most basic textbooks on Riemannian geometry. By $H^2_{\iso}(S)$ we denote 
the $H^2$ isometries of the surface $S$. Notice that 
$$ u \in H^2_{\iso}(S) \Leftrightarrow \t u \in H^2_g (U),$$
where $\t u =u \circ \xi$. We repeat the geometrical framework we used in \cite{Hornungvel12}.
\begin{itemize}
	%
	\item The volume element on $S$ is denoted by $d\vol_S$.
	\item The scalar product on a vector space $V$ is denoted by $\al{x, y}_V$,
	and we define $x\cdot y = \al{x, y}_{\R^k}$, for $x, y \in \R^k$.
	\item We denote by $TS$ the tangent bundle over $S$, i.e., the collection
	of tangent spaces $T_x S$ with $x\in S$.
	A basis of $T_x S$ is given by $\tau_1(x), \tau_2(x)$, defined by
	$$
	\tau_{\a}(x) = (\d_{\a}\xi)(r(x))\mbox{ for all }x\in S.
	$$
	We can regard $T_xS$ as a subspace of $\R^3$; then we write
	$\sigma\cdot\tau = \al{\sigma, \tau}_{T_xS}$.
	\item By $T^*S$ we denote the cotangent bundle. A basis of $T^*_x S$
	is given by $(\tau^1(x), \tau^2(x))$ dual to $(\tau_1, \tau_2)$. It is
	uniquely determined by the condition
	$$
	\tau^{\a}(\tau_{\b}) = \delta_{\a\b},
	$$
	where $\delta_{\a\b}$ is the Kronecker symbol. Observe that $\tau^{\a}$
	is more commonly denoted by $dx^{\a}$, but we will not use that notation.
	We can identify $T_xS$ with $T_x^*S$ via $\tau \mapsto \al{\tau, \cdot}$.
	Via this identification we can identify $\tau\in T^*S$ with the unique vector
	$v\in T_xS\subset\R^3$ with the property that $\tau = \al{v, \cdot}_{\R^3}$
	on $T_x S$. Notice that
	 \begin{equation} \label{nadopuna1}
	 \nabla r =\left(\begin{array}{c} \tau^1 \\ \tau^2 \end{array}\right).  
	 \end{equation}           
	%
	\item By $n : S\to\mathbb{S}^2$ we denote the unit normal to $S$, i.e.,
	$$
	n(x) = \frac{\tau_1(x)\times\tau_2(x)}{|\tau_1(x)\times\tau_2(x)|}\mbox{ for all }x\in S.
	$$
	We define $\tau^3 = \tau_3 = n$. The normal bundle of $S$ is denoted by $NS$ and by definition has fibers $N_x S$ 
given by the
	span of $n(x)$. We denote by
	$$
	T_S(x) = I - n(x)\otimes n(x)
	$$
	the orthogonal projection from $\R^3$ onto $T_x S$.
	We will frequently deal with  vector fields $V : S\to\R^3$ on the surface.

	%
	\item The tensor product bundles $TS\otimes TS$ etc. are defined fiberwise.
	If $T_x S$ is regarded as a subspace of $\R^3$, then
	$T_x^*S\otimes T_x^*S$ can be regarded as a subspace of $\R^{3\times 3}$.
	\\
	The symmetric product $E\odot F$ of two vector spaces (or bundles) $E$ and $F$
	by definition consists of elements of the form
	$$
	a\odot b := \frac{1}{2}\left( a\otimes b + b\otimes a\right)
	$$
	with $a\in E$ and $b\in F$.
	\\
	Sections of the bundle $T^*S\odot T^* S$ are called quadratic forms on $S$.
	Sections $B$ of $T^*S\otimes T^*S$ can be regarded as maps from $S$ into
	$\R^{3\times 3}$ via the embedding $\iota$ defined by
	$
	\iota(B) = B(T_S, T_S).
	$
	(On the right-hand side and elsewhere we identify $(\R^3)^*\otimes (\R^3)^*$ with $\R^{3\times 3}$).
	By definition, $B(T_S, T_S) : S\to\R^{3\times 3}$ takes the vector fields
	$v, w : S\to\R^3$ into the function $x\mapsto B(x)(T_S(x)v(x), T_S(x)w(x))$.
	%
	\item For any vector bundle $E$ over $S$ we denote by $L^2(S; E)$ the space
	of all $L^2$-sections of $E$. The spaces $H^1(S; E)$ etc. are defined similarly.
	Explicitly, e.g. for $E = TS$ we have
	$$
	L^2(S, TS) = \{f\in L^2(S, \R^3) : f(x)\in T_x S \mbox{ for a.e. }x\in S\}.
	$$
	\item For any vector bundle $E$ over $S$ with fibers $E_x$,
	we denote by $L^2(\Y, E)$ the vector bundle over $S$ with
	fibers $L^2(\Y, E_x)$. The bundles $H^1(\Y, E)$ etc. are defined similarly.
	For example, $L^2$-sections of the bundle $H^1(\Y, TS)$ are given by
\begin{eqnarray*}
	& & L^2(S, H^1(\Y, TS)) = \\ & &\hspace{+5ex}\{Z\in L^2(S, H^1(\Y, \R^3)) : Z(x)\in H^1(\Y, T_xS)\mbox{ for 
a.e. }x\in S\}.
\end{eqnarray*}
	%
	\item For a scalar function $f : S\to\R$ its gradient field along $S$
	will be denoted by $df$, which is also the notation for the corresponding
	$1$-form. In other words,
	$$
	df(x)(\tau) = \D_{\tau} f(x)\mbox{ for all }\tau\in T_xS.
	$$
	Here and elsewhere $\D$ denotes the usual gradient on $\R^3$ (or on $\R^2$) of the extension of $f$,
	and $\D_{\tau} f = \tau\cdot\D f = \sum_i \tau_i\d_i f$ . % By $D$ we denote the Levi-Civita connection.
	We extend these definitions componentwise to maps into $\R^3$.
	%
	
	%
	\item For functions $f\in L^2(S, H^2(\Y))$ we define $\Hessy f$ to be the section of
	the bundle $L^2\left( \Y; T^*S\odot T^* S \right)$ over $S$ given by
	$$
	(\Hessy f)(x, y) = (\D_y^2 f)_{\a\b}(x, y)\tau^{\a}(x)\otimes\tau^{\b}(x),
	$$
	where $(\D_y^2 f)_{\a\b} = \d_{y_{\a}}\d_{y_{\b}}f$.
	%
	\item For $v\in L^2(S, H^1(\Y; \R^2))$ we define
	the section $\Defy v$ of the bundle $L^2(\Y, T^*S\odot T^*S)$ by
	$$
	(\Defy v)(x, y) = (\sym\D_y v(x, y))_{\a\b}\tau^{\a}(x)\otimes\tau^{\b}(x).
	$$
	Here and elsewhere $\D_y$ is gradient in $\Y$ with respect to the variable $y$.
	%
	\item The Weingarten map $\AA$ of $S$ is given by $\AA = dn$, i.e.,
	$$
	\AA(x)\tau = (\D_{\tau} n)(x) \mbox{ for all }x\in S,\ \tau\in T_x M.
	$$
	We extend $\AA$ to a linear map on $TS\oplus NS \stackrel{\sim}{=} \R^3$
	by setting $\AA = \AA\circ T_S$, i.e., we define $\AA(x)n(x) = 0$.
	\\

\end{itemize}
\subsubsection{The nearest point retraction}

The nearest point retraction of a tubular neighbourhood of $S$ onto $S$ will be denoted by $\pi$.
Hence
$$
\pi (\hx + t\na(\hx)) = \hx \quad  \mbox{ whenever $|t|$ is small enough.}
$$
After rescaling the ambient space, we may assume that the curvature of $S$ is as small as we please.
In particular, we may assume without loss of generality that $\pi$ is
well-defined on a domain containing the closure of
$\{\hx+t\na (\hx); \ \hx \in S, -1 < t < 1 \}$,
and that
$
|Id + t\AA(\hx)|\in (1/2, 3/2)
$
for all $t\in (-\tfrac{1}{2}, \tfrac{1}{2})$ and all $\hx\in S$.
\\
For a subset $A \subset S$ and $h\in (0,1]$ we define
$
A^h=\{\hx + t\na (\hx): \ \hx \in S,-h/2<t<h/2 \}.
$
In particular, the shell is given by
$$
S^h=\{\hx+t\na (\hx): \ \hx \in S,\ t\in (-h/2, h/2) \}.
$$
We introduce the function $t : S^1\to\R$ by
\begin{equation}
\label{deft}
t(x) = (x - \pi(x))\cdot n(x)\mbox{ for all }x\in S^1.
\end{equation}

We extend all maps $f : S \to \R^k$ trivially from $S$ to $S^1$, simply by
defining
\begin{equation}
\label{fasin}
f(x) = f(\pi(x)) \mbox{ for all }x\in S^1.
\end{equation}
In particular, we extend $r$, $T_S$ and $\AA$ trivially to $S^1$, i.e.,
we have $\AA(x) = \AA(\pi(x))$ and $T_S(x) = T_S(\pi(x))$ and $r(x) = r(\pi(x))$
for all $x\in S^1$.
The following  identity is easy to verify and is given in \cite{Hornungvel12}
\begin{equation} \label{id1}
	(\nabla \pi)(x) = T_S(x) \left( I + t(x) \AA(x)\right)^{-1}.	
\end{equation}	
By using the chain rule we have for $f : S\to\R^k$ the following identity
	\begin{equation}\label{dpi-2}
	\nabla  (f\circ\pi) = df\circ \pi T_S (I + t \AA)^{-1}.
	\end{equation}

\subsubsection{Thin films}




To deal with thin films, we introduce the map $\Xi : \omega\times\R\to\R^3$ by setting
$$
\Xi(z', z_3) = \xi(z') + z_3n(\xi(z')) \mbox{ for all $z'\in\omega$ and }z_3\in\R.
$$
As in \cite{FJM-02} we will use the diffeomorphism $\t\Theta^h : \Omega_h\to\Omega$
given by  $\t\Theta^h(z_1, z_2, z_3) = (z_1, z_2, z_3/h)$,
and for a map $\t y : \Omega \to\R^3$
we introduce the scaled gradient
$
\t\D_h y = (\d_1 y, \d_2 y, \frac{1}{h}\d_3 y).
$
The  counterpart of $\t\Theta^h$ on the shell
is the diffeomorphism $\Theta^h : S^h \to S^1$ given by
\begin{equation} \label{defphih}
\Theta^h(x) = \pi(x) + \frac{t(x)}{h}n(x).
\end{equation}
It is easy to see that
\begin{equation}
\label{comu}
\Theta^h\circ\Xi = \Xi\circ\t\Theta^h \mbox{ on }\Omega_h.
\end{equation}
For given $u : S^h\to\R^3$ we define its
pulled back version $\t u : \Omega_h\to\R^3$ by $\t u = u\circ\Xi$.
We also define its rescaled version $y : S^1\to\R^3$ by
$ 
\ya(\Theta^h) = \ua \mbox{ on }S^h
$ 
and we define the pulled back version $\t y$ of this map
by $\t y = y\circ\Xi$. Then it is easy to see that
\begin{equation}
\label{nablahas}
(\t\D_h\t y)\circ\t\Theta^h = \D\t u\mbox{ on }\Omega_h.
\end{equation}
We define the rescaled gradient $\D_h y$ of $y$ by the condition
\begin{equation}
\label{defDh}
(\nabla_h y)\circ \Theta^h = \nabla u \mbox{ on }S^h.
\end{equation}
This corresponds to the fact that the rescaled gradient on th canonical domain is just the physical gradient (on the 
physical domain) translated to the canonical domain. 
Using \eqref{comu} and  \eqref{nablahas} it is easy to see that
\begin{equation}
\label{nablahas-2}
\t\D_h\t y = \nabla u(\Xi)\left((\D\Xi)\circ (\t\Theta^h)^{-1}\right).
\end{equation}
Since $\nabla t = n$, formula (\ref{id1}) and formula \eqref{dpi-2} show (recall that $n$ is extended trivially to 
$S^1$):
\begin{equation*}
\nabla\Theta^h = \nabla\pi + \frac{t}{h}\nabla n + \frac{1}{h}n\otimes n
= T_S(I + t\AA)^{-1} + \frac{t}{h}\AA T_S(I + t\AA)^{-1} + \frac{1}{h}n\otimes n.
\end{equation*}
%
Since $T_S$ clearly commutes with $\AA$, we see that $T_S$ commutes with $(I + t\AA)^{-1}$ as well. Hence
\begin{equation}
\label{dphi}
\nabla\Theta^h = (I_h + \frac{t}{h}\AA)(I + t\AA)^{-1}\mbox{ on }S^h,
\end{equation}
where $I_h=T_S+\tfrac{1}{h} n \otimes n$.
%
To express $\nabla_h y$ in terms of $\D y$,
insert the definition of $y$ into \eqref{defDh} and use \eqref{dphi} to find
\begin{equation}
\label{fladh}
\nabla_h y =  \nabla y\ (I_h + t\AA)(I + th\AA)^{-1} \mbox{ on }S^1.
\end{equation}
%


\subsubsection{Two-scale convergence}



Recall that we extend the chart $r$ trivially from $S$ to $S^1$. We make the following definitions:



\begin{itemize}
	\item A sequence $(f^h)\subset L^2(S^{1})$ is said to converge weakly two-scale
	on $S^1$ to the function $f\in L^2(S^{1},L^2(\calY))$ as $h\to 0$, provided
	that the sequence $(f^h)$ is bounded in $L^2(S^{1})$ and
	\begin{equation}\label{2skalen}
	\lim\limits_{h\to 0}\int_{S^{1}} f^h(x)\,\rho(x, r(x)/\e)\ud
	x=\int_{S^{1}}\int_{\Y}f(x,y)\,\rho(x,y)\ud y\ud x.
	\end{equation}
	for all $\rho\in C_c^0(S^{1},C^0(\calY))$.
	\item  We say that $f^h$ strongly two-scale converges to $f$ if, in addition,
	\begin{equation*}
	\lim\limits_{h\to 0}\|f^h\|_{L^2(S^{1})}=\|f\|_{L^2(S^{1}\times Y)}.
	\end{equation*}
\end{itemize}


%
We write $f^h\wtto f$ to denote weak two-scale convergence
and  $f^h\stto f$ to denote strong two-scale convergence.
%
If $f^h \wtto f$ then $f^h\rightharpoonup \int_{\calY} f(\cdot,y)dy$ weakly in $L^2$.
If $f^h$ is bounded in $L^2(S^{1})$ then it has a
subsequence which converges weakly two-scale to some $f \in
L^2(S^{1};L^2(\calY))$. These and other facts can be deduced from the corresponding
results on planar domains (cf. \cite{Allaire-92, Visintin-06}) by means of the
following simple observations.
\\
Defining $\t f^h = f^h\circ\Xi$ and $\t f(z, y) = f(\Xi(z), y)$,
and taking
$$
\t\rho(z, y) = \rho(\Xi(z), y)\left( \det\D\Xi^T(z)\D\Xi(z) \right)^{1/2},
$$
a change of variables shows that \eqref{2skalen} is equivalent to
%
\begin{equation}\label{2skalen-2}
\int_{\Omega} \t f^h(z)\t\rho(z, z'/\e)\ dz \to
\int_{\Omega} \int_{\Y} \t f (z, y)\t\rho(z,y)\ dy\ dz,
\end{equation}
%
where $z'$ is the projection of $z$ onto $\R^2$. Hence $f^h\wtto f$ on $S^1$ if and only if
$\t f^h\wtto \t f$ on $\Omega$ in the usual sense.
\\
When $f^h : S\to\R$, then $f^h\wtto f$ on $S$ means, by definition, that the trivial
extensions converge weakly two-scale on $S^1$. In particular, $f^h\wtto f$
on $S$ if and only if $\t f^h\wtto \t f$ on $\omega$.
All these definitions are extended componentwise
to vector-valued maps. For quadratic forms $q$, $q^h$ on $S$ we say $q^h\wtto q$
if $q^h(\tau, \sigma)\wtto q(\tau, \sigma)$ for all $\tau, \sigma\in C^1(S, TS)$.
A similar definition applies to other bundles.
In the end we give the compactness lemma which gives the information about two-scale limit of scaled gradients. For 
the proof see \cite[Lemma 4.3]{Hornungvel12}. 

\begin{lemma}
	\label{lm:50}
	Let $\wa^h \in H^1(S^{1},\R^3)$ be such that
	\begin{equation*}
	\limsup\limits_{h\to 0}\left( \,\|\wa^h\|_{L^2(S^1)}+\|\nabla_h\wa^h\|_{L^2(S^1)}\,\right)<\infty.
	\end{equation*}
	Then there exists a map $\wa_0 \in H^1(S,\R^3)$ and a field
	$\bs H_\gamma \in L^2(S \times I \times \calY, \R^{3 \times 3})$ of the form
	\begin{equation}\label{P:comp1}
	\bs H_\gamma =
	\left\{\begin{aligned}
	&\left(\,\nabla_y\wa_1,\wa_2 \right)
	&&\text{for some }
	\left\{\begin{aligned}
	&\wa_1\in L^2(S;\dot{H}^1(\mathcal Y,\R^3))\\
	&\wa_2\in L^2(S \times \Y \times I, \R^3))
	\end{aligned}\right\}\\
	&&&\text{if $\gamma=0$},\\
	&\left(\,\nabla_y\wa_1,\tfrac{1}{\gamma}\partial_3\wa_1\,\right)
	&&\text{for some $\wa_1\in L^2(S;\dot{H}^1(I\times\mathcal Y, \R^3))$}\\
	&&&\text{if $\gamma\in(0,\infty)$},\\
	&\left(\,\nabla_y\wa_1,\wa_2\,\right)
	&&\text{for some }
	\left\{\begin{aligned}
	&\wa_1\in L^2(S \times I, \dot{H}^1(\mathcal Y, \R^3))\\
	&\wa_2\in L^2(S \times I,\R^3))
	\end{aligned}\right\}\\
	&&&\text{if $\gamma=\infty$},\\
	\end{aligned}\right.
	\end{equation}
	such that, up to a subsequence, $\wa^h\to\wa_0$ in $L^2$ and
	\begin{equation*}
	\nabla_h\wa^h\wtto d\wa_0\circ T_S + \sum_{i, j = 1}^3(\hat{H}_\gamma)_{ij}\tau^i\otimes\tau^j \qquad\text{weakly
		two-scale on $S^1$}.
	\end{equation*}
	Here, $\wa_0$ is the weak limit in $H^1(S)$ of
	$\int_I \wa^h(x+tn(x))dt$ and $\hat{H}_\gamma \in L^2(S^1 \times \calY; \R^{3 \times 3})$ is defined by
	$\hat{H}_\gamma(x,y) =H_{\gamma}(\pi(x),t(x),y)$.
\end{lemma}	

\subsubsection{Elasticity framework}
Throughout this paper we assume that the limit
$$
\gamma := \lim_{h \to 0} \tfrac{h}{\e(h)}
$$
exists in $[0,\infty]$. We will frequently write $\e$ instead of $\e(h)$,
but always with the understanding that $\e$ depends on $h$. In the case $\gamma=0$ we will additionally assume that $
\e(h)^2 \ll h$. 
\\
From now on we fix a Borel measurable energy density
\begin{equation*}
W:S^{1}\times\R^2\times \R^{3 \times 3}\to [0,\infty]
\end{equation*}
with the following properties:
\begin{itemize}
	\item[(W1)] (continuity) $W(\cdot,y,\boldsymbol F)$ is continuous for almost every $y\in\R^2$ and $\boldsymbol F
\in\R^{3 \times 3}$.
	\item[(W2)] (periodicity) $W(x,\cdot,\boldsymbol F)$ is $\Y$-periodic for all $x\in S^1$ and almost every
	$\boldsymbol F\in\R^{3 \times 3}$.
	\item[(W3)] (objectivity) For all $(x, y)\in S^1\times\Y$ we have $W(x,y, I) = 0$ and
	$W(x, y, \Ra\boldsymbol F) = W(x, y, \boldsymbol F)$
	for all $\boldsymbol F\in\R^{3 \times 3}$, $\boldsymbol R\in\SO 3$.
	%
	\item[(W4)] (uniform quadratic growth near identity) There exist constants $0 < \alpha\leq\beta$ and $\rho > 0$ 
such that
	for all $(x, y)\in S^1\times\Y$ we have
	\begin{align*}
	W(x, y, F) &\geq \alpha\dist^2(\boldsymbol F,\SO 3)\quad\text{ for all }F\in\R^{3 \times 3}
	\\
	W(x, y, F) &\leq \beta\dist^2(\boldsymbol F,\SO 3)\quad\text{ for all }F\in\R^{3\times 3}\mbox{ with }
	\dist^2(\boldsymbol F,\SO 3)\leq\rho.
	\end{align*}
	\item[(W5)] (uniform expansion)For each $(x, y)\in S^1\times\Y$ there exists a
	quadratic form $\mathcal Q(x, y, \cdot) : \R^{3\times 3}\to\R$
	such that
	\begin{equation}
	\label{esup}
	\esssup_{(x,y) \in S^1 \times \calY}
	\frac{|W(x,y,I+G)-\mathcal{Q}(x,y,G)|}{|G|^2}\to 0 \mbox{ as }G\to 0.
	\end{equation}
\end{itemize}

In the following lemma we collect the properties of $\mathcal{Q}(\cdot,\cdot,\cdot)$. They are direct consequence of 
the properties of $W$. For a proof see \cite[Lemma~2.7]{Neukamm-11}.
\begin{lemma}
	\label{lem:1}
	Let $W$ be as above and let $\mathcal{Q}$ be the quadratic form
	associated to $W$ through the expansion \eqref{esup}. Then
	\begin{enumerate}[(i)]
		\item[(Q1)] $\mathcal{Q}(\cdot,y,\cdot)$ is continuous for almost every $y\in\R^2$,
		\item[(Q2)] $\mathcal{Q}(x,\cdot, G)$ is $Y$-periodic
		and measurable for all $x\in\Omega$ and all
		$ G\in\R^{3\times 3}$,
		\item[(Q3)] for all $x\in\Omega$ and almost every
		$y\in\R^2$ the map $\mathcal{Q}(x,y,\cdot)$ is quadratic and
		satisfies
		\begin{equation*}
			\alpha|\sym \boldsymbol G|^2\leq \mathcal{Q}(x,y,\boldsymbol G)=\mathcal{Q}(x,y,\sym \boldsymbol G)\leq 
\beta|\sym \boldsymbol G|^2\qquad\text{for all $ \boldsymbol G\in\R^{3\times 3} $.}
		\end{equation*}
	\end{enumerate}
\end{lemma}



The elastic energy per unit thickness
of a deformation $u^h \in H^1(S^h;\R^3)$ of the shell $S^h$ is given by
$$
J^h(u^h)=\frac{1}{h}\int_{S^h} W\left( \Theta^h(x), r(x)/\e, \nabla u^h(x) \right)\ dx.
$$
In order to express the elastic energy in terms of the new variables, we associate with $y : S^1\to\R^3$
the energy
\begin{eqnarray*}
	I^h(y) &=& \int_{S^1} W\left( x, r(x)/\e, \nabla_h y(x) \right)
	\det\left(I+t(x) \AA(x) \right)^{-1}\ dx \\
	&=& \int_S \int_I W\left( x+t n(x), r(x)/\e, \nabla_h y(x + tn(x)) \right)\ dt\ d\vol_S(x).
\end{eqnarray*}
By a change of variables we have
$$
J^h(u^h) = \frac{1}{h} \int_{S^1} W\left( x, r(x)/\e, \nabla_h y^h(x) \right)\ \left|\det\nabla (\Theta^h)^{-1}(x)
\right|\ dx,
$$
where $y^h (\Xi^h)=u^h$. 
Using \eqref{dphi} it is easy to see  that
\begin{equation} \label{procjena1}
J^h(u^h) = I^h(y^h)\left( 1 + O(h) \right)\mbox{ as }h\to 0,
\end{equation}
where $|O(h)|\leq Ch$.





















\subsection{Asymptotic energy functionals and main result}

Next we will introduce the asymptotic energy functionals.
In order to do so, we need the definition of the relaxation fields and the cell formulas.
\\
Recall that $a\odot b = \frac{1}{2}(a\otimes b + b\otimes a)$.
We now define relaxation operators with range in the space of $L^2$-sections of
the vector bundle over $S$ with fibers given for each $x\in S$ by
\begin{equation}
	\label{bundle}
	L^2\left(I\times\Y,
	(T_x^*S \odot T_x^*S) \oplus (T_x^*S\odot N_x^* S) \oplus (N_x^*S \odot N_x^*S)\right).
\end{equation}
Of course each of these fibers is isomorphic to $L^2(I\times\Y; \R^{3\times 3}_{\sym})$.
We now make the following definitions:

Set $D(\mathcal U_0)  =
\dot{H}^1(\mathcal Y,\R^2)\times \dot H^2(\Y)\times L^2(I\times\mathcal Y,\R^3)$
and define
\begin{align*}
	\mathcal{U}_0 (\zetaa,\varphi, g) &= \Defy\zeta + 2g_{\a} \tau^{\a}\odot n + g_3 n\otimes n - t\Hessy\varphi
	\mbox{ for all }(\zeta, \varphi, g)\in L^2(S, D(\mathcal U_0)).
\end{align*}
%
Set $D\left( \mathcal U_{\infty} \right) =
L^2(I, \dot{H}^1(\mathcal Y,\R^2))\times L^2(I, \dot H^1(\Y))\times L^2(I, \R^3)$
and define
$$
\mathcal{U}_\infty(\zeta,\rho, c) = \Defy\zeta +
2(\d_{y_{\a}}\rho + c_{\a})\tau^{\a} \odot n + c_3 n\otimes n
\mbox{ for all }(\zeta, \rho, c)\in L^2\left(S, D\left( \mathcal U_{\infty} \right)\right).
$$
Set $D(\mathcal U_{\gamma}) =
\dot{H}^1(I\times \mathcal Y;\R^2)\times \dot{H}^1(I\times \mathcal Y)$
and define
$$
\mathcal{U}_\gamma (\zeta, \rho) = \Defy\zeta +
(\d_{y_{\a}}\rho + \frac{1}{\gamma}\d_3\zeta_{\a}) \tau^{\a}\odot n
+ (\frac{1}{\gamma}\d_3\rho) n\otimes n
$$
for all $(\zeta, \rho)\in L^2\left(S, D(\mathcal U_{\gamma})\right).$
\\
By trivially embedding $D(\U_0)$ as constant maps into $L^2(S, D(\U_0))$,
we can regard $\U_0$ also as a map from $D(\U_0)$ itself into \eqref{bundle}.
For each $x\in S$ the fiberwise action $\U_0^{(x)}$ of $\U_0$ is
$$
\U_0^{(x)}(\zeta, g) = (\Defy\zeta)(x) + 2g_{\a} \tau^{\a}(x)\odot n(x) + g_3 n(x)\otimes n(x)
\mbox{ for all }(\zeta, g)\in D(\mathcal U_0).
$$
For each $x\in S$ we define $L_0^{(x)}(I\times\Y) = \U_0^{(x)}(D(\U_0))$, i.e.,
$$
L_0^{(x)}(I\times\Y) = \left\{ \U_0^{(x)}(\zeta, g) : (\zeta, g)\in D(\mathcal U_0)\right\}.
$$
This is a subspace of \eqref{bundle}, i.e., of $L^2(I\times\Y; \R^{3\times 3}_{\sym})$.
We denote by
$L_0(I\times\Y)$ the vector bundle over $S$ with fibers $L_0^{(x)}(I\times\Y)$;
in what follows we will frequently omit the index $(x)$ for the fibers.
The bundles $L_{\gamma}(I\times \Y)$ , for $\gamma \in (0,\infty]$ are defined analogously.
The elements of these spaces are the {\em relaxation fields}.
For $\gamma \in [0,\infty]$ and $x \in S$ we define
$
 \mathcal{Q}_{\gamma} (x) : T_x^*S \otimes T_x^* S \to\R,
$
by setting
%
\begin{equation} \label{defQg}
\mathcal{Q}_{\gamma}(x, q) = \inf_{\bs U \in L^{(x)}_{\gamma} (I \times \calY ) \atop p \in T_x^*S \otimes T_x^* S}
\int_I\int_{\calY} \mathcal{Q}\Big(x + tn(x), y, p + tq + U(t, y) \Big)\ dy \ dt,
\end{equation}
for each $x\in S$ and $q\in T_x^*S \otimes T_x^* S$.
%
For $x\in S$ and $q\in T_x^*S\odot T_x^* S$ define  the homogeneous
relaxation (cf. \cite{Lewicka1}):
\begin{equation}\label{lewicka1}
	\mathcal{Q}_2(x, t, q) =
	\min_{M\in \R^{3\times 3}_{\sym}} \{ \mathcal{Q}(x+tn(x), M): M(T_S, T_S) = q(T_S, T_S)\}.
\end{equation}

Then it is easy to see that
%
\begin{align}
	\label{eq:iggg1111}
	\mathcal{Q}_0(x,q) &= \inf \int_{I \times \mathcal{Y}}
	\mathcal{Q}_2\Big(x+tn(x),y, p + tq+\Defy\zeta - t\Hessy\varphi \Big)\ dt\ dy,	
\end{align}
where both infima are taken over all $\zeta\in \dot H^1(\Y,\R^2)$, all $\varphi\in\dot H^2(\Y)$ and all $p \in T^*S 
\odot T^* S$.
%
In the case when $\mathcal{Q}$ does not depend on $t$ , i.e., the material is homogeneous
in the thickness direction we find:
$$
\mathcal{Q}_0(x,q) =
\frac{1}{12} \inf_{\varphi \in \dot{H}^2 (\mathcal Y)}
\int_{\mathcal{Y}} \mathcal{Q}_2 (x,y,q+ \Hessy\varphi ) \ud y.
$$
\begin{remark}
The formulae \eqref{lewicka1} and \eqref{lewicka1} correspond to the fact that in the regime when dimensional 
reduction dominates, one can, at least on the level of energies, first do the dimensional reduction \eqref{lewicka1} 
and then the homogenization \eqref{eq:iggg1111}. However, doing homogenization of the bending shell model is a 
different problem, since one has to deal with the restriction of being an isometry (see \cite{NeuOlb-13}  for the 
planar case).
\end{remark} 
As in \cite{NeuVel-12}, one can prove pointwise continuity of the energies in the parameter $\gamma$, i.e.,  that for 
all $x \in S$, 
$q \in T_x^*S\odot T_x^*S$ we have
\begin{equation*} \lim_{\gamma \to \infty} \mathcal{Q}_{\gamma}(\hx,q)=\mathcal{Q}_{\infty}(\hx,q)
	\mbox{ and }  \lim_{\gamma \to 0} \mathcal{Q}_{\gamma}(\hx,q) =\mathcal{Q}_0 (\hx,q).
\end{equation*}
For $u \in H^2_{\iso}$, by $\AA^r_u$  we denote the relative change of Weingarten maps 
\begin{eqnarray}\label{izWm}
	\AA^r_u&=&(R\circ \Xi^{-1})^t  \nabla (R \circ \Xi^{-1})T_S \cdot n\\
	&=&(R\circ \Xi^{-1})^t  \nabla ((R \circ \Xi^{-1}) n)T_S-\AA \\
	&=& (R\circ \Xi^{-1})^t  \AA_{u( S)}(R\circ \Xi^{-1})T_S-\AA,
\end{eqnarray}
where $R=\nabla \t u \in H^1(\omega;\SO 3)$ and  we have denoted by $\nabla (\t R^h \circ \Xi^{-1}) T_S\cdot n$ the 
operator on $\R^3$ given by
$$  (\nabla (\t R^h \circ \Xi^{-1}) T_S\cdot n)v=\nabla (\t R^h \circ \Xi^{-1}) T_S v \ n, \textrm{ for all } v \in 
\R^3.  $$
Notice that $\AA^r_u \in  L^2(S; T^*S \odot T^* S)$. 



For $\gamma \in [0,\infty]$ we define the functionals
$I_\gamma : H^1(S;\R^3) \to \R$ by setting
$$
I_\gamma (u) =
\begin{cases}
\int_S \mathcal{Q}_\gamma\left(x,\AA^r_u(x) \right) d\vol_S(x), & \quad \textrm{if } u \in H^2_{\iso}, \\
+\infty, & \quad \textrm{otherwise}.
\end{cases}
$$

We are now ready to state the main result of this part.
\begin{theorem} \label{tm:glavni}
	Let $W$ be as above and assume that
	$\ua^h \in H^1(S^h;\R^3)$ satisfy
	\begin{equation} \label{uuuvjet}
		\limsup_{h \to 0} h^{-2}J^h(\ua^h) < \infty.
	\end{equation}
	Define $\ya^h : S^1 \to \R^3$ by $\ya^h(\Theta^h) = \ua^h$. Put $\gamma=\lim_{h \to 0} \frac{h}{\e(h)}$ and in 
the case $\gamma=0$ assume additionally $\e(h)^2 \ll h$. 
	Then the following are true:
	\begin{enumerate}[(i)]
		\item
		\textit{(lower bound)}. If $y^h$ is such that $\ya^h-\fint_{S^1} y^h \to u$, strongly in $L^2$, then we have 
		$$
		\liminf_{h \to 0} h^{-2} J^h (\ua^h) \geq I_{\gamma}(u).
		$$
		%
		\item \textit{(recovery sequence)}. Assume additionally that $\omega$ is star shaped with respect to some 
ball $B \subset \omega$ and that $S$ is convex, i.e.,  there exists $C>0$ such that for every $x\in S$ we have
		$$  \frac{1}{C} |\tau|^2  \leq \AA(x) \tau\cdot \tau \leq C |\tau|^2,\quad \forall \tau \in T_xS. $$
		Then we have that for any  $u \in H^2_{\iso}$  there exist $\ua^h\in H^1(S^h;\R^3)$
		satisfying (\ref{uuuvjet}), and such that $y^h \to u$, stronly in $H^1$. Moreover, we have
		$$
		\lim_{h \to 0} h^{-2} J^h (\ua^h) =I_{\gamma}(u). 
		$$
	\end{enumerate}
\end{theorem}
This is complemented by the compactness result, given in \cite[Theorem 1]{FJMMshells03}.
\begin{theorem}
	\label{T:FJM-compactness}
	Suppose a sequence $u^h\in H^1(S^h,\R^3)$ has finite
	bending energy, that is (\ref{uuuvjet}) is satisfied. Define $y^h: S^1 \to \R^3$ as above. 
	Then there exists $u\in
	H^2_{\iso}(S)$ such that on a subsequence
	\begin{align*}
	y^h-\fint_{S^1} y^h\,dx&\to u,&\qquad&\text{strongly in }H^1(S^1,\R^3),\\
	\nabla_h y^h&\to R,&\qquad&\text{strongly in
	}L^2(S^1,\R^{3\times 3}),
	\end{align*}
	as $h\to 0$ after passing to subsequences and extending $u$ and $n$ trivially to $S^1$. Here $R \in H^1(S,\SO 3)$ 
is defined by $RT_S=\nabla u T_S$ and $Rn=R\tau_1 \times R\tau_2$. 
\end{theorem}

\subsection{Two scale compactness and lower bound}
We impose that the sequence $(u^h)\subset  H^1 (S^h,\R^3)$ satisfies 
\begin{equation}
\label{uuuvjet} \limsup_{h \to 0} h^{-2} J^h(u^h)<\infty,
\end{equation}
and  for a given sequence $(u^h)\subset  H^1 (S^h;\R^3)$ we define the sequence $(y^h) \subset  H^1 (S^1;\R^3) $ by 
the expression $y^h(\Theta^h)=u^h$. 
The following lemma gives us the compactness result we need. It
is a direct consequence of \cite[Theorem 3.1]{FJM-02} and of the arguments in
\cite{FJM-06}. We refer also to  \cite{FJMMshells03}. The result is obtained by dividing 
the set $\omega$ into the small cuboids $(C^h_n)_{i=1}^n$ of size $\delta (h)$ in the in-plane direction and height 
$h$  and then applying the theorem on geometric rigidity on the domains $\Xi(C^h_i)$ for every $i$ and $h>0$ and by 
smoothening.  


\begin{lemma}
	\label{t6f2}
	There exist constants $C,c>0$, depending only
	on $\omega$, such that the following is true for every $h>0$ sufficiently small:
	if $u\in H^1(S^h, \R^3)$ then there exists a map $R : \omega\to SO(3)$
	which is piecewise constant on each cube $x + \delta(h)Y$ with $x\in\delta(h)\Z^2$ and there exists
	$\t{R}\in H^1(\omega, \R^{3 \times 3} )$ such that for each $\xi \in \R^2$ which satisfy $|\xi|_\infty=\max\{|\xi 
\cdot e_1|,|\xi \cdot e_2|\}\leq h$ and for each $\t \omega \subset \omega$  which satisfy $\dist(\t \omega,\partial 
\omega)>ch$ we have:
	\begin{eqnarray*}
		&& \|\nabla u\circ (\Theta^h)^{-1}\circ \Xi - R\|^2_{L^2(\t \omega \times I)} + \|R - \t{R}\|^2_{L^2(\t 
\omega)}+h^2 \|R - \t{R}\|^2_{L^\infty(\t \omega)} \\
		&& + h^2\|\nabla'\t{R}\|^2_{L^2(\t \omega)}+\|\tau_{\xi} R-R\|^2_{L^2(\t \omega)} \leq C \left\|\dist 
\left(\nabla u\circ (\Theta^h)^{-1}\circ \Xi, \SO 3\right) \right\|^2_{L^2(\Omega)},
	\end{eqnarray*}
	where $(\tau_{\xi} R)(z'):=R(z'+\xi)$. Here 
	\begin{equation*}
	\delta(h):=  \left\{ \begin{aligned} \e(h),&&  \text{if }  \lim_{h \to 0} \frac{h}{\e(h)}>0 \\ \lceil\tfrac{h}
{\e(h)}\rceil \e(h), && \text{if } \lim_{h \to 0} \frac{h}{\e(h)}=\infty,\\ h, &&  \text{otherwise.}
	\end{aligned} \right. 
	\end{equation*} 
\end{lemma}
For a sequence $(u^h)$ that satisfies (\ref{uuuvjet}), we define the sequence $(G^h)\subset L^2 (S^1;\R^{3 \times 
3} )$ of approximate strains with 
\begin{equation} \label{apstrain}
G^h =\frac{(R^h \circ \Xi^{-1})^t \nabla u^h \circ(\Theta^h)^{-1}-I}{h},
\end{equation}
where $R^h: \omega \to \SO 3$ is a sequence of piecewise constant maps given by Lemma \ref{t6f2}.
Notice that $\nabla u^h \circ(\Theta^h)^{-1}=\nabla_h y^h$, where the operator $\nabla_h$ is given by (\ref{defDh}).

For  $\t \omega \Subset \omega$  we denote by $\t S= \Xi(\omega)$ and by $\t S^1=\t \omega \times I$. 
We are interested in identifying two-scale limits of the sequence $(G^h)$. 
We will not give the complete proof, only reduce the case to the planar situation , when the rest of the proof goes 
by not so difficult adaptation to the proofs given in \cite{Horneuvel12,Vel13}.
\begin{proposition}\label{dvoskalnilimesi}
Let $(u^h)$ be a sequence of deformations that satisfies (\ref{uuuvjet}), let $G^h$ be defined with (\ref{apstrain}) 
and let $u \in H^2_{\iso}(S)$ be determined by Theorem \ref{T:FJM-compactness}. Then for every $\t \omega \Subset 
\omega$  such that $\partial \t \omega$ is $C^{1,1}$  we have the following convergence on a subsequence:
\begin{itemize}
\item 
case $\gamma \in (0,\infty)$: there exist
$B \in L^2 (\t S, T^* S \odot T^*S)$ and $ (\zeta, \rho)\in L^2(\t S, D(\mathcal U_{\gamma}))$ such that
\begin{eqnarray*}
	\sym G^h \wtto B+t\AA^r_{u}+ \U_{\gamma}(\zeta,\rho);
\end{eqnarray*}	
\item case $\gamma=\infty$: there exist 
$B \in L^2 (S, T^* S \odot T^*S)$ and 
$(\zeta,\rho,c) \in L^2(\t S; \DD(\U_{\infty}))$
such that
$$
\sym G^h \wtto B+t\AA^r_{u}+ \U_{\infty}(\zeta,\rho,c),
$$
\item case $\gamma=0$, $\varepsilon^2 \ll h \ll \varepsilon$: 
there exist $B \in L^2 (\t S; T^* S \odot T^*S)$ and $ (\zeta, \varphi, g)\in L^2(\t S, D(\mathcal U_0))$ such that
\begin{eqnarray*}
	\sym G^h \wtto B+t\AA^r_{u}+ \U_{0}(\zeta,\varphi,g).
\end{eqnarray*}

\end{itemize}
\end{proposition}	
\begin{proof}
Take $\t \omega \Subset \omega$ such that $\t \omega$ has $C^{1,1}$ boundary 
We decompose the deformation $y^h$ on the set $\t S^1=\t \omega \times I$, where $\t \omega \Subset \omega$ is such 
that $\partial \t \omega$ is $C^{1,1}$.  
\begin{equation} \label{dekompoz1}
y^h(x+tn)=\bar{y}^h(x)+t h(\t R^h \circ \Xi^{-1}) n +h z^h(x+tn), 
\end{equation}
 where $\t R^h$ is given by Lemma \ref{t6f2} and $\bar{y}^h (x)= \int_I y^h(x+tn(x)) dt$. 
 Notice that 
 \begin{equation} \label{usrednjenje}
 \int_I z^h(x+tn) dt=0.
 \end{equation} 
 We compute using \eqref{dpi-2} and \eqref{fladh} 
\begin{eqnarray} \label{rastav}
\nabla_h y^h &=& \nabla \bar{y}^h T_S(I+th\AA)^{-1}+(\t R^h \circ \Xi^{-1}) n \otimes n\\ \nonumber & &+th \nabla (\t 
R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1}\cdot n +  th(\t R^h \circ \Xi^{-1}) \AA T_S(I+th\AA)^{-1}\\ \nonumber & &  +h 
\nabla_h z^h.
\end{eqnarray}

Notice that
\begin{equation} \label{borisbye}
 \nabla \bar{y}^h=\int_I \nabla y^hT_S (I+t\AA) dt.  
 \end{equation} 
From Lemma \ref{uuuvjet} and \eqref{t6f2}  we conclude that
$$ \| \nabla_h y^h-\t R^h \circ \Xi^{-1} \|_{L^2(\t S^1)}\leq Ch. $$
From the identity \eqref{fladh} and \eqref{borisbye} we also conclude that 
\begin{equation} \label{occcjena}
 \|\nabla \bar{y}^h-\t (R^h \circ \Xi^{-1})T_S\|_{L^2(\t S^1)} \leq Ch. 
\end{equation} 
From the last two expressions and (\ref{rastav}) we can conclude that
$$ \limsup_{h \to 0} \| \nabla_h z^h\|_{L^2(\t S^1)} < \infty.  $$
By using Poincare inequality and (\ref{usrednjenje}) we also conclude that
$$ \limsup_{h \to 0} \|z^h\|_{L^2(\t S^1)} < \infty. $$
In order to identify the two scale limit of the approximate strain we decompose it in the following form 
\begin{eqnarray*}
G^h  &=& (R^h \circ \Xi^{-1})^t \frac{\nabla \bar{y}^h T_S(I+th\AA)^{-1}-(R^h\circ \Xi^{-1}) T_S}{h}\\ & &+(R^h\circ 
\Xi^{-1})^t \frac{(\t R^h \circ \Xi^{-1})-( R^h \circ \Xi^{-1})}{h}  n \otimes n\\ 
& &+t(R^h\circ \Xi^{-1})^t\nabla(\t R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1}\cdot n \\ & &+  t(R^h\circ \Xi^{-1})^t (\t 
R^h \circ \Xi^{-1}) \AA T_S(I+th\AA)^{-1}\\ \nonumber & &  +(R^h \circ \Xi^{-1})^t \nabla_hz^h,
\end{eqnarray*}
where, similarly as before, by $\nabla (\t R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1}\cdot n$ we have denoted the operator 
on $\R^3$ given by
$$  (\nabla (\t R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1}\cdot n)v=\nabla (\t R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1} v \ 
n, \textrm{ for all } v \in \R^3.  $$
Notice that 
\begin{eqnarray*}
\frac{\nabla \bar{y}^h T_S(I+th\AA)^{-1}-(R^h\circ \Xi^{-1}) T_S}{h}&=& \frac{\nabla \bar{y}^h T_S-(R^h\circ 
\Xi^{-1}) T_S}{h}-t\nabla \bar{y}^h \AA T_S\\ & & +O(h^2),
\end{eqnarray*}
where $\|O(h^2)\|_{L^2(\t S^1)} \leq Ch^2$. 
Notice that $\nabla \bar{y}^h \tau_{\alpha}=\partial_{\alpha}  \t{\bar{y}}^h $, for $\alpha=1,2$. 
Since now we can, by pulling back, easily translate the problem to the planar one by using the techniques developed 
in \cite{Horneuvel12} (see the proof of Proposition 3.2) and Lemma \ref{lm:50} we can conclude the following
\begin{itemize}
\item case $\gamma \in (0,\infty)$ \\
There exist $w^0 \in L^2 (\t S,\R^3)$, $w,v \in L^2(\t S, \dot{H}^1(\calY,\R^3))$, $ \t B\in L^2(\t S, \R^{3 \times 
2}) $, $ z \in L^2(\t S ,\dot{H}^1(I\times\mathcal Y, \R^3))$ such that
\begin{eqnarray*}
\frac{\nabla \bar{y}^h T_S-(R^h\circ \Xi^{-1}) T_S}{h}  &\wtto& \sum_{i\leq3;j \leq2} \t B_{ij}(x)   \tau^i \otimes 
\tau^j\\ & &+\frac{1}{\gamma}(y \cdot  \nabla')R \circ \Xi^{-1}(x) T_S\\ & &+\sum_{i\leq3;j \leq2}  \partial_{y_j} 
v_i(x,y) \tau^i \otimes \tau^j;\\
\frac{(\t R^h \circ \Xi^{-1})-( R^h \circ \Xi^{-1})}{h} n &\wtto & \tfrac{1}{\gamma} \sum_{i=1}^3(w^0_i(x)+w_i(x,y))
\tau^i\\ & &+\tfrac{1}{\gamma} (y\cdot \nabla') R \circ \Xi^{-1}(x)n ;\\
\nabla(\t R^h \circ \Xi^{-1}) T_S(I+th \AA)^{-1} \cdot n &\wtto& \nabla (R \circ \Xi^{-1}(x))T_S \cdot n\\ & &+
\sum_{i \leq 3; j\leq 2} \partial_{y_j}  w_i(x,y) \tau^i \otimes \tau^j;\\
\nabla_h z^h &\wtto& \sum_{i,j=1}^3\left(\,\nabla_y z(x,t,y),\tfrac{1}{\gamma}\partial_3 z(x,t,y)\,\right)_{ij} 
\tau^i \otimes \tau^j. 
\end{eqnarray*}
Here $R$ is $L^2$ limit of the sequence $\t R^h$, i.e., $R^h$ and  we denoted by $(y \cdot \nabla'R)= y_1 \partial_1 
R+y_2 \partial_2 R$. Using the fact that 
$R^t\partial_\alpha R$ for $\alpha=1,2$ is skew symmetric matrix we conclude that there exist
$B \in L^2 (\t S, T^* S \odot T^*S)$ and $ (\zeta, \rho)\in L^2(\t S, D(\mathcal U_{\gamma}))$ such that
\begin{eqnarray*}
\sym G^h \wtto B+t\AA^r_{u}+ \U_{\gamma}(\zeta,\rho),
\end{eqnarray*}
where
\begin{eqnarray*}
B&=& \sum_{\alpha,\beta=1,2} \left((M \t B)_{\alpha \beta}+(M \t B)_{\beta \alpha}\right)\tau^{\alpha} \odot 
\tau^{\beta}.         \\
\left(\begin{array}{c} \zeta \\ \rho \end{array} \right)&=&M(z+v+tw^0+tw)+\gamma t \left(\begin{array}{c} (M \t 
B)_{31} \\ (M \t B)_{32} \\ 0  \end{array} \right),            
\end{eqnarray*} 
and where $M \in L^{\infty}(S;\R^{3 \times 3})$; $M_{ij}=(R \circ \Xi^{-1})^t\tau^j \cdot \tau_i $.
\item case $\gamma=\infty$. In the same way as before we can conclude that
there exist $w^0 \in L^2 (\t S,\R^3)$, $w,v \in L^2(\t S, \dot{H}^1(\calY,\R^3))$, $ \t B\in L^2(\t S , \R^{3 \times 
2}) $, $ z \in L^2(\t S \times I ,\dot{H}^1(\mathcal Y, \R^3))$, $d \in L^2(\t S \times I, \R^3)$ such that
\begin{eqnarray*}
	\frac{\nabla \bar{y}^h T_S-(R^h\circ \Xi^{-1}(x)) T_S}{h}  &\wtto& \sum_{i\leq3;j \leq2} \t B_{ij}(x) \tau^i 
\otimes \tau^j\\ & &+\sum_{i\leq3;j \leq2}  \partial_{y_j} v_i(x,y) \tau^i \otimes \tau^j;\\
	\frac{(\t R^h \circ \Xi^{-1})-( R^h \circ \Xi^{-1})}{h} n &\wtto & \sum_{i=1}^3 w^0_i(x)\tau^i;\\
	\nabla(\t R^h \circ \Xi^{-1}) T_S(I+th \AA)^{-1} \cdot n &\wtto& \nabla (R \circ \Xi^{-1}(x))T_S \cdot n\\ & &+
\sum_{i \leq 3; j\leq 2} \partial_{y_j}  w_i(x,y) \tau^i \otimes \tau^j;\\
	\nabla_h z^h &\wtto& \sum_{i,j=1}^3\left(\,\nabla_y z(x,t,y),d(x,t)\,\right)_{ij} \tau^i \otimes \tau^j. 
\end{eqnarray*}
From this we conclude that there exist 
$B \in L^2 (S; T^* S \odot T^*S)$ and 
$(\zeta,\rho,c) \in L^2(\t S; \DD(\U_{\infty}))$
such that
$$
	\sym G^h \wtto B+t\AA^r_{u}+ \U_{\infty}(\zeta,\rho,c),
$$
where 
\begin{eqnarray*}
 B&=& \left((M \t B)_{\alpha \beta}+(M \t B)_{\beta \alpha}\right) \tau^{\alpha} \odot \tau^{\beta}, \\
 \left(\begin{array}{c} \zeta \\ \rho \end{array} \right) &=& M\left(\begin{array}{c} \frac{1}{2}(w_1+v_1) \\ 
\frac{1}{2}(w_2+v_2) \\ w_3+v_3  \end{array} \right)+Mz, \\
 c &=& M\left(\begin{array}{c} \frac{1}{2}(d_1+w^0_1) \\ \frac{1}{2}(d_2+w^0_2) \\ d_3  \end{array} \right)+ 
\left(\begin{array}{c} \frac{1}{2}(M \t B)_{31} \\ \frac{1}{2}(M \t B)_{32} \\ 0  \end{array} \right),
\end{eqnarray*}
and $M$ is defined above. 
\end{itemize}
The  case $\gamma=0$, $\varepsilon(h)^2 \ll h \ll \varepsilon(h)$ requires different approach. 
We will explain how to adapt the approach from \cite{Vel13}. Firstly by using (\ref{occcjena})
we conclude that
$$\|\nabla'\t{\bar{y}}^h-(\t R^h (\tau_1\circ \xi),\t R^h (\tau_2 \circ \xi))\|_{L^2(\t \omega; \R^{3 \times 2})} 
\leq Ch. $$ 
Denote by $p^h=(\t R^h (\tau_1\circ \xi),\t R^h (\tau_2 \circ \xi)) $. We look the following minimization problem
\begin{equation*}
	\min_{\t v^h\in H^1(\t \omega,\R^3)\atop\int_{\t \omega} \t v^h=0}\int_{\t S}|\nabla' \t v^h-p^h  |^2\,d x'.
\end{equation*}
 The
associated Euler-Lagrange equation reads
\begin{equation*}
	\left\{\begin{aligned}
		-\triangle' \t v^h=\,&\,-\nabla\cdot p^h \qquad&&\text{in } \t \omega\\
		\partial_\nu \t v^h=\,&\, p^h \cdot\nu\qquad&&\text{on }\partial \t \omega,
	\end{aligned}\right.
\end{equation*}
subject to $\int_{\t{\omega}}\t  v^h\,dx=0$. Above, $\nu$ denotes the normal on
$\partial \t{\omega}$. We obtain by
standard  regularity estimates that  $\t v^h \in H^2 (\t \omega;\R^3)$ under the assumption that $\partial\t \omega$ 
is $C^{1,1}$. In this case it holds
\begin{eqnarray*} \label{balicka}
\|\t v^h\|_{H^2(\t \omega;\R^3)}&\leq& C( \|\nabla\cdot p^h\|_{L^2(\t \omega;\R^3)}+\|p^h\|_{L^2(\t \omega;\R^3)}) \\ 
&\leq& C h,\\
\|\nabla' \t v^h-\nabla' \t{\bar{y}}^h\|_{L^2(\t \omega;\R^3)} &\leq& Ch, \\
\|\nabla'\t v^h-(\t R^h (\tau_1\circ \xi ),\t R^h (\tau_2\circ \xi ) )\|_{L^2(\t \omega;\R^{3\times 2})} & \leq& Ch,
\end{eqnarray*}
for some $C>0$. We now decompose the sequence of the deformations $(y^h)$ in the following way
\begin{equation}
y^h(x+tn)=v^h(x)+(\bar{y}^h(x)-v^h(x))+t h(\t R^h \circ \Xi^{-1}) n +h z^h(x+tn),
\end{equation}
where $v^h= \t v^h \circ \xi$. We also decompose the sequence of strains $(G^h)$ in the following way
\begin{eqnarray*}
G^h&=&
\frac{1}{h}\sum_{\alpha,\beta=1,2}  \left( (R^h \circ \Xi^{-1}) \tau_{\alpha} \cdot \nabla v^h T_S (I+th\AA)^{-1} 
\tau_{\beta} \right) \tau^{\alpha} \otimes \tau^{\beta}\\ & &+\frac{1}{h}\sum_{\alpha=1,2}\left((R^h \circ \Xi^{-1}) 
n \cdot \nabla v^h T_S (I+th\AA)^{-1} \tau_{\alpha}\right) n \otimes \tau^\alpha \\ \nonumber & &+\frac{1}{h} 
\left(R^h\circ \Xi^{-1})^t(\nabla \bar{y}^hT_S(I+th\AA)^{-1}-\nabla v^hT_S(I+th\AA)^{-1}\right) 
\\ & &+ \frac{(R^h\circ \Xi^{-1})^t(\t R^h \circ \Xi^{-1})-I}{h} n \otimes n\\ 
& &+t(R^h\circ \Xi^{-1})^t\nabla(\t R^h \circ \Xi^{-1}) T_S(I+th\AA)^{-1}\cdot n \\ & &+  t(R^h\circ \Xi^{-1})^t (\t 
R^h \circ \Xi^{-1}) \AA T_S(I+th\AA)^{-1}\\ \nonumber & &  +(R^h \circ \Xi^{-1})^t \nabla_hz^h.
\end{eqnarray*}
By using the tools developed in \cite{Vel13} for the planar case, adapted to this framework,  we can conclude that 
there exist $B^1 \in L^2 (\t S, T^* S \odot T^* S)$, $B^2 \in L^2(\t S,\R^{3 \times 2} )$, $ w \in  L^2(\t S,\dot{H}
^1(\mathcal Y,\R^2))$, $p \in L^2(\t S\times \mathcal Y,\R^2)$, $k \in L^2(\t S; \dot{H}^1(\calY,\R^3))$, $ r\in 
L^2(\t S, H^1(\mathcal{Y},\R^3))$, $\varphi \in  \dot H^2(\Y)$, $\phi \in L^2(\t S \times I, \dot{H}^1(\mathcal Y, 
\R^3))$, $d \in L^2(\t S \times I\times\mathcal Y,\R^3)$ such that


\begin{eqnarray*}
\frac{1}{h}\sum_{\alpha,\beta=1,2}  \left( (R^h \circ \Xi^{-1}) \tau_{\alpha} \cdot \partial_{\tau_{\beta}}v^h 
\right) \tau^{\alpha} \otimes \tau^{\beta} &\wtto& B^1(x)+\Defy w, \\
\frac{1}{h}\sum_{\alpha=1,2}\left((R^h \circ \Xi^{-1}) n \cdot \partial_{\tau_{\alpha}}v^h \right) n \otimes \tau^
\alpha &\wtto& p_{\alpha}(x,y) n \otimes \tau^\alpha, \quad \textrm{for } \alpha=1,2,\\
\frac{1}{h} \left(R^h\circ \Xi^{-1})^t(\nabla \bar{y}^hT_S-\nabla v^hT_S\right) &\wtto& \sum_{i\leq3;j \leq2} 
B^2_{ij}(x) \tau^i \otimes \tau^j\\ & &+\sum_{i\leq3;j \leq2}  \partial_{y_j} k_i(x,y) \tau^i \otimes \tau^j, \\
\frac{(R^h\circ \Xi^{-1})^t(\t R^h \circ \Xi^{-1})-I}{h} n &\wtto& \sum_{i=1}^3 l_i(x,y)\tau^i;\\
\nabla(\t R^h \circ \Xi^{-1}) T_S\cdot n &\wtto& R \Hessy \varphi, \\
\nabla_h z^h &\wtto&  \sum_{i,j=1}^3\left(\,\nabla_y \phi(x,y),d(x,y,t)\,\right)_{ij} \tau^i \otimes \tau^j.
\end{eqnarray*}
From this we conclude that there exist $B \in L^2 (\t S, T^* S \odot T^*S)$ and $ (\zeta, \varphi, g)\in L^2(\t S, 
D(\mathcal U_0))$ such that
\begin{eqnarray*}
	\sym G^h \wtto B+t\AA^r_{u}+ \U_{0}(\zeta,\varphi,g),
\end{eqnarray*}
where
\begin{eqnarray*}
B &=& B^1+\left((M  B^2)_{\alpha \beta}+(M  B^2)_{\beta \alpha}\right) \tau^{\alpha} \odot \tau^{\beta} \\
\zeta &=& w+ \left((Mk)_1, (Mk)_2\right)^t+(\phi_1,\phi_2)^t \\
g&=& \left(\begin{array}{c} \frac{1}{2}(p_1+l_1+d_1) \\ \frac{1}{2}(p_2+l_2+d_2)\\ l_3+d_3  \end{array} \right)+
\left(\begin{array}{c} \frac{1}{2}(M B^2)_{31} \\ \frac{1}{2}(M  B^2)_{32} \\ 0  \end{array} \right)+\frac{1}
{2}\left(\begin{array}{c} \partial_{y_1} (Mk)_3 \\ \partial_{y_2} (Mk)_3 \\ 0  \end{array} \right).
\end{eqnarray*}
\end{proof} 

\begin{remark}\label{boris1}
We were able to completely identify oscillatory part of two-scale limit of strain without using the convexity of the 
shell. The weak limit contained in the field $B$ remains unknown. We will use the convexity in the construction of 
the recovery sequence by showing that in that case an arbitrary matrix field $B$ can be recovered. In the plate case 
this is not completely true, but it is true that some appropriate class of the fields $B$ can be recovered. However, 
it is commented that the class contains the minimizers that we need for the relaxation.	
\end{remark}
We will just give the sketch of the proof of the lower bound since it follows standard arguments (truncation, Taylor 
expansion and lower semicontinuity of integral functional with respect to two-scale convergence).
\subsubsection*{Proof of Theorem \ref{tm:glavni} - lower bound}  
Without loss of generality we may assume that
$\fint_{S^1}y^h\,dx=0$ and $\limsup_{h\to
	0}$ $h^{-2} J^{h}(u^h)<\infty$. We only consider the case
$\gamma\in(0,\infty)$. The proof in other cases is similar.
By using Theorem \ref{T:FJM-compactness} we conclude that $u \in H^2_{\iso}(S)$ and that stronger convergence, given 
in that theorem are satisfied. 
Take an arbitrary $\t \omega \Subset \omega$ with $C^{1,1}$ boundary.
By using Proposition \ref{dvoskalnilimesi}  we have that there exists $B \in L^2 (\t S; T^* S \odot T^*S)$ and $ 
(\zeta, \rho)\in L^2(\t S, D(\mathcal U_{\gamma}))$ such that
\begin{eqnarray*}
	\sym G^h \wtto B+t\AA^r_{u}+ \U_{\gamma}(\zeta,\rho);
\end{eqnarray*}	
As explained in \cite{Neukamm-11}
(cf. \cite{FJM-02} for the corresponding argument in the homogeneous case), a careful Taylor expansion of $W(x
+tn(x),r(x)/\e ,\id+h G^h(x))$
combined with the lower semi-continuity of convex integral functionals
with respect to  weak two-scale convergence (see e.g. \cite[Proposition~1.3]{Visintin-07}) and using relations 
(\ref{esup}), \eqref{procjena1}  yields the lower bound
\begin{align*}
\liminf\limits_{h\to 0}h^{-2} J^h(u^h) & =
\liminf\limits_{h\to 0}h^{-2} I^h(y^h)
\\
&= \int_{\t S^1}\int_{I \times Y}\mathcal{Q}\left(x,t,y,B+t\AA^r_u (x)+\mathcal{U}_{\gamma}(\zeta,\rho)\right)\,dt\, 
dy\,dvol_S.
\end{align*}
Minimization over $B\in L^2(\t S,\R^{2\times 2})$ and $ (\zeta, \rho)\in L^2(\t S, D(\mathcal U_{\gamma}))$ yields
\begin{equation*}
\liminf\limits_{h\to 0} h^{-2} J^h(u^h)\geq  \int_{\t S^1} \mathcal{Q}_{\gamma}(x, \AA^r_u (x)) dvol_S.
\end{equation*}
By using the properties of $\mathcal{Q}_{\gamma}$ (see Lemma \ref{lem:2b}) and exhausting $\omega$ with  the sequence 
$(\t \omega_n)_n$ such that for each $n \in \N$ we have $\t \omega_n \Subset \omega$ and $\t \omega_n$ has $C^{1,1}$ 
boundary we have the claim. 
\subsection{Upper bound}
The proof of upper bound has the analogy with the planar cases. The crucial ingredient here is, as already commented 
in Remark \ref{boris1}  that every matrix field $B$ can be recovered.  
It is easy to see, by using Korn's inequality, that each fiber of
$L_\gamma(I \times\mathcal Y)$ 
for $\gamma \in [0,\infty]$  is a closed subspace of
$L^2 (I \times\mathcal Y,\R^{3 \times 3}_{\sym})$.
Also by Korn's inequality it is easy to see
(see also \cite{Neukamm-10,Neukamm-11}) that the following
coercivity bound is satisfied:
%
$$
\|\mathcal{U}_0(\zetaa,\bs g)\|^2_{L^2}\geq C(\|\zetaa\|_{H^1}^2+\|\bs g\|_{L^2}^2)
\mbox{ for all }(\zeta, g)\in D(\U_0),
$$
where the constant $C$ depends on the embedding $\xi$.
Analogous bounds are satisfied by $\U_{\gamma}$, for $\gamma \in (0,\infty]$
with the obvious norms on their respective domains of definition.


\begin{comment}
\begin{eqnarray*}
\|\zetaa\|_{H^1}^2+\|\varphi\|_{H^2}+\|\bs g\|_{L^2}^2 &\lesssim&  \|\mathcal{U}_0^0(\zetaa,\varphi, \bs g)\|
^2_{L^2},\\ && \hspace{-15ex} \forall \zetaa \in \dot{H}^1(\calY;\R^2),\varphi \in \dot{H}^2(\calY), \bs g \in L^2(I
\times \calY;\R^3) \text{ and }  \\ && \\
\|\zetaa\|_{H^1}^2+\|\varphi\|_{H^2}+\|\bs g\|_{L^2}^2 &\lesssim&  \|\mathcal{U}_{0,\gamma_1}^{1} (\zetaa,\varphi,\bs 
g)\|^2_{L^2},\\ && \hspace{-15ex} \forall \zetaa \in \dot{H}^1(\calY;\R^2),\varphi \in \dot{H}^2(\calY), \bs g \in 
L^2(I\times \calY;\R^3) \text{ and } \gamma_1 \in (0,\infty) \\ && \\
\|\zetaa\|_{H^1}^2+\|\xi\|_{H^1}+\|\bs c\|_{L^2}^2 &\lesssim&  \|\mathcal{U}_\infty (\zetaa,\xi,\bs c)\|^2_{L^2},\\ 
&& \hspace{-10ex} \forall \zetaa \in \dot{H}^1(\calY;\R^2),\xi \in L^2(I;\dot{H}^1(\calY)), \bs c \in L^2(I;\R^3),\\ 
&& \\
\|\phia\|_{H^1}^2 &\leq& C(\gamma) \|\mathcal{U}_\gamma (\phia)\|_{L^2}^2, \ \forall \phia \in \dot{H}^1(I \times 
\calY;\R^3).
\end{eqnarray*}
Here the constant absorbed into the symbol $\lesssim$ depends on $\psi_1,\psi_2$.
\end{comment}










The following  lemma is analogous to \cite[Lemma~2.10, 2.11]{NeuVel-12}.

\begin{lemma}
	\label{lem:2b}
	For $\gamma \in [0,\infty]$ we have the following
	\begin{enumerate}[(a)]
	\item the function
	$\mathcal{Q}_\gamma:S\times TS \otimes TS\to [0, \infty)$ is continuous,
	and there exists a constant $C>0$ depending only on the energy density
	$W$ and the surface $S$ such that, for all $x\in S$ and all $q \in T^*_xS\odot T^*_xS$,
	\begin{equation}\label{boris2}
	C^{-1}|q|^2 \leq \mathcal{Q}_\gamma( x,q)
	\leq C|q|^2.
	\end{equation} 
	For fixed $x \in S$, the function $\mathcal{Q}_\gamma(x):T^*_xS\odot T^*_xS\to [0, \infty)$ is a quadratic form. 
	\item 
	there exists a bounded linear operator
	\begin{equation*}
		{\bs \Pi_\gamma}: L^2(S, T^{*}S \otimes T^{*}S)\to L^2(S,T^{*}S \odot T^{*}S) \times L^2(S,L_{\gamma}(I 
\times \mathcal{Y})),
	\end{equation*}
	such that for almost every $x \in S$ we have
	$$
	\mathcal{Q}_{\gamma}(x, q) =
	\int_I\int_{\calY} \mathcal{Q}\Big(x + tn(x), y, 	{\bs \Pi_\gamma^1}[q](x) + tq(x) +
	{\bs\Pi_\gamma^2}[q](x, t, y) \Big) \ud y \ud t,
	$$
	where ${\bs \Pi_\gamma^1}[q]$, ${\bs \Pi_\gamma^2}[q]$  are projections of 	${\bs \Pi_\gamma}[q]$ on 
	$L^2(S,TS \odot TS)$ and $L^2(S,L_{\gamma}(I \times \mathcal{Y})$ respectively. 
	Moreover, if $q \in C^0(S,TS \otimes TS)$ then
	$
	{\bs \Pi_\gamma}[q]\in C^0( S, T^{*}S \odot T^{*}S) \times C^0(S,L_{\gamma}(I \times \mathcal{Y})).
	$
\end{enumerate}
\end{lemma}

\begin{proof} We will just give the sketch of the proof for $\gamma \in (0,\infty)$. We introduce the space $
\mathcal{V}_{x}=T^{*}S \odot T^{*}S+L_{\gamma}(I \times \mathcal{Y})$ (the sum is orthogonal in the $L^2$-norm and 
thus the projections on the components are well defined and linear). Notice that for fixed $x \in S$ and $q \in T^{*}
_xS \otimes T^{*}_xS$   we can look the minimization equation \eqref{defQg} as  projecting  the function $t\sym q$ on 
the linear space $\mathcal{V}_{x}$ in the norm determined by the quadratic form $\mathcal{Q}$ (which is equivalent to 
the $L^2$- norm). From this we obtain the uniqueness and the linearity of the minimizer for $q \in T^{*}_xS \odot 
T^{*}_xS$. (a) can be now easily proved (see also \cite{NeuVel-12}). Also the part (b) is easily proved, firstly for 
$q \in  C^0(S,TS \otimes TS)$ and then  for $q \in L^2(S,TS \otimes TS)$ by extension,  using the inequalities in 
\eqref{boris2}.  The continuity of ${\bs \Pi_\gamma}[q]$ for $q \in C^0(S,TS \otimes TS)$ follows 
from the property (Q1) stated in Lemma \ref{lem:1}. 
\end{proof}
The proof of the following lemma is given in \cite[Lemma 5.6]{Lewicka1}. 
It uses the results in \cite{Nirenberg1}.
We just augment it with the regularity statement that can be easily seen by the arguments given there and standard 
elliptic regularity. 
\begin{lemma}\label{pomocnalema}
Let $\omega \subset \R^2$ be simply connected with $C^2$ boundary. Let $s: \overline{\omega} \to \R^3$ be of class 
$C^{2,1}(\overline{\omega},\R^3)$. Assume additionally that $s(\omega)$ defines the local surface such  the 
Weingarten map $\AA_s$ is strictly positive (or negative) definite  up to the boundary, i.e., there exists $C>0$ such 
that  
\begin{equation*}
\forall z' \in \omega, \forall v \in \R^2, \quad \frac{1}{C} |v|^2 \leq \AA_s(z')v \cdot v \leq C|v|^2,
\end{equation*}
where $(\AA_s)_{\alpha\beta}=\partial_{\alpha} n_s \partial_{\beta}s$, for $\alpha, \beta=1,2$; $n_s = \tfrac{\d_1 s
\times\d_2 s}{|\d_1 s\times\d_2 s|}$. Then for every $B \in L^2(\omega, \R^{2\times 2}_{\sym})$ the system 

$$q^s_w=B, \quad (q^s_w)_{\alpha \beta}=\tfrac{1}{2}(\partial_{\alpha}s \partial_{\beta}w+\partial_{\alpha}w 
\partial_{\beta}s), \textrm{ for }  \alpha, \beta=1,2,$$ has a solution $w \in H^1(\omega, \R^3)$. If we assume that 
$s \in C^{3,\alpha}(\omega,\R^3)$, $B \in C^{2,\alpha}(\omega,\R^{2\times 2}_{\sym})$, then we have that $w \in 
C^{3,\alpha}(\omega,\R^3)$. 
\end{lemma}
\begin{remark}
The system $q_w^s=B$ arises naturally in the geometrical question of the problem of continuation of the infinitesimal 
bendings (see, e.g., \cite{HorLewRez13,LewMorPak11}). In \cite{Nirenberg1} this system arises as one step 
linearization of solving the Weyl's problem (finding a convex surface with prescribed metric) by a cotinuation 
method. 
\end{remark}

\subsubsection*{Proof of Theorem \ref{tm:glavni} - upper bound} 
\begin{itemize}
\item case  $\gamma \in (0,\infty)$:  
Without loss of generality we may assume that $\omega$ is star-shaped with respect to some ball $B(0,r)$ centered at 
the origin. Notice that, as a consequence of this, we have that for $\omega^{\lambda}:=\lambda \omega$, $(0<
\lambda<1)$, it holds $\omega^{\lambda} \Subset \omega$. 
Take an arbitrary  $u \in H^2_{\iso} (S) $. 
 By Remark \ref{regulrem1} we have that $u \in C^{3,\alpha}(S,\R^3)$.
 Define $\tilde u \in C^{3,\alpha}(\omega,\R^3)$ by $\t u =u\circ r$. 
Let $R \in C^{1,\alpha}(S,\SO 3 )$ be such that $RT_S= \nabla u$. Take arbitrary $B \in C_0^0 (S, T^*S \odot T^*S)$, 
$(\zeta,\rho) \in C_0^1(S,\dot{C}^1(I \times \mathcal{Y},\R^2)) \times  C_0^1(S,\dot{C}^1(I \times \mathcal{Y})) $.  
Define $\tilde  u \in C^{3,\alpha}(S,\R^3)$, $\t B \in C_0^0 (\omega, \R^{2 \times 2}_{\sym})$, $\t R \in 
C^{1,\alpha}(\omega,\SO 3 )$,
$(\hat{\zeta},\hat{\rho}) \in C_0^1(S,\dot{C}^1(I \times \mathcal{Y},\R^2)) \times  C_0^1(S,\dot{C}^1(I \times 
\mathcal{Y})) $
 by 
$$\t u =u\circ r,\, \t B_{ij}=(B\circ \xi)\tau_i \cdot \tau_j, \,   \t R=R \circ r.$$
For fixed $0<\lambda_0<1$, sufficiently close to $1$, define $ \t w^0 \in C^{3,\alpha} (\lambda_0 \omega,\R^3)$ as 
the solution of the system $q^{\t u}_w=\t B|_{\lambda_0 \omega}$. For $0<\lambda<\lambda_0$, define 
$\t u^{\lambda},\t w^{\lambda} \in C^{3,\alpha}(\overline \omega, \R^3)$, $\t B^{\lambda}  \in C_0^0 (\omega, \R^{2 
\times 2}_{\sym})$, $\t R^{\lambda}  \in C^{1,\alpha} (\omega, \SO 3)$ by
$$\t u^{\lambda}(z')=\tfrac{1}{\lambda} \t u(\lambda z'), \t w^{\lambda}(z')=\tfrac{1}{\lambda} \t w(\lambda z'), \t 
B^{\lambda}(z')=B(\lambda z'), \t R^{\lambda}(z')=R(\lambda z').$$ In an obvious way, by pushing forward, define 
$u^{\lambda}, w^{\lambda} \in C^{3,\alpha}(\overline S, \R^3)$, $B^{\lambda} \in C_0^0 (S, T^*S \odot T^*S)$, 
$R^{\lambda} \in  C^{1,\alpha} (S, \SO 3)$. Notice that 
\begin{eqnarray} \nonumber
&&\lim_{\lambda<\lambda_0 \atop \lambda,\lambda_0 \to 1} \| \t u^\lambda-\t u\|_{H^2(\omega,\R^3)}=0, \, 
\lim_{\lambda<\lambda_0 \atop \lambda,\lambda_0 \to 1} \| \t B^\lambda-\t B\|_{L^2(\omega,\R^{2 \times 2}_{\sym})}
=0,\\ \label{rela1}& &\hspace{+10ex}\lim_{\lambda<\lambda_0 \atop \lambda,\lambda_0 \to 1} \| \t R^\lambda-\t R\|
_{H^1(\omega,\SO 3)}=0 ,
\end{eqnarray}
i.e., 
\begin{eqnarray}\nonumber
& &\lim_{\lambda<\lambda_0 \atop \lambda,\lambda_0 \to 1} \|  u^\lambda- u\|_{H^2(S,\R^3)}=0,\quad \lim_{\lambda<
\lambda_0 \atop \lambda,\lambda_0 \to 1} \|  B^\lambda- B\|_{L^2(S,T^*S \odot T^*S)}=0, \\  \label{rela2} & &
\hspace{+10ex} \lim_{\lambda<\lambda_0 \atop \lambda,\lambda_0 \to 1} \|  R^\lambda- R\|_{H^1(S,\SO 3)}=0.
\end{eqnarray}
We define the recovery sequence
\begin{eqnarray*}
y^h(x+tn(x))&=& u^{\lambda}(x)+thR^{\lambda}(x)n(x)+hw^{\lambda}(x)\\ & &-th^2\left(\partial_{\tau_\alpha} 
w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)\\  & & +h\varepsilon R^{\lambda}(x)
\left(\zeta_{\alpha}(x,t,r(x)/\e) \tau^{\alpha}(x)+\rho(x,t,r(x)/\e) n(x)\right). 
\end{eqnarray*}
Notice that
\begin{equation}
\lim_{h \to 0} \|y^h-u^{\lambda}\|_{H^1(S,\R^3)}=0.
\end{equation}
After a short computation we obtain
\begin{eqnarray} \label{zadnje1}
(R^{\lambda})^t\nabla_h y^h&=& I+th \AA^r_{u^{\lambda}}(x)+h(R^{\lambda}(x))^t\nabla w^{\lambda}(x)T_S\\ \nonumber & 
&-h\left(\partial_{\tau_\alpha} w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)\otimes n(x)\\ 
\nonumber
& &+h\mathcal{U}_{\gamma}(\zeta,\rho)(x,t, r(x)/\varepsilon)+o(h),
\end{eqnarray}
where $\lim_{h \to 0} \| \tfrac{o(h)}{h}\|_{L^{\infty}}=0$. 
Here we used \eqref{nadopuna1}, \eqref{dpi-2}, \eqref{fladh}, \eqref{izWm}. 

Notice that
$$\sym  \left( (R^{\lambda})^t\nabla w^{\lambda}T_S-\left(\partial_{\tau_\alpha} w^{\lambda} (x)\cdot R^{\lambda}(x) 
n(x)\right) \tau^{\alpha}(x)\otimes n(x) \right)= B^{\lambda}(x). $$
By using the objectivity property (W3), \eqref{esup} we easily obtain from \eqref{zadnje1}
\begin{eqnarray*}
& &\lim_{h \to 0}\Bigg|\frac{1}{h^2} W\left(x+t n(x), r(x)/\e, \nabla_h y(x + tn(x)) \right)\\ & & \hspace{+5ex}-
\mathcal{Q}\left(x+t n(x), r(x)/\e,\AA^r_{u^{\lambda}}(x)+B^{\lambda}(x)+\mathcal{U}_{\gamma}(\zeta,\rho)(x,t, r(x)/
\varepsilon)\right)\Bigg|=0.  
\end{eqnarray*}
From this we easily obtain 
\begin{eqnarray*}
& & \lim_{h \to 0} h^{-2} I^h(y^h)=\\
& & \int_{S}\int_{I \times \mathcal{Y}} \mathcal{Q}\left(x+t n(x), y,\AA^r_{u^{\lambda}}(x)+B^{\lambda}(x)+
\mathcal{U}_{\gamma}(\zeta,\rho)(x,t, y)\right)dydt d\vol_S(x).
\end{eqnarray*}
By using diagonalization argument and letting $\lambda, \lambda_0 \to 1$, we can construct a sequence $(y^h)$ such 
that $\lim_{h \to 0} \|y^h-u\|_{H^1(S,\R^3)}=0$ and 
\begin{eqnarray*}
 & &\lim_{h \to 0} h^{-2} I^h(y^h)=\\
 & & \int_{S}\int_{I \times \mathcal{Y}} \mathcal{Q}\left(x+t n(x), y,\AA^r_{u}(x)+B(x)+\mathcal{U}_{\gamma}(\zeta,
\rho)(x,t, y)\right)dydt d\vol_S(x).
\end{eqnarray*}
Next we take $B_l \in L^2(S,T^*S \odot T^*S)$, $(\zeta_l, \rho_l) \in L^2(S, \mathcal{D}(\mathcal{U}_{\gamma}))$ by 
defining
$$B_l=  {\bs \Pi_\gamma^1}[\AA^r_{u^{\lambda}}],\quad \mathcal{U}_{\gamma}(\zeta_l, \rho_l)={\bs \Pi_\gamma^2}
[\AA^r_{u^{\lambda}}].$$
By using approximation with  the sequence $(B^n) \subset C_0^0 (S, T^*S \odot T^*S)$, $(\hat{\zeta}^n,\hat{\rho}^n)_n 
\subset C_0^1(S,\dot{C}^1(I \times \mathcal{Y},\R^2)) \times  C_0^1(S,\dot{C}^1(I \times \mathcal{Y})$ such that 
$$\lim_{n \to \infty}\|B^n-B_l\|_{L^2(S,T^*S \odot T^*S)}=0,\,  \lim_{n \to \infty}\|(\zeta^n,\rho^n)-(\zeta_l,
\rho_l)\|_{L^2(S,D(\mathcal{U}_{\gamma}))}=0.$$
and the diagonalization argument we easily obtain the recovery sequence. 
\item case $\gamma=\infty$ the only difference is in the relaxation part of the recovery sequence. We will just give 
the recovery sequence. For  $(\zeta,\rho,c) \in C^1_0(S,C^1_0(I,\dot{C}^1(\mathcal Y,\R^2)))\times C^1_0(S, C^1_0(I, 
\dot C^1(\Y)))\times C^1_0(S,C^1_0(I, \R^3)) $ we define 
\begin{eqnarray*}
	y^h(x+tn(x))&=& u^{\lambda}(x)+thR^{\lambda}(x)n(x)+hw^{\lambda}(x)\\ & &-th^2\left(\partial_{\tau_\alpha} 
w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)\\  & & +h\varepsilon R^{\lambda}(x)
\left(\zeta_{\alpha}(x,t,r(x)/\e) \tau^{\alpha}(x)+\rho(x,t,r(x)/\e)\right)\\ & &+h^2 R^{\lambda}(x)\left(2\int_0^{t} 
c_\alpha(x,s)\ ds \taub^\alpha+ \int_0^{t} c_3(x,s)\ ds n\right). 
\end{eqnarray*}
The results from the previous case can be easily repeated. 
\item case $\gamma=0$, $\e^2 \ll h \ll \e$. 
For
$(\zeta,\varphi,g) \in C^1_0 (S,\dot{C}^1(\mathcal Y,\R^2)) \times C^2_0(S,\dot C^2(\Y))\times C^1_0(S,C^1_0(I\times
\mathcal Y,\R^3))$ we define
\begin{eqnarray*}
	y^h(x+tn(x))&=& u^{\lambda}(x)+thR^{\lambda}(x)n(x)+hw^{\lambda}(x)\\ & &
	+\e^2 R^{\lambda}(x) n(x)\varphi(x,r(x)/\e)-th\e  R^{\lambda} (x) \left(\partial_{y_{\alpha}} \varphi(x, r(x)/\e) 
\tau^{\alpha}  \right)\\ & &
	-th^2\left(\partial_{\tau_\alpha} w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)\\  & &+h
\varepsilon R^{\lambda}(x) \left(\zeta_{\alpha}(x, r(x)/\e)\tau^{\alpha}(x) \right) \\ & &
+h^2 R^{\lambda}(x) \Bigg(2 \int_0^{t} g_{\alpha}(x, s, r(x)/\e)\ ds\ \tau^\alpha(x)\\ & &+\int_0^{t} g_3(x, s, r(x)/
\e)\ ds\ n(x)\Bigg).
\end{eqnarray*}
In this case we have 
\begin{eqnarray} \label{zzzzadnje1}
(R^{\lambda})^t\nabla_h y^h&=& I+th \AA^r_{u^{\lambda}}(x)+h(R^{\lambda}(x))^t\nabla w^{\lambda}(x)T_S\\ \nonumber & 
&-h\left(\partial_{\tau_\alpha} w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)\otimes n(x)\\ 
\nonumber
& &+\e \partial_{\alpha} \varphi(x,r(x)/\e) n(x) \otimes\tau^{\alpha}(x)-\e \partial_{\alpha} \varphi(x,r(x)/\e) 
\tau^{\alpha}(x) \otimes n(x)
\\ \nonumber
& &+h\mathcal{U}_{0}(\zeta,\varphi,g)(x,t, r(x)/\varepsilon)+o(h).
\end{eqnarray}
Although in the expansion we have the dangeorus term of order $\e\gg h$, the symmetric part of this term vanishes 
(see \cite{Vel13} for the planar case). This means that
\begin{eqnarray} \label{zzzzadnje1}
	 (R^{\e})^t(R^{\lambda})^t\nabla_h y^h&=& I+th \AA^r_{u^{\lambda}}(x)+h(R^{\lambda}(x))^t\nabla w^{\lambda}(x)T_S
\\ \nonumber & &-h\left(\partial_{\tau_\alpha} w^{\lambda} (x)\cdot R^{\lambda}(x) n(x)\right) \tau^{\alpha}(x)
\otimes n(x)
	\\ \nonumber
	& &+h\mathcal{U}_{0}(\zeta,\varphi,g)(x,t, r(x)/\varepsilon)+o(h),
\end{eqnarray}
where $R^{\e}=\exp(A^{\e})$ and 
$$A^{\e}=\e \partial_{\alpha} \varphi(x,r(x)/\e) n(x) \otimes\tau^{\alpha}(x)-\e \partial_{\alpha} \varphi(x,r(x)/\e) 
\tau^{\alpha}(x) \otimes n(x).$$
Again we can repeat the results from the first case (by using the objectivity property (W3) and \eqref{esup}). 
\end{itemize}
\vspace{1cm}



\def\cprime{$'$}

\begin{thebibliography}{FJMM03}

\bibitem[ABP91]{AcBuPe}
Emilio Acerbi, Giuseppe Buttazzo, and Danilo Percivale.
\newblock A variational definition of the strain energy for an elastic string.
\newblock {\em J. Elasticity}, 25(2):137--148, 1991.

\bibitem[All92]{Allaire-92}
Gr{\'e}goire Allaire.
\newblock Homogenization and two-scale convergence.
\newblock {\em SIAM J. Math. Anal.}, 23(6):1482--1518, 1992.

\bibitem[BFF00]{Braides-Fonseca-Francfort-00}
Andrea Braides, Irene Fonseca, and Gilles Francfort.
\newblock 3{D}-2{D} asymptotic analysis for inhomogeneous thin films.
\newblock {\em Indiana Univ. Math. J.}, 49(4):1367--1404, 2000.

\bibitem[BN95]{BrezisNirenberg1}
H.~Brezis and L.~Nirenberg.
\newblock Degree theory and {BMO}. {I}. {C}ompact manifolds without boundaries.
\newblock {\em Selecta Math. (N.S.)}, 1(2):197--263, 1995.

\bibitem[BN96]{BrezisNirenberg2}
H.~Brezis and L.~Nirenberg.
\newblock Degree theory and {BMO}. {II}. {C}ompact manifolds with boundaries.
\newblock {\em Selecta Math. (N.S.)}, 2(3):309--368, 1996.
\newblock With an appendix by the authors and Petru Mironescu.

\bibitem[Bra85]{Braides-85}
Andrea Braides.
\newblock Homogenization of some almost periodic coercive functional.
\newblock {\em Rend. Accad. Naz. Sci. XL Mem. Mat. (5)}, 9(1):313--321, 1985.

\bibitem[Caf90a]{Caff2}
L.~A. Caffarelli.
\newblock Interior {$W^{2,p}$} estimates for solutions of the
  {M}onge-{A}mp\`ere equation.
\newblock {\em Ann. of Math. (2)}, 131(1):135--150, 1990.

\bibitem[Caf90b]{Caff1}
L.~A. Caffarelli.
\newblock A localization property of viscosity solutions to the
  {M}onge-{A}mp\`ere equation and their strict convexity.
\newblock {\em Ann. of Math. (2)}, 131(1):129--134, 1990.

\bibitem[Caf91]{Caff3}
L.~A. Caffarelli.
\newblock Some regularity properties of solutions of {M}onge {A}mp\`ere
  equation.
\newblock {\em Comm. Pure Appl. Math.}, 44(8-9):965--969, 1991.

\bibitem[Cia00]{Ciarletshell00}
Philippe~G. Ciarlet.
\newblock {\em Mathematical elasticity. {V}ol. {III}}, volume~29 of {\em
  Studies in Mathematics and its Applications}.
\newblock North-Holland Publishing Co., Amsterdam, 2000.
\newblock Theory of shells.

\bibitem[Cla76]{Clarke}
F.~H. Clarke.
\newblock On the inverse function theorem.
\newblock {\em Pacific J. Math.}, 64(1):97--102, 1976.

\bibitem[FJM02]{FJM-02}
Gero Friesecke, Richard~D. James, and Stefan M{\"u}ller.
\newblock A theorem on geometric rigidity and the derivation of nonlinear plate
  theory from three-dimensional elasticity.
\newblock {\em Comm. Pure Appl. Math.}, 55(11):1461--1506, 2002.

\bibitem[FJM06]{FJM-06}
Gero Friesecke, Richard~D. James, and Stefan M{\"u}ller.
\newblock A hierarchy of plate models derived from nonlinear elasticity by
  gamma-convergence.
\newblock {\em Arch. Ration. Mech. Anal.}, 180(2):183--236, 2006.

\bibitem[FJMM03]{FJMMshells03}
Gero Friesecke, Richard~D. James, Maria~Giovanna Mora, and Stefan M{\"u}ller.
\newblock Derivation of nonlinear bending theory for shells from
  three-dimensional nonlinear elasticity by {G}amma-convergence.
\newblock {\em C. R. Math. Acad. Sci. Paris}, 336(8):697--702, 2003.

\bibitem[GT01]{GT01}
David Gilbarg and Neil~S. Trudinger.
\newblock {\em Elliptic partial differential equations of second order}.
\newblock Classics in Mathematics. Springer-Verlag, Berlin, 2001.
\newblock Reprint of the 1998 edition.

\bibitem[Hei60]{Heinz-MZ1959}
E.~Heinz.
\newblock \"{U}ber die {D}ifferentialungleichung {$0<\alpha \leq rt-s^{2}\leq
  \beta <\infty $}.
\newblock {\em Math. Z}, 72:107--126, 1959/1960.

\bibitem[HLP13]{HorLewRez13}
Peter Hornung, Marta Lewicka, and Mohammad~Reza Pakzad.
\newblock Infinitesimal isometries on developable surfaces and asymptotic
  theories for thin developable shells.
\newblock {\em J. Elasticity}, 111(1):1--19, 2013.

\bibitem[HNV14]{Horneuvel12}
Peter Hornung, Stefan Neukamm, and Igor Vel\v{c}i\'{c}.
\newblock Derivation of a homogenized nonlinear plate theory from 3d
  elasticity.
\newblock {\em Calculus of Variations and Partial Differential Equations},
  pages 1--23, 2014.

\bibitem[Hor11a]{Hornung-arma2}
Peter Hornung.
\newblock Approximation of flat {$W^{2,2}$} isometric immersions by smooth
  ones.
\newblock {\em Arch. Ration. Mech. Anal.}, 199(3):1015--1067, 2011.

\bibitem[Hor11b]{Hornung-arma1}
Peter Hornung.
\newblock Fine level set structure of flat isometric immersions.
\newblock {\em Arch. Ration. Mech. Anal.}, 199(3):943--1014, 2011.

\bibitem[Horon]{h-sverak}
P.~Hornung.
\newblock Strong solutions of {M}onge-{A}mp\`ere without convexity assumption.
\newblock {\em Preprint}, in preparation.

\bibitem[HV14]{Hornungvel12}
Peter Hornung and Igor Vel\v{c}i\'{c}.
\newblock Derivation of a homogenized von-k\'arm\'an shell theory from 3d
  elasticity.
\newblock {\em Annales de l'Institut Henri Poincare (C) Non Linear Analysis},
  (0):--, 2014.

\bibitem[I{\v{S}}93]{IwaniecSverak}
T.~Iwaniec and V.~{\v{S}}ver{\'a}k.
\newblock On mappings with integrable dilatation.
\newblock {\em Proc. Amer. Math. Soc.}, 118(1):181--188, 1993.

\bibitem[LDR95]{LDRa95}
Herv{\'e} Le~Dret and Annie Raoult.
\newblock The nonlinear membrane model as variational limit of nonlinear
  three-dimensional elasticity.
\newblock {\em J. Math. Pures Appl. (9)}, 74(6):549--578, 1995.

\bibitem[LDR96]{LDRa96}
H.~Le~Dret and A.~Raoult.
\newblock The membrane shell model in nonlinear elasticity: a variational
  asymptotic derivation.
\newblock {\em J. Nonlinear Sci.}, 6(1):59--84, 1996.

\bibitem[LMP10]{Lewicka1}
Marta Lewicka, Maria~Giovanna Mora, and Mohammad~Reza Pakzad.
\newblock Shell theories arising as low energy {$\Gamma$}-limit of 3d nonlinear
  elasticity.
\newblock {\em Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}, 9(2):253--295, 2010.

\bibitem[LMP11]{LewMorPak11}
Marta Lewicka, Maria~Giovanna Mora, and Mohammad~Reza Pakzad.
\newblock The matching property of infinitesimal isometries on elliptic
  surfaces and elasticity of thin shells.
\newblock {\em Arch. Ration. Mech. Anal.}, 200(3):1023--1050, 2011.

\bibitem[LMP15]{LewickaMahadevanPakzad}
M.~Lewicka, L.~Mahadevan, and R.~Pakzad.
\newblock The {M}onge-{A}mpere constraint: matching of isometries, density and
  regularity and elastic theories of shallow shells.
\newblock {\em preprint}, 2015.

\bibitem[M{\"u}l87]{Mueller-87}
Stefan M{\"u}ller.
\newblock Homogenization of nonconvex integral functionals and cellular elastic
  materials.
\newblock {\em Arch. Rational Mech. Anal.}, 99(3):189--212, 1987.

\bibitem[MV]{MarVel-14}
Marohni\'{c} Maroje and Igor Vel{\v{c}}i{\'{c}}.
\newblock Non-periodic homogenization of bending-torsion theory for
  inextensible rods from 3d elasticity.
\newblock {\em accepted in Annali di Matematica, Pura ed Applicata, DOI:
  10.1007/s10231-015-0504-0}.

\bibitem[Neu10]{Neukamm-10}
Stefan Neukamm.
\newblock {\em Homogenization, linearization and dimension reduction in
  elasticity with variational methods}.
\newblock Phd thesis, Tecnische Universit{\"a}t M{\"u}nchen, 2010.

\bibitem[Neu12]{Neukamm-11}
Stefan Neukamm.
\newblock Rigorous derivation of a homogenized bending-torsion theory for
  inextensible rods from three-dimensional elasticity.
\newblock {\em Arch. Ration. Mech. Anal.}, 206(2):645--706, 2012.

\bibitem[Nir53]{Nirenberg1}
Louis Nirenberg.
\newblock The {W}eyl and {M}inkowski problems in differential geometry in the
  large.
\newblock {\em Comm. Pure Appl. Math.}, 6:337--394, 1953.

\bibitem[NO15]{NeuOlb-13}
Stefan Neukamm and Heiner Olbermann.
\newblock Homogenization of the nonlinear bending theory for plates.
\newblock {\em Calc. Var. Partial Differential Equations}, 53(3-4):719--753,
  2015.

\bibitem[NS85]{NikolaevShefel}
I.~G. Nikolaev and S.~Z. Shefel{\cprime}.
\newblock Convex surfaces with positive bounded specific curvature, and a
  priori estimates for {M}onge-{A}mp\`ere equations.
\newblock {\em Sibirsk. Mat. Zh.}, 26(4):120--136, 205, 1985.

\bibitem[NV13]{NeuVel-12}
Stefan Neukamm and Igor Vel{\v{c}}i{\'{c}}.
\newblock Derivation of a homogenized von {K}\'arm\'an plate theory from 3{D}
  elasticity.
\newblock {\em M3AS}, 23(14):2701--2748, 2013.

\bibitem[Res89]{Reshetnyak}
Yu.~G. Reshetnyak.
\newblock {\em Space mappings with bounded distortion}, volume~73 of {\em
  Translations of Mathematical Monographs}.
\newblock American Mathematical Society, Providence, RI, 1989.
\newblock Translated from the Russian by H. H. McFaden.

\bibitem[Sch69]{schwartz69}
J.~T. Schwartz.
\newblock {\em Nonlinear Functional Analysis}.
\newblock Gordon \& Breach, New York, 1969.

\bibitem[Sch82]{Schulz-MZ1982}
F.~Schulz.
\newblock \"{U}ber die {D}ifferentialgleichung {$rt-s^{2}=f$} und das
  {W}eylsche {E}inbettungsproblem.
\newblock {\em Math. Z.}, 179(1):1--10, 1982.

\bibitem[Sch07]{Schmidt-07}
Bernd Schmidt.
\newblock Plate theory for stressed heterogeneous multilayers of finite bending
  energy.
\newblock {\em J. Math. Pures Appl. (9)}, 88(1):107--122, 2007.

\bibitem[{\v S}ve]{sverak}
V.~{\v S}ver\'ak.
\newblock On regularity for the {M}onge-{A}mp\`ere equation without convexity
  assumptions.
\newblock {\em unpublished notes}.

\bibitem[Vel]{Velcic-13}
Igor Vel{\v{c}}i{\'{c}}.
\newblock On the general homogenization for the equations of von k\'arm\'an
  plate.
\newblock {\em accepted in Analysis and Applications,
  DOI:10.1142/S0219530515500244}.

\bibitem[Vel15]{Vel13}
Igor Vel{\v{c}}i{\'c}.
\newblock On the derivation of homogenized bending plate model.
\newblock {\em Calc. Var. Partial Differential Equations}, 53(3-4):561--586,
  2015.

\bibitem[Vis06]{Visintin-06}
Augusto Visintin.
\newblock Towards a two-scale calculus.
\newblock {\em ESAIM Control Optim. Calc. Var.}, 12(3):371--397 (electronic),
  2006.

\bibitem[Vis07]{Visintin-07}
Augusto Visintin.
\newblock Two-scale convergence of some integral functionals.
\newblock {\em Calc. Var. Partial Differential Equations}, 29(2):239--265,
  2007.

\end{thebibliography}



\end{document}
1506.02571.extracted_bib
[ABP91]{AcBuPe Emilio Acerbi, Giuseppe Buttazzo, and Danilo Percivale. \newblock A variational definition of the strain energy for an elastic string. \newblock {\em J. Elasticity, 25(2):137--148, 1991.
1506.02571.extracted_bib
[All92]{Allaire-92 Gr{\'egoire Allaire. \newblock Homogenization and two-scale convergence. \newblock {\em SIAM J. Math. Anal., 23(6):1482--1518, 1992.
1506.02571.extracted_bib
[BFF00]{Braides-Fonseca-Francfort-00 Andrea Braides, Irene Fonseca, and Gilles Francfort. \newblock 3{D-2{D asymptotic analysis for inhomogeneous thin films. \newblock {\em Indiana Univ. Math. J., 49(4):1367--1404, 2000.
1506.02571.extracted_bib
[BN95]{BrezisNirenberg1 H.~Brezis and L.~Nirenberg. \newblock Degree theory and {BMO. {I. {Compact manifolds without boundaries. \newblock {\em Selecta Math. (N.S.), 1(2):197--263, 1995.
1506.02571.extracted_bib
[BN96]{BrezisNirenberg2 H.~Brezis and L.~Nirenberg. \newblock Degree theory and {BMO. {II. {Compact manifolds with boundaries. \newblock {\em Selecta Math. (N.S.), 2(3):309--368, 1996. \newblock With an appendix by the authors and Petru Mironescu.
1506.02571.extracted_bib
[Bra85]{Braides-85 Andrea Braides. \newblock Homogenization of some almost periodic coercive functional. \newblock {\em Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 9(1):313--321, 1985.
1506.02571.extracted_bib
[Caf90a]{Caff2 L.~A. Caffarelli. \newblock Interior {$W^{2,p$ estimates for solutions of the {Monge-{Amp\`ere equation. \newblock {\em Ann. of Math. (2), 131(1):135--150, 1990.
1506.02571.extracted_bib
[Caf90b]{Caff1 L.~A. Caffarelli. \newblock A localization property of viscosity solutions to the {Monge-{Amp\`ere equation and their strict convexity. \newblock {\em Ann. of Math. (2), 131(1):129--134, 1990.
1506.02571.extracted_bib
[Caf91]{Caff3 L.~A. Caffarelli. \newblock Some regularity properties of solutions of {Monge {Amp\`ere equation. \newblock {\em Comm. Pure Appl. Math., 44(8-9):965--969, 1991.
1506.02571.extracted_bib
[Cia00]{Ciarletshell00 Philippe~G. Ciarlet. \newblock {\em Mathematical elasticity. {Vol. {III, volume~29 of {\em Studies in Mathematics and its Applications. \newblock North-Holland Publishing Co., Amsterdam, 2000. \newblock Theory of shells.
1506.02571.extracted_bib
[Cla76]{Clarke F.~H. Clarke. \newblock On the inverse function theorem. \newblock {\em Pacific J. Math., 64(1):97--102, 1976.
1506.02571.extracted_bib
[FJM02]{FJM-02 Gero Friesecke, Richard~D. James, and Stefan M{\"uller. \newblock A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. \newblock {\em Comm. Pure Appl. Math., 55(11):1461--1506, 2002.
1506.02571.extracted_bib
[FJM06]{FJM-06 Gero Friesecke, Richard~D. James, and Stefan M{\"uller. \newblock A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. \newblock {\em Arch. Ration. Mech. Anal., 180(2):183--236, 2006.
1506.02571.extracted_bib
[FJMM03]{FJMMshells03 Gero Friesecke, Richard~D. James, Maria~Giovanna Mora, and Stefan M{\"uller. \newblock Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by {Gamma-convergence. \newblock {\em C. R. Math. Acad. Sci. Paris, 336(8):697--702, 2003.
1506.02571.extracted_bib
[GT01]{GT01 David Gilbarg and Neil~S. Trudinger. \newblock {\em Elliptic partial differential equations of second order. \newblock Classics in Mathematics. Springer-Verlag, Berlin, 2001. \newblock Reprint of the 1998 edition.
1506.02571.extracted_bib
[Hei60]{Heinz-MZ1959 E.~Heinz. \newblock \"{Uber die {Differentialungleichung {$0<\alpha \leq rt-s^{2\leq \beta <\infty $. \newblock {\em Math. Z, 72:107--126, 1959/1960.
1506.02571.extracted_bib
[HLP13]{HorLewRez13 Peter Hornung, Marta Lewicka, and Mohammad~Reza Pakzad. \newblock Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells. \newblock {\em J. Elasticity, 111(1):1--19, 2013.
1506.02571.extracted_bib
[HNV14]{Horneuvel12 Peter Hornung, Stefan Neukamm, and Igor Vel\v{ci\'{c. \newblock Derivation of a homogenized nonlinear plate theory from 3d elasticity. \newblock {\em Calculus of Variations and Partial Differential Equations, pages 1--23, 2014.
1506.02571.extracted_bib
[Hor11a]{Hornung-arma2 Peter Hornung. \newblock Approximation of flat {$W^{2,2$ isometric immersions by smooth ones. \newblock {\em Arch. Ration. Mech. Anal., 199(3):1015--1067, 2011.
1506.02571.extracted_bib
[Hor11b]{Hornung-arma1 Peter Hornung. \newblock Fine level set structure of flat isometric immersions. \newblock {\em Arch. Ration. Mech. Anal., 199(3):943--1014, 2011.
1506.02571.extracted_bib
[Horon]{h-sverak P.~Hornung. \newblock Strong solutions of {Monge-{Amp\`ere without convexity assumption. \newblock {\em Preprint, in preparation.
1506.02571.extracted_bib
[HV14]{Hornungvel12 Peter Hornung and Igor Vel\v{ci\'{c. \newblock Derivation of a homogenized von-k\'arm\'an shell theory from 3d elasticity. \newblock {\em Annales de l'Institut Henri Poincare (C) Non Linear Analysis, (0):--, 2014.
1506.02571.extracted_bib
[I{\v{S93]{IwaniecSverak T.~Iwaniec and V.~{\v{Sver{\'ak. \newblock On mappings with integrable dilatation. \newblock {\em Proc. Amer. Math. Soc., 118(1):181--188, 1993.
1506.02571.extracted_bib
[LDR95]{LDRa95 Herv{\'e Le~Dret and Annie Raoult. \newblock The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. \newblock {\em J. Math. Pures Appl. (9), 74(6):549--578, 1995.
1506.02571.extracted_bib
[LDR96]{LDRa96 H.~Le~Dret and A.~Raoult. \newblock The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. \newblock {\em J. Nonlinear Sci., 6(1):59--84, 1996.
1506.02571.extracted_bib
[LMP10]{Lewicka1 Marta Lewicka, Maria~Giovanna Mora, and Mohammad~Reza Pakzad. \newblock Shell theories arising as low energy {$\Gamma$-limit of 3d nonlinear elasticity. \newblock {\em Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9(2):253--295, 2010.
1506.02571.extracted_bib
[LMP11]{LewMorPak11 Marta Lewicka, Maria~Giovanna Mora, and Mohammad~Reza Pakzad. \newblock The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. \newblock {\em Arch. Ration. Mech. Anal., 200(3):1023--1050, 2011.
1506.02571.extracted_bib
[LMP15]{LewickaMahadevanPakzad M.~Lewicka, L.~Mahadevan, and R.~Pakzad. \newblock The {Monge-{Ampere constraint: matching of isometries, density and regularity and elastic theories of shallow shells. \newblock {\em preprint, 2015.
1506.02571.extracted_bib
[M{\"ul87]{Mueller-87 Stefan M{\"uller. \newblock Homogenization of nonconvex integral functionals and cellular elastic materials. \newblock {\em Arch. Rational Mech. Anal., 99(3):189--212, 1987.
1506.02571.extracted_bib
[MV]{MarVel-14 Marohni\'{c Maroje and Igor Vel{\v{ci{\'{c. \newblock Non-periodic homogenization of bending-torsion theory for inextensible rods from 3d elasticity. \newblock {\em accepted in Annali di Matematica, Pura ed Applicata, DOI: 10.1007/s10231-015-0504-0.
1506.02571.extracted_bib
[Neu10]{Neukamm-10 Stefan Neukamm. \newblock {\em Homogenization, linearization and dimension reduction in elasticity with variational methods. \newblock Phd thesis, Tecnische Universit{\"at M{\"unchen, 2010.
1506.02571.extracted_bib
[Neu12]{Neukamm-11 Stefan Neukamm. \newblock Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity. \newblock {\em Arch. Ration. Mech. Anal., 206(2):645--706, 2012.
1506.02571.extracted_bib
[Nir53]{Nirenberg1 Louis Nirenberg. \newblock The {Weyl and {Minkowski problems in differential geometry in the large. \newblock {\em Comm. Pure Appl. Math., 6:337--394, 1953.
1506.02571.extracted_bib
[NO15]{NeuOlb-13 Stefan Neukamm and Heiner Olbermann. \newblock Homogenization of the nonlinear bending theory for plates. \newblock {\em Calc. Var. Partial Differential Equations, 53(3-4):719--753, 2015.
1506.02571.extracted_bib
[NS85]{NikolaevShefel I.~G. Nikolaev and S.~Z. Shefel{\cprime. \newblock Convex surfaces with positive bounded specific curvature, and a priori estimates for {Monge-{Amp\`ere equations. \newblock {\em Sibirsk. Mat. Zh., 26(4):120--136, 205, 1985.
1506.02571.extracted_bib
[NV13]{NeuVel-12 Stefan Neukamm and Igor Vel{\v{ci{\'{c. \newblock Derivation of a homogenized von {K\'arm\'an plate theory from 3{D elasticity. \newblock {\em M3AS, 23(14):2701--2748, 2013.
1506.02571.extracted_bib
[Res89]{Reshetnyak Yu.~G. Reshetnyak. \newblock {\em Space mappings with bounded distortion, volume~73 of {\em Translations of Mathematical Monographs. \newblock American Mathematical Society, Providence, RI, 1989. \newblock Translated from the Russian by H. H. McFaden.
1506.02571.extracted_bib
[Sch69]{schwartz69 J.~T. Schwartz. \newblock {\em Nonlinear Functional Analysis. \newblock Gordon \& Breach, New York, 1969.
1506.02571.extracted_bib
[Sch82]{Schulz-MZ1982 F.~Schulz. \newblock \"{Uber die {Differentialgleichung {$rt-s^{2=f$ und das {Weylsche {Einbettungsproblem. \newblock {\em Math. Z., 179(1):1--10, 1982.
1506.02571.extracted_bib
[Sch07]{Schmidt-07 Bernd Schmidt. \newblock Plate theory for stressed heterogeneous multilayers of finite bending energy. \newblock {\em J. Math. Pures Appl. (9), 88(1):107--122, 2007.
1506.02571.extracted_bib
[{\v Sve]{sverak V.~{\v Sver\'ak. \newblock On regularity for the {Monge-{Amp\`ere equation without convexity assumptions. \newblock {\em unpublished notes.
1506.02571.extracted_bib
[Vel]{Velcic-13 Igor Vel{\v{ci{\'{c. \newblock On the general homogenization for the equations of von k\'arm\'an plate. \newblock {\em accepted in Analysis and Applications, DOI:10.1142/S0219530515500244.
1506.02571.extracted_bib
[Vel15]{Vel13 Igor Vel{\v{ci{\'c. \newblock On the derivation of homogenized bending plate model. \newblock {\em Calc. Var. Partial Differential Equations, 53(3-4):561--586, 2015.
1506.02571.extracted_bib
[Vis06]{Visintin-06 Augusto Visintin. \newblock Towards a two-scale calculus. \newblock {\em ESAIM Control Optim. Calc. Var., 12(3):371--397 (electronic), 2006.
1506.02571.extracted_bib
[Vis07]{Visintin-07 Augusto Visintin. \newblock Two-scale convergence of some integral functionals. \newblock {\em Calc. Var. Partial Differential Equations, 29(2):239--265, 2007.