Large Time Existence for Thin Vibrating Plates
H. Abels
M.G.\ Mora
and S.\ M\"uller
We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling of the initial values such that the limit system as $h\to 0$ is either the nonlinear von K\'arm\'an plate equation or the linear fourth order Germain-Lagrange equation. In the case of the linear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.
0909.2796.tex
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\begin{document}
\begin{titlepage}
\title{Large Time Existence for Thin Vibrating Plates}%\\[1ex] Draft version!! -- Do not distribute!}
\author{H. Abels, M.G.\ Mora, and S.\ M\"uller}

\end{titlepage}
\maketitle
\begin{abstract}
We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large times under appropriate scaling of the initial values such that the limit system as $h\to 0$ is either the nonlinear von K\'arm\'an plate equation or the linear fourth order Germain-Lagrange equation. In the case of the linear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.
\end{abstract}
\noindent{\bf Key words:} Wave equation, plate theory, von K\'arm\'an, nonlinear elasticity, dimension reduction, singular perturbation 


\noindent{\bf AMS-Classification:} Primary: 74B20, %%    	Nonlinear elasticity
Secondary:
35L20, %%    	Boundary value problems for second-order, hyperbolic equations
35L70,  %%%   	Nonlinear second-order PDE of hyperbolic type
74K20 %%    	Plates
\section{Introduction}% and Main Result}

In the present contribution we study the nonlinear wave equation for a thin vibrating plate (or beam if $d=2$). The plate is assumed to be of small but positive thickness $h>0$ and satisfies the equations of three-dimensional nonlinear elastodynamics.

In order to explain the result and the model under consideration, let us start by recalling some facts and results for the corresponding variational problems, see \cite{FrieseckeJamesMueller} for further details. We consider the elastic energy
\begin{equation*}
  \tilde{E}^h(z)= \frac1h \int_{\Omega_h} \left(W(\nabla z(x)) - f^h\cdot (z(x)-x) \right)\sd x,
\end{equation*}
where $\Omega_h= \Omega'\times (-\frac{h}2,\frac{h}2)$ is the reference configuration of the thin plate, $\Omega'\subset \R^{d-1}$, $d=2,3$, is a suitable bounded domain, and $z\colon \Omega_h \to \R^d$ is the deformation of the plate. For simplicity, we will restrict ourselves to the case $d=3$ in this introduction. Rescaling $\Omega_h$ to $\Omega= \Omega'\times (-\frac{1}2,\frac{1}2)$, we obtain the rescaled energy
\begin{equation*}
  E^h(y)=  \int_{\Omega} \left(W(\nabla_h y(x)) - f^h\cdot \left(y(x)-
      \begin{pmatrix}
        x_1\\ x_2\\ h x_3
      \end{pmatrix}
\right) \right)\sd x,
\end{equation*}
where $y(x)= z(x',hx_3)$ with $x'=(x_1,x_2)$ and $\nabla_h = (\partial_{x_1},\partial_{x_2},\frac1h \partial_{x_3})$. The limit as $h\to 0$ depends on the asymptotic behaviour of $f^h$. More precisely, let $f^h$ be of order $h^\alpha$. If $\alpha=2$, then the energy $E^h$ is of order $h^\beta$ with $\beta=2$. The rescaled energy  $\frac1{h^2}E^h$ converges as $h\to 0$ to the elastic energy from the geometrically fully nonlinear Kirchhoff theory in the sense of $\Gamma$-convergence. To the authors' knowledge there are no results on existence of solutions for the corresponding dynamic wave equation or on regularity of non-minimizing equilibria. Indeed even the precise definition of equilibrium is not completely clear since the isometry constraint $\nabla \ol{y}^T\nabla \ol{y}= \Id $ for the limit map $\ol{y}\colon \Omega'\to \R^3$ makes the problem very rigid; see Hornung~\cite{Hornung6,Hornung7} for recent progress. If $\alpha>2$ and $\beta= 2\alpha-2$, then the limit energy can be described as
\begin{equation*}
  \frac{\Lambda_\alpha}2 \int_{\Omega'} Q_2\left(\eps(U)+ \frac{\nabla V\otimes \nabla V}2\right)\sd x' + \frac1{24} \int_{\Omega'} Q_2(\nabla^2 V)\sd x',
\end{equation*}
where $\eps(U)= \sym(\nabla U)$,
\begin{eqnarray}
  \label{eq:1}
  U&=& \lim_{h\to 0} \frac1{h^\gamma}
  \left(
\begin{pmatrix}
    y_1^h\\y_2^h
  \end{pmatrix}
  - \operatorname{Id}'
\right),\quad
  V= \lim_{h\to 0} \frac1{h^\delta} y_3^h,\\\label{eq:2}
\delta&=&\alpha-2,\quad \gamma=
\begin{cases}
2(\alpha-2) &\text{if}\ 2<\alpha\leq 3\\
  \alpha-1 &\text{if}\ \alpha> 3
\end{cases}
, 
\end{eqnarray}
where $\operatorname{Id}'(x)= (x_1,x_2)^T$
and $Q_2\colon \R^{2\times 2}\to \R$ is related to $Q_3(F):= D^2W(\Id)(F,F)$ by
\begin{equation*}
  Q_2(G)= \min_{a\in\R^3} Q_3(G+a\otimes e_3+ e_3\otimes a).
\end{equation*}
Here
\begin{equation*}
  \Lambda_\alpha =
  \begin{cases}
    +\infty &\text{if}\ 2 <\alpha <3,\\
   1 &\text{if}\ \alpha =3,\\
 0 &\text{if}\ \alpha >3.
  \end{cases}
\end{equation*}
Thus for $2<\alpha<3$ one has the ``geometrically linear'' constraint $2\eps(U)+\nabla V\otimes \nabla V=0$, which again has so far prevented the rigorous study of the associated dynamic wave equation or non-minimizing equilibria.
For $\alpha=3$ (and therefore $\beta=4$) one obtains the von K\'arm\'an plate theory and for $\alpha>3$ (and therefore $\beta>4$) one obtains a linear Euler-Lagrange equation (linear Germain-Lagrange theory), 
which for isotropic materials reduces to the biharmonic equation.

Here we study the cases $\alpha=3, \beta=4$  and $\alpha>3, \beta=2\alpha-2>4$ in the dynamic situation. The equations of elastodynamics arise from the Lagrangian
\begin{equation*}
  \frac1h \int_{\Omega_h} \left(\frac{|z_t|^2}2- W(\nabla z(x))+ f^hz\right)\sd x=  \int_{\Omega} \left(\frac{|y_t|^2}2- W(\nabla_h y(x))+f^hy\right)\sd x
\end{equation*}
and solutions formally preserve the total energy
\begin{equation}\label{eq:3}
  \int_{\Omega} \left(\frac{|y_t|^2}2+ W(\nabla_h y(x))- f^hy\right)\sd x %% f_k in Lagrangian
\end{equation}
In view of (\ref{eq:1})-(\ref{eq:2}) we expect that
\begin{alignat*}{3}
  y_3 &\sim h, &\quad
  \begin{pmatrix}
    y_1\\y_2
  \end{pmatrix}
 - \operatorname{Id}'
&\sim h^2 &\quad& \text{for}\ \alpha=3,\beta=4\\
  y_3 &\sim h^{\alpha-2}, &\quad
  \begin{pmatrix}
    y_1\\y_2
  \end{pmatrix}
-\operatorname{Id}'
 &\sim h^{\alpha-1} &\quad& \text{for}\ \alpha>3,\beta=2\alpha-2>4
\end{alignat*}
%%% Wirklich so? Auch im Fall \alpha>3, also \Lambda_\alpha=0?
The idea to balance the kinetic and potential energy in (\ref{eq:3}) suggests to rescale time as $\tau=ht$ if $\alpha=3$. Then the total energy becomes
\begin{equation*}
  E_{\rm tot} = h^4\int_{\Omega} \left(\frac{|\partial_\tau \frac{y}h|^2}2+ \frac1{h^4}W(\nabla_h y(x))- \frac{f^h_3}{h^3}\frac{y_3}h\right)\sd x
\end{equation*}
and with $f_h= h^{-3}f_3^he_3$ the evolution equations is 
\begin{equation*}
 \frac1{h^2}\partial_\tau^2y-\frac1{h^4} \Div_h DW(\nabla_h y)= \frac1h f_h.
\end{equation*}
or equivalently
\begin{equation}\label{eq:4}
 \partial_\tau^2y-\frac1{h^2} \Div_h DW(\nabla_h y)= h f_h,
\end{equation}
where $f_h \sim 1$ as $h\to 0$.
In the case $\alpha=3$ we will show existence of strong solutions of (\ref{eq:4}) for well-prepared and small data in a natural scaling with respect to $h$ and  time $\tau \in (0,T_0)$ with $T_0>0$ sufficiently small. In particular we assume that the rescaled $f_h$ is small, cf. Section~\ref{sec:MainResult} below. -- Note that the small time interval $(0,T_0)$ for $\tau$ turns over to a large time interval $(0,T_0h^{-1})$ in the original times scale for $t$. In the case $\alpha>3$, we will use the same time scale. Then we are able to show existence of strong solutions for $\tau \in (0,T)$ for any $T>0$ provided that $f_h \sim h^{\alpha-3}$ and suitable initial data, cf. Section~\ref{sec:MainResult} below. We note that this time scale is subcritical for this scaling of $f_h$. In this case we are even able to construct the leading term of the solution $y=y_h$ as $h\to 0$ provided $W(F)=\dist(F,SO(3))^2$, cf. Section~\ref{sec:Asymptotics}. %%% Allgemeiner Fall?

Together with \cite{AbelsMoreMuellerGammaConv} this shows that after the natural time rescaling and for well prepared data of the correct size solutions of the $3$-d nonlinear elastodynamics converge to solutions of the dynamic von K\'arm\'an equation or linear von K\'arm\'an equation depending on the size of the data. %Moreover, in the cas $\alpha>3$, the results of Section~\ref{sec:Asymptotics} show convergence of solution of the $3$-d nonlinear elastodynamics converge to solutions of the dynamic linear von K\'arm\'an equation.
We note that a similar result in the case of stationary solutions was shown  by Monneau~\cite{MonneauJustification} if the limit system are the von K\'arm\'an plate equations. 
Ge, Kruse and Marsden~\cite{GeKruseMarsden} have taken an alternative and very general
approach to study the limit from three-dimensional
elasticity to shells and rods by establishing convergence of the
underlying Hamiltionian structure.
This suggests, but does not prove the convergence of the corresponding
dynamical problems
(see e.g. recent work by Mielke~\cite{MielkeWeakConvMethods} for the question on the relation of the
convergence of the Hamiltonian
and the convergence of the resulting dynamical problems).
General information and many further references
on the dynamics of lower-dimensional nonlinear elastic structures can be
found in the book by Antman~\cite{Antman}.
For results on existence of weak and strong solutions of the non-stationary von K\'arm\'an plate equations we refer to e.g. Chen and Wahl~\cite{ChenWahl}, Koch and Lasiecka~\cite{KochLasiecka}, Lasiecka~\cite{LasieckaStabilization}, Koch and Stahel~\cite{KochStahel}. For a survey on results and open problem of nonlinear elasticity, stationary and non-stationary, we refer to Ball~\cite{BallOpenProblems}.
 

Let us explain the strategy of our proof and the main difficulties. Basically, the strong solutions are constructed by the usual energy method as presented e.g. in the books by Majda~\cite{MajdaCompFluidFlow} and Dafermos~\cite{Dafermos}. (For a more abstract and general version see e.g. the classical paper by Hughes et al.~\cite{MarsdenEtAl}.) The starting point in the method is the conservation of energy:
\begin{equation*}
  \frac{d}{dt} \left(\frac12\|\partial_t y(t)\|_{L^2(\Omega)}^2+\frac1{h^2}\int_\Omega W(\nabla_y y)\sd x + \int_\Omega hf_h\cdot y(t)\sd x\right) = 0
\end{equation*}
which follows from (\ref{eq:4}) by multiplication with $\partial_t y$ under appropriate boundary conditions. (Here and in the following we replace $\tau$ by $t$.) Moreover, differentiating (\ref{eq:4}) with respect to $x$ one gets a control of 
\begin{equation}\label{eq:5}
  \frac{d}{dt} \left(\frac12\|\partial_t \partial_x^\beta y(t)\|_{L^2(\Omega)}^2+\frac1{h^2}\int_\Omega D^2W(\nabla_h y)\partial_x^\beta \nabla_h y:\partial_x^\beta\nabla_h y \sd x\right) = R_\beta,
\end{equation}
where the remainder term $R_\beta$ can be controlled with the aid of the Gronwall inequality once the left hand side controls $\partial_x^\beta \nabla_h y$ suitably. To this end it is essential to have the coercive estimate  
\begin{equation}\label{eq:coercive}
  \frac1{h^2}\int_\Omega D^2W(\nabla_h y)\nabla_h w:\nabla_h w\sd x\geq c_0 \left\|\frac1h \eps_h(w)\right\|_{L^2(\Omega)}^2 
\end{equation}
where $\eps_h(w) = \sym (\nabla_h w)$, cf. (\ref{eq:UniformCoercive'}) below. By Korn's inequality in the present $h$-dependent version we have
\begin{equation*}
  \|\nabla_h w\|_{L^2(\Omega)} \leq C\left\|\frac1h\eps_h (w)\right\|_{L^2(\Omega)},
\end{equation*}
cf. Lemma~\ref{lem:Korn} below. Therefore we will have one order of $h$ better decay of the symmetric part of $\nabla_h y$ than for the full gradient/the skew-symmetric part. To obtain (\ref{eq:coercive}) (and similar estimates) it will be essential that 
$$\frac1h\|\eps_h (y)-I\|_{L^\infty}+ \|\nabla_h y-I\|_{L^\infty}\leq \eps h$$ 
for some sufficiently small $\eps>0$ and to treat the symmetric and asymmetric part carefully in a Taylor expansion of $D^2W(\nabla_h y)$ around $I$, cf. Sections~\ref{sec:Preliminaries} and \ref{sec:Linear} for the details.

Several technical difficulties arise from the fact that we are dealing with natural boundary conditions at the upper and lower boundary $x_d=\pm \frac12$. In tangential direction we assume periodic boundary conditions. First of all, in this situation it is easy to differentiate in tangential and temporal direction to obtain (\ref{eq:5}) with $\partial_x^\beta w$ replaced by $\partial_z^\beta w$, where $z=(x',t)$ and $x'=(x_1,\ldots ,x_{d-1})$. Therefore we are using anisotropic $L^2$-Sobolev spaces of sufficiently high order to control $\nabla_h y$ in $L^\infty$. In particular, one of the basic spaces is 
\begin{equation*}
  \tilde{V}(\Omega)=\left\{u\in L^2(\Omega): \nabla u, \partial_{x_j}\nabla u\in L^2(\Omega), j=1,\ldots, d-1\right\}\hookrightarrow L^\infty(\Omega)
\end{equation*}
if $d=2,3$. Note that $\tilde{V}(\Omega)$ is slightly larger than $H^2(\Omega)$ and that $u\in H^2(\Omega)$ if and only if $u\in \tilde{V}(\Omega)$ and $\partial_{x_d}^2 u\in L^2(\Omega)$. Moreover, since we are dealing with natural boundary conditions, we want to keep the equation in divergence form. Therefore we do not use the identity
\begin{equation*}
  \Div_h DW(\nabla_hy)= D^2W(\nabla_hy)\cdot \nabla_h^2y
\end{equation*}
to obtain a quasi-linear system. Instead we differentiate (\ref{eq:4}) with respect to time and solve
\begin{equation*}
  \partial_t w -\frac1{h^2}\Div_h \left(D^2W(\nabla_hy)\nabla_hw\right)=h\partial_t f,
\end{equation*}
where $w=\partial_t y$. Unfortunately we cannot solve the latter equation directly since we are missing suitable control of $\partial_{x_j} y$, $j=1,\ldots,d-1$. Therefore we first replace $D^2W(\nabla_h y)$ by suitably smoothed coefficients $A_n(\nabla_hy)$ and construct a solution $w_n$,  $y_n$, respectively, for the smoothed systems. Once we have one solution $y_n$ at hand, we can differentiate it with respect to $x_j$, $j=1,\ldots, d-1$ and get a solutions of
\begin{equation*}
  \partial_t w_j^n -\frac1{h^2}\Div_h A_n(\nabla_hy_n)\nabla_hw_j^n=h\partial_{x_j} f,
\end{equation*}
where $w_j^n=\partial_{x_j} y_n$. These solutions $w_j^n$ satisfy the same estimates as $w_n=\partial_t y_n$ (uniformly in $0<h\leq h_0$ and the smoothing parameter $n\in\N$). Then we can pass to the limit $n\to\infty$ to obtain a solution of the original system.

The structure of the article is as follows: In Section~\ref{sec:Preliminaries} we introduce some notation and derive some preliminary results. Our main result is presented in Section~\ref{sec:MainResult}. Afterwards we introduce our approximate system used for the construction of strong solutions in Section~\ref{sec:Approx}. The essential results for the linearized system are derived in Section~\ref{sec:Linear}, which are applied in Section~\ref{sec:LocalInTime} to obtain a strong solution first locally in time for fixed $n\in\N$. Then uniform bounds in $T$, $h$ and $n\in\N$ are shown in Section~\ref{sec:uniform} and our main result is shown by first extending the solution for small times in $[0,T]$ for some $T>0$ given and then passing to the limit $n\to\infty$. Finally, in Section~\ref{sec:Asymptotics} we derive a first order asymptotic expansion as $h\to 0$ in the case that the limit system is linear, i.e., $\beta>4$, and $W(F)= \dist(F,SO(d))^2$.

\noindent
{\bf Acknowledgements:} This work was partially supported by GNAMPA, through the project ``Problemi di riduzione di dimensione per strutture elastiche sottili'' 2008.

\section{Notation and Preliminaries}\label{sec:Preliminaries}

For any measurable set $M\subseteq \R^N$ the inner product of $L^2(M)$ (w.r.t. to Lebesgue measure) is denoted by $(.,.)_M$. Moreover, $H^k(\Omega)$, $k\in\N_0$, denotes the usual $L^2$-Sobolev spaces. If $X$ is a Banach space, then the vector-valued variants of $L^2(M)$ and $H^k(M)$ are denoted by $L^2(M;X)$, $H^k(M;X)$, respectively. Furthermore, $C^k([0,T];X)$, $k\in \N_0$, denotes the space of all $k$-times continuously differentiable functions $f\colon [0,T]\to X$. 

For the following $\Omega= (-L,L)^{d-1}\times(-\frac12,\frac12)$, $\Omega'=(-L,L)^{d-1}$, $d=2,3$, $x=(x',x_d)$,
where $x'\in \R^{d-1}$, let $\nabla_h = (\nabla_{x'},
\frac1h \partial_{x_d})^T$, $\nabla_{x,t}=(\partial_t, \nabla_x)$ and let
\begin{equation*}
  \eps_h (w) = \sym(\nabla_h w),\qquad \eps(w)=\eps_1(w),
\end{equation*}
if $w\colon M\subset \R^d\to \R^d$ is a suitable vector field. Here $\sym A=
\frac12 (A+A^T)$ and we denote $\skw A:= \frac12 (A-A^T)$. Moreover, we denote $z=(t,x')$, where $z_0 = t$ and $z_j = x_j$ for $j=1,\ldots, d-1$.

For $s> 0$, $s\not\in \N_0$, we define $L^2$-Bessel potential spaces 
\begin{equation*}
  H^s(\Omega) = \{f\in L^2(\Omega): f= F|_{\Omega}\ \text{for some}\ F\in H^s(\R^d)\}
\end{equation*}
as usual by restriction, equipped with the quotient norm.
Since $\Omega$ is a Lipschitz domain, there is a continuous extension operator $E$ such that $E\colon H^k(\Omega)\to H^k(\R^d)$ for all $k\in \N$, cf. Stein~\cite[Chapter VI, Section 3.2]{Stein:SingInt}. Hence $H^s(\Omega)$, $s\geq 0$, is retract of $H^s(\R^d)$ and we obtain the usual interpolation properties, cf. e.g. \cite{Triebel1}. In particular, we have
\begin{equation}\label{eq:InterpolHs}
  (H^{s_0}(\Omega), H^{s_1}(\Omega))_{\theta,2}= H^s(\Omega),\qquad s= (1-\theta)s_0+ \theta s_1, 
\end{equation}
for all $\theta \in (0,1)$, $s\geq 0$, where  $(.,.)_{\theta,p}$ denotes the real interpolation method.

If $0<T\leq \infty$ and $X$ is a Banach space, then  $BUC([0,T];X)$ is the space of all bounded and uniformly continuous functions $f\colon [0,T)\to X$.
Now let $X_0,X_1$ be Banach spaces such that $X_1\hookrightarrow X_0$ densely. 
Then
\begin{equation}
  \label{eq:BUCEmbedding}
   W^1_p(0,T;X_0) \cap L^p(0,T;X_1) \hookrightarrow BUC([0,T];(X_0,X_1)_{1-\frac1p,p})
\end{equation}
for all $1\leq p <\infty$
continuously, cf. Amann~\cite[Chapter III, Theorem 4.10.2]{Amann}.
% Moreover, there is a continuous extension operator 
% \begin{equation}\label{eq:Extension}
% E\colon (X_0,X_1)_{1-\frac1p,p}\to W^1_p(0,\infty;X_0) \cap L^p(0,\infty;X_1)\quad \text{such that}\ Eu_0|_{t=0}=u_0,  
% \end{equation}
%  cf. \cite[Chapter~III, Theorem 4.10.2]{Amann}. Actually (\ref{eq:BUCEmbedding}) and (\ref{eq:Extension}) follow directly from the trace method for the real interpolation, cf. \cite[Corollary~3.12.3]{Interpolation}. 
If $X_0=H$ is a Hilbert space and $H$ is identified with its dual, then $X_1\hookrightarrow H\hookrightarrow X_1'$ and
\begin{equation}
  \label{eq:HNormDifferential}
  \frac12\frac{d}{dt} \|f\|_{H}^2 = \weight{\frac{d}{dt} f(t), f(t)}_{X_1',X_1} \qquad \text{for almost all}\ t\in [0,T] 
\end{equation}
provided that $f\in L^p(0,T;X_1)$ and $\frac{d}{dt} f\in L^{p'}(0,T;X_1')$, $1< p <\infty$, cf. Zeidler~\cite[Proposition 23.23]{ZeidlerIIa}. 
In particular, (\ref{eq:HNormDifferential}) implies 
\begin{equation}\label{eq:StrongBUCConv}
  \sup_{t\in [0,T]}\|f(t)\|_{H}^2
 \leq 2\left(\|\partial_t f\|_{L^2(0,T;X_1')}\|f\|_{L^2(0,T;X_1)} + \|f(0)\|_H^2\right).
\end{equation}
Replacing $f(t)$ by $tf(t)$ and $(T-t)f(T-t)$, one easily derives from the latter estimate
\begin{equation}\label{eq:StrongBUCConv'}
  \sup_{t\in [0,T]}\|f(t)\|_{H}
 \leq C_T\|f\|_{H^1(0,T;X_1')}^{\frac12}\|f\|_{L^2(0,T;X_1)}^{\frac12} 
\end{equation}
for some $C_T>0$ depending on $T>0$.
 


In the following $\mathcal{L}^n(V)$, $n\in\N$, denotes the space of all
$n$-linear mappings $A\colon V^n\to \R$ for a vector space $V$. Moreover, if $A\in
\mathcal{L}^n(V)$, $n\geq 2$, and $x_1,\ldots, x_k\in V$, $1\leq k\leq n$, then $A[x_1,\ldots, x_k] \in
\mathcal{L}^{n-k}(V)$ is defined by $A[x_1,\ldots, x_k](x_{k+1},\ldots, x_n)= A(x_1,\ldots,
x_n)$ for all $x_{k+1},\ldots,x_n\in V$. 

We introduce the scaled inner product 
\begin{equation*}
  A:_h B = \frac1{h^2} \sym A : \sym B + \skw A:\skw B, \quad A,B\in \R^{d\times d}, 0<h\leq 1,
\end{equation*}
and $|A|_h= \sqrt{A:_hA}$ where $A:B= \sum_{i,j=1}^d a_{ij} b_{ij}$. This choice of inner product is motivated by the Korn inequality in thin domains, see Lemma~\ref{lem:Korn} below. Of
course, $:_1$ coincides with the usual inner product $:$ on $\R^{d\times d}$
and therefore $|A|_1=|A|$. For $W\in
\mathcal{L}^n(\R^{d\times d}) $ we define the induced scaled norm by
\begin{equation*}
  |W|_{h} = \sup_{|A_j|_h\leq 1, j=1,\ldots, n} |W(A_1,\ldots,A_n)|.
\end{equation*}
Note that, since $|A|_h\geq |A|_1=|A|$ for all $A\in \R^{d\times d}$, we have $|W|_h\leq |W|_1=:|W|$ for any $W\in
\mathcal{L}^n(\R^{d\times d}) $ and $0< h\leq 1$.

As usual we
identify $\mathcal{L}^1(\R^{d\times d})= (\R^{d\times d})'$ with $\R^{d\times
  d}$. But one has to be careful whether this representation is taken with
respect to the usual scalar product $:$ on $\R^{d\times d}$ or with respect to
$:_h$, i.e., $W\in \mathcal{L}^1(\R^{d\times d})$ is identified with $A\in
\R^{d\times d}$ such that 
\begin{equation*}
  W(B) = A:_hB\qquad \text{for all}\ B\in \R^{d\times d}.
\end{equation*}
If nothing else is mentioned, we identify $(\R^{d\times d})'$ and $\R^{d\times
  d}$ using the standard inner product $:$. In particular, if $W\in C^1(U)$, $U\subset \R^{d\times d}$ and $A\in U$, then
$DW(A)\in (\R^{d\times d})'\cong \R^{d\times d}$ coincides with
\begin{equation*}
  DW(A): B= \left. \frac{d}{dt}W(A+tB)\right|_{t=0}\qquad \text{for all}\
  B\in\R^{d\times d}. 
\end{equation*}
Furthermore, $W\in \mathcal{L}^2(\R^{d\times d})$ is usually identified with the linear mapping $\tilde{W}\colon \R^{d\times d}\to \R^{d\times d}$ defined by
\begin{equation*}
  \tilde{W}A: B = W(A,B)\qquad \text{for all}\ A,B\in \R^{d\times d}.
\end{equation*}
% For some purposes it will be useful to take $:_h$ for the
% representation. Therefore we define $D_hW(A)\in\R^{d\times d}$ by
% \begin{equation*}
%   D_hW(A):_h B= \left. \frac{d}{dt}W(A+tB)\right|_{t=0}\qquad \text{for all}\
%   B\in\R^{d\times d}. 
% \end{equation*}
% Similarly, if $W\in C^{k}(U)$, $k\geq 2$ and $A\in U$, we define
% \begin{equation*}
%   D_h^kW(A)(B_1,\ldots,B_{k-1}):_h B_k= \left. \frac{d}{dt}D^{k-1}W(A+tB_k)(B_1,\ldots, B_{k-1})\right|_{t=0}
% \end{equation*}
% for all $B_1,\ldots,B_k$. Here $D^{k-1} W(A+tB_k)$ is understood as element
% in $\mathcal{L}^{k-1}(\R^{d\times d})$.
Finally, we denote by
\begin{equation*}
  \|W\|_{L^p_h(M;\mathcal{L}^n(\R^{d\times d}))}\equiv \|W\|_{L^p_h(M)} =
  \left(\int_M |W(x)|_h^p\sd x\right)^{\frac1p}
\end{equation*}
if $1\leq p<\infty$ and with the obvious modifications if $p=\infty$. Here
$M\subseteq \R^d$ is measurable. Moreover, for $f\in L^p(M;\R^{d\times d})$ the scaled norm
$\|f\|_{L^p_h(M;\R^{d\times d})}\equiv \|f\|_{L^p_h(M)}$ is defined in the same way.
 
We now state the relevant  Korn inequality in thin domains.
\begin{lemma}\label{lem:Korn}
  There is a constant $C$ such that
  \begin{equation}\label{eq:Korn}
    \|\nabla_h u \|_{L^2(\Omega)} \leq C \left\|\frac1h\eps_h(
    u)\right\|_{L^2(\Omega)}
  \end{equation}
  for all $0<h\leq 1$ and $u\in H^1(\Omega)^d$ such that $u|_{x_j=-L}=u|_{x_j=L}$,
  $j=1,\ldots, d-1$.
\end{lemma}
\begin{proof}
For clamped boundary conditions the Korn inequality in thin domains was
 proved by Kohn and Vogelius~\cite[Prop. 4.1]{KohnVogeliusII}. They mention that
 the result also holds without boundary conditions, modulo infinitesimal
 rigid motions.
 For the convenience of the reader we provide a proof of Lemma~\ref{lem:Korn}.

%Since the scaling of the Korn constant like $\frac1h$ is crucial for our arguments, we provide a detailed proof for the convenience of the reader, even though the result is well-known to experts.
First we prove the case $d=2$. Let $\Omega_h:=(-L,L)^{d-1}{\times}(-\frac{h}{2},\frac{h}{2})$ and let $u\in
H^1(\Omega_h;\R^2)$ satisfy the boundary conditions  $u|_{x_j=-L}=u|_{x_j=L}$,
  $j=1,\ldots, d-1$.
First of all by a simple scaling in $x_d$, (\ref{eq:Korn}) is equivalent to
  \begin{equation}\label{eq:Korn2}
    \|\nabla u \|_{L^2(\Omega_h)} \leq \frac{C}h \|(\nabla u)_{sym}\|_{L^2(\Omega_h)}
  \end{equation}
Let $N_h$ be the integer part of $\frac{2L}{h}$ and let $\ell_h:=\frac{2L}{N_h}$.
We set $J_h:=\{-L+k\ell_h: k=0,\dots,N_h-1\}$. 
By applying Korn inequality on the set $(a, a+\ell_h){\times}(-\frac{h}{2},\frac{h}{2})$
for every $a\in J_h$, we can construct a piecewise constant function $A\colon(-L,L)\to \mtwo$ such that $A(x_d)$ is skew-symmetric and
\begin{equation}\label{pwkorn}
\int_{\Omega_h}|\nabla u- A|^2\, dx \leq C \int_{\Omega_h}|\eps(u)|^2\, dx. 
\end{equation}
Note that, since $\frac{\ell_h}h$ is bounded from above and from below, we can use the same Korn inequality constant on each set 
$(a, a+\ell_h){\times}(-\frac{h}{2},\frac{h}{2})$.

We claim that
\begin{equation}\label{Aest}
\int_{\Omega_h}|A(x_1)-A_0|^2\, dx\leq \frac{C}{h^2}\int_{\Omega_h}|\eps(u)|^2\, dx. 
\end{equation}
where $A_0:=A(-L)$.

Let us fix $a\in J_h$ and let $b:=a+\lambda\ell_h$, with $\lambda\in\{0,1\}$.
By applying Korn inequality on the set $(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})$ 
we have that there exists $\tilde A\in\mtwo$ such that
$$
\int_{(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\nabla u- \tilde A|\, dx \leq C 
\int_{(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\eps(u)|^2\, dx.
$$
From this inequality we deduce
\begin{eqnarray*}
h\ell_h|A(b)-\tilde A|^2
& \leq &
2  \int_{(b, b+\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\nabla u- A(x_1)|^2\, dx
\\
& & + 2 \int_{(b, b+\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\nabla u-\tilde A|^2\, dx
\\
& \leq & C \int_{(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\eps(u)|^2\, dx.
\end{eqnarray*}
Combining the previous inequality for $\lambda=0$ and $\lambda=1$, we obtain
\begin{eqnarray*}
h\ell_h|A(a)-A(b)|^2
& \leq & 2h\ell_h( |A(a)-\tilde A|^2 + |A(b)-\tilde A|^2)
\\
& \leq & C \int_{(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\eps(u)|^2\, dx.
\end{eqnarray*}
As $A$ is constant on each interval $(a, a+\ell_h)$, this is equivalent to say that
\begin{equation}\label{diffquot}
\int_{(a, a+\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|A(x_1+\ell_h)-A(x_1)|^2\, dx
\leq C \int_{(a, a+2\ell_h){\times}(-\frac{h}{2},\frac{h}{2})}|\eps(u)|^2\, dx.
\end{equation}

Let us set $I_{k,j}:=-L+\ell_h(k,k+j)$.
By convexity we have the following estimate: 
\begin{eqnarray*}
\lefteqn{\int_{\Omega_h}|A(x_1)-A_0|^2\, dx =
h\sum_{k=0}^{N_h-1}\int_{I_{k,1}}|A(x_1)-A_0|^2\, dx_1}
\\
& = & h\sum_{k=0}^{N_h-1}\int_{I_{k,1}}
\Big|\sum_{m=0}^{k-1}\big( A(x_1-m\ell_h)-A(x_1-(m+1)\ell_h)\big)\Big|^2\, dx_1
\\
& \leq & h\sum_{k=0}^{N_h-1} k \sum_{m=0}^{k-1} \int_{I_{k,1}}
\Big|A(x_1-m\ell_h)-A(x_1-(m+1)\ell_h)\Big|^2\, dx_1.
\end{eqnarray*}
By \eqref{diffquot} we deduce 
$$
\int_{\Omega_h}|A(x_1)-A_0|^2\, dx \leq \sum_{k=0}^{N_h-1} k \sum_{m=0}^{k-1} 
C\int_{I_{k-m-1, 2}{\times}(-\frac{h}{2},\frac{h}{2})}
|\eps(u)|^2\, dx.
$$
It is easy to see that for every $k=0,\dots, N_h-1$
$$
\sum_{m=0}^{k-1} 
\int_{I_{k-m-1, 2}{\times}(-\frac{h}{2},\frac{h}{2})}
|\eps(u)|^2\, dx
\leq 2 \int_{\Omega_h} |\eps(u)|^2\, dx.
$$
Therefore, we conclude that
$$
\int_{\Omega_h}|A(x_1)-A_0|^2\, dx \leq C N_h^2\int_{\Omega_h} |\eps(u)|^2\, dx,
$$
which proves claim \eqref{Aest}.

Combining \eqref{pwkorn} and \eqref{Aest}, we conclude that for every $u\in H^1(\Omega_h;\R^2)$
there exists a constant skew-symmetric $A_0\in\mtwo$ such that
$$
\int_{\Omega_h}|\nabla u-A_0|^2\, dx\leq \frac{C}{h^2}\int_{\Omega_h}|\eps(u)|^2\, dx.
$$

Since 
$$
\int_{\Omega_h}\Big|\frac1{|\Omega_h|}\fint_{\Omega_h}({\rm skw}\nabla u)\, dx -A_0 \Big|^2\, dx \leq
\int_{\Omega_h}|({\rm skw}\nabla u)-A_0|^2\,dx ,
$$ 
we also have that
\begin{equation}\label{korn2}
\int_{\Omega_h}\Big|\nabla u-\frac1{|\Omega_h|}\fint_{\Omega_h}({\rm skw}\nabla u)\Big|^2\, dx
\leq \frac{C}{h^2}\int_{\Omega_h}|\eps(u)|^2\, dx 
\end{equation}
for every $u\in H^1(\Omega_h;\R^2)$.

Now, if $u$ is periodic in tangential direction, then
\begin{eqnarray*}
\int_{\Omega_h}\Big|\frac1{|\Omega_h|}\fint_{\Omega_h}({\rm skw}\nabla u)\Big|^2\,dx 
& = &
\int_{\Omega_h}\Big|\frac1{|\Omega_h|}\fint_{\Omega_h}\partial_2 u_1\Big|^2\, dx
\\
& = &\int_{\Omega_h}\Big|\frac1{|\Omega_h|}\fint_{\Omega_h}(\partial_2 u_1+\partial_1 u_2)\Big|^2\, dx
\\
& \leq & \int_{\Omega_h}|\eps(u)|^2\, dx,
\end{eqnarray*}
which, together with \eqref{korn2}, provides us with the desired inequality.

In order to prove the case $d=3$, we use that (\ref{eq:Korn}) for $d=2$
implies
\begin{equation*}
  \left\|
    \begin{pmatrix}
      \partial_{x_j}\\ \frac1{h} \partial_{x_3}
    \end{pmatrix}
    \begin{pmatrix}
      u_j\\ u_3
    \end{pmatrix}
\right\|_{L^2(\Omega)} \leq \frac{C}h   \left\|
    \left(\begin{pmatrix}
      \partial_{x_j}\\ \frac1{h} \partial_{x_3}
    \end{pmatrix}
    \begin{pmatrix}
      u_j\\u_3
    \end{pmatrix}
\right)_{sym} 
\right\|_{L^2(\Omega)} \leq \frac{C}h\|(\nabla_h u)_{sym}\|_{L^2(\Omega)} 
\end{equation*}
for $j=1,2$ and any $u\in H^1(\Omega)^3$. Moreover, applying Korn's inequality in
$(-L,L)^2$ with periodic boundary conditions, we obtain
\begin{equation*}
  \|\nabla_{x'} u'\|_{L^2(\Omega)} \leq C \|(\nabla_{x'}
  u')_{sym}\|_{L^2(\Omega)}
  \leq C\|(\nabla_x u)_{sym}\|_{L^2(\Omega)},
\end{equation*}
where $u'= (u_1,u_2)^T$. Altogether this proves (\ref{eq:Korn}) for $d=3$.
\end{proof}
\begin{remark}
  The latter lemma shows that $\left\|\frac1h\eps_h(u)\right\|_{L^2(\Omega)}$ is
    equivalent to $\|\nabla_h u\|_{L^2_h(\Omega)}$ with constants independent
    of $0<h\leq 1$.
\end{remark}
We denote by $H^1_h(\Omega)$ the space $H^1_{per}(\Omega)^d\cap \{u: \int_\Omega u(x)\sd x=0\}$ equipped with the norm
\begin{equation*}
 \|u\|_{H^1_h(\Omega)}= \left\|\frac1h \eps_h(u)\right\|_{L^2(\Omega)}, \qquad u \in H^1(\Omega)^d
\end{equation*}
and $H^{-1}_h(\Omega)$ its dual space with norm
\begin{equation*}
  \|f\|_{H^{-1}_h(\Omega)}= \sup \left\{ \left|\weight{f,\varphi}_{H^{-1}_h,H^1_h}\right|:u\in H^1_h(\Omega)\ \text{with}\ \left\|\frac1h\eps_h(u)\right\|_{L^2}=1\right\}.
\end{equation*}
Furthermore, we denote
\begin{equation*}
  H^m_{per}(\Omega)= \left\{f \in H^m(\Omega): \partial_x^\alpha f|_{x_j=-L}= \partial_x^\alpha f|_{x_j=L}, j=1,\ldots,d-1, |\alpha|\leq m-1\right\}.
\end{equation*}

 Throughout this contribution the following anisotropic variant of $H^m_{per}(\Omega)$ will be important:
   \begin{eqnarray*}
     H^{m_1,m_2}(\Omega) &=& \left\{u\in L^2(\Omega): \nabla_{x'}^k\partial_{x_d}^l u\in
       L^2(\Omega),k=0,\ldots,m_1, l=0,\ldots, m_2\right.,\\
      && \left. \ \partial_{x'}^\alpha \partial_{x_d}^lu|_{x_j=-L}=\partial_{x'}^\alpha \partial_{x_d}^lu|_{x_j=L}, j \leq d-1, |\alpha|\leq m_1-1, l\leq m_2\right\}
   \end{eqnarray*}
   where $m_1\in \N,m_2\in\N_0$. The spaces are equipped with the inner product
   \begin{equation*}
     (f,g)_{H^{m_1,m_2}}= \sum_{|\alpha|\leq m_1,k=0,\ldots, m_2} (\partial_{x'}^\alpha \partial_{x_d}^k f,\partial_{x'}^\alpha \partial_{x_d}^k g)_{L^2(\Omega)}
   \end{equation*}
   Please note that periodic boundary conditions are included in the spaces $H^{m_1,m_2}(\Omega)$ in contrast to the space $H^m(\Omega)$, where we denote them by a subscript ``$per$'' in order to be consistent with the usual definition of $H^m(\Omega)$.

% \begin{lemma}\label{lem:LInfty} 
%   Let $\theta\in (0,\frac12]$ if $d=3$ and let $\theta=0$ if $d=2$. Then there is some $C=C(\Omega,\theta)$ such
%   that
%   \begin{equation}\label{eq:LinftyEstim}
%     \|f\|_{L^\infty(\Omega)}\leq C
%     \|f\|_{H^{1,1}(\Omega)}^{\frac12} \|f\|_{H^{2,0}(\Omega)}^{\theta} \|f\|_{H^{1,0}(\Omega)}^{\frac12-\theta} 
%   \end{equation}
% %   \begin{equation}\label{eq:LinftyEstim}
% %     \|f\|_{L^\infty(\Omega)}\leq
% %     \begin{cases}
% %    C\left\|\left(\partial_{x_1}\partial_{x_2} f,
% %     \nabla f, f\right)\right\|_{L^2(\Omega)}& \text{if}\ d=2\\
% %    C\left\|\left(\nabla_{x'}\partial_{x_3} f, \nabla_{x'}^2 f,
% %     \nabla f, f\right)\right\|_{L^2(\Omega)}& \text{if}\ d=3
% %     \end{cases}
% %   \end{equation}
%   for all $f\in H^1_{per}(\Omega)$  
% such that the right-hand sides are well-defined
%   and finite.
% \end{lemma}
% \begin{proof}
%   First let $d=3$. Then $\|f\|_{H^{1+2\theta,0}(\Omega)}$ and $\|f\|_{H^{1,1}(\Omega)}$ are
%   equivalent to the norms of $f$ in $L^2(-\frac12,\frac12; H^{1+2\theta}((-L,L)^2))$, $
%   H^1(-\frac12,\frac12;H^1((-L,L)^2)))$, respectively, where
%   \begin{eqnarray*}
%     \lefteqn{L^2(-\frac12,\frac12; H^{1+2\theta}((-L,L)^2))\cap
%   H^1(-\frac12,\frac12;H^1((-L,L)^2))}\\
% && \hookrightarrow BUC([-\frac12,\frac12];
%   H^{1+\theta}((-L,L)^2))\hookrightarrow C^0(\ol{\Omega})
%   \end{eqnarray*}
%   because of (\ref{eq:BUCEmbedding}) and Sobolev
%   embedding.  More precisely,
%   \begin{equation}\label{eq:BUCInterpol}
%     \|f\|_{BUC([-L,L];(H_0,H_1)_{\frac12,2})}\leq C\|f\|_{H^1(-L,L;H_0)}^\frac12\|f\|_{L^2(-L,L;H_1))}^\frac12,
%   \end{equation}
%   for general Hilbert space $H_0,H_1$ and
%   $\|f\|_{H^{1+2\theta}(\Omega')}\leq C\|f\|_{H^2(\Omega')}^{2\theta} \|f\|_{H^1(\Omega')}^{1-2\theta}$ due to (\ref{eq:StrongBUCConv'}) and (\ref{eq:InterpolHs}).
%   This implies the estimate (\ref{eq:LinftyEstim}). 
% Here $BUC([0,T];X)$ is the space of all bounded and uniformly continuous $f\colon [0,T]\to X$.

% Similarly, if $d=2$, then we use (\ref{eq:BUCInterpol}) with $H_0=H_1=H^1(\Omega')= H^1(-L,L)\hookrightarrow C^0([-L,L])$.
% \end{proof}

Similarly, an anisotropic variant of $L^p$ will be useful:
\begin{eqnarray*}
  L^{p,q}(\Omega)&=& \left\{u\colon \Omega\to \R: \|u(x_1,.)\|_{L^q(-\frac12,\frac12)}\in L^p((-L,L)^{d-1})\right\}
\end{eqnarray*}
where $1\leq p,q\leq \infty$
equipped with the norm
\begin{equation*}
  \|f\|_{L^{p,q}} = \left\|\|u(x_1,.)\|_{L^q(-\frac12,\frac12)}\right\|_{L^p((-L,L)^{d-1})}.
\end{equation*}
We note that from the usual H\"older inequality it follows that
\begin{equation*}
  \|fg\|_{L^{p,q}(\Omega)}\leq \|f\|_{L^{p_1,q_1}(\Omega)}\|g\|_{L^{p_2,q_2}(\Omega)},
\end{equation*}
for all $1\leq p_1,q_1,p_2,q_2\leq \infty$ such that
\begin{equation*}
  \frac1p = \frac1{p_1} +\frac1{p_2},\qquad   \frac1q = \frac1{q_1} +\frac1{q_2}.
\end{equation*}
\begin{lemma}\label{lem:Embeddings}
  Let $d=2,3$. Then
  \begin{equation*}
    H^{1,0}(\Omega) \hookrightarrow L^{p,2}(\Omega), \quad
    H^{2,0}(\Omega) \hookrightarrow L^{\infty,2}(\Omega), \quad
    H^1(\Omega) \hookrightarrow L^{4,\infty}(\Omega) \quad
  \end{equation*}
continuously for $p=\infty$ if $d=2$ and any $1\leq p<\infty$ if $d=3$. Finally, let
\begin{equation*}
  V(\Omega):= H^{1,1}(\Omega)\cap H^{2,0}(\Omega).
\end{equation*}
Then $V(\Omega)\hookrightarrow C^0(\ol{\Omega})$ continuously.
\end{lemma}
\begin{proof}
  The first embedding follows from $H^1(\Omega')\hookrightarrow L^p(\Omega')$ and the  second from $H^2(\Omega')\hookrightarrow L^\infty(\Omega')$ since $d=2,3$ and $\Omega'= (-L,L)^{d-1}$. The third embedding follows from
\begin{equation*}
  H^1(-\tfrac12,\tfrac12;L^2(\Omega'))\cap L^2(-\tfrac12, \tfrac12; H^1(\Omega'))\hookrightarrow BUC([-\tfrac12, \tfrac12];H^{\frac12}(\Omega'))
\end{equation*}
and $H^{\frac12}(\Omega')\hookrightarrow L^4(\Omega')$. Finally, the last embedding follows from
  \begin{eqnarray*}
    \lefteqn{L^2(-\tfrac12,\tfrac12; H^{1+k}((-L,L)^{d-1}))\cap
  H^1(-\tfrac12,\tfrac12;H^1((-L,L)^{d-1}))}\\
&& \hookrightarrow BUC([-\tfrac12,\tfrac12];
  H^{1+\frac{k}2}((-L,L)^{d-1}))\hookrightarrow C^0(\ol{\Omega})
  \end{eqnarray*}
where $k=d-2$
  because of (\ref{eq:BUCEmbedding}) and Sobolev
  embeddings.
\end{proof}
\begin{remark}
  The spaces $H^{1,0}(\Omega)$ and $V(\Omega)$ are the fundamental spaces, which will be used to solve the evolution equation. We note that
  \begin{equation*}
    f\in V(\Omega)\quad \Leftrightarrow \quad f, \nabla f \in H^{1,0}(\Omega).
  \end{equation*}
Most of the time we will estimate $f\in V(\Omega)$ by the $h$-dependent norm
\begin{equation*}
\|f\|_{V_h}:=  \|(f,\nabla_h f)\|_{H^{1,0}(\Omega)}.
\end{equation*}
\end{remark}
 Because of the embedding $V(\Omega)\hookrightarrow L^\infty(\Omega)$, we are able to show that $V(\Omega)$ is an algebra with respect to point-wise multiplication. More precisely, we
   obtain:
   \begin{corollary}\label{cor:Algebra} 
Let $d=2,3$. Then there is some $C=C(\Omega)>0$ such that
     \begin{equation}\label{eq:ProdEstim}
       \|(u_1\cdot u_2, \nabla_h (u_1\cdot u_2)\|_{H^{1,0}(\Omega)}\leq C\|(u_1, \nabla_h u_1)\|_{H^{1,0}(\Omega)}\|(u_2, \nabla_h u_2)\|_{H^{1,0}(\Omega)}
     \end{equation}
for all $u_1,u_2\in V(\Omega)$ uniformly in $0<h\leq 1$. Moreover,
     if $F\in C^{2}(\ol{U})$ for some open $U\subset \R^N$,
     $N\in\N$, and $u\in V(\Omega)^N$, then for every $R>0$ there is some $C(R)$ independent of $u$ such that
     \begin{equation}\label{eq:CompEstim}
       \|(F(u), \nabla_h F(u))\|_{H^{1,0}(\Omega)} \leq C(R)\qquad \text{if}\ \|(u,\nabla_h u)\|_{H^{1,0}(\Omega)}\leq R
     \end{equation}
     uniformly in $0<h\leq 1$ and if $u(x)\in \ol{U}$ for all $x\in\overline{\Omega}$.
   \end{corollary}
   \begin{proof}
     First of all (\ref{eq:ProdEstim}) follows from (\ref{eq:CompEstim}) by
     first considering $\|u_1\|_{V_h}, \|u_2\|_{V_h}\leq 1$ and
     $F(u_1,u_2) = u_1\cdot u_2$ together with a scaling argument.

     Hence it only remains to prove (\ref{eq:CompEstim}).      
     First of all, 
     \begin{eqnarray*}
       \partial_{x_j} F(u) &=& DF(u) \partial_{x_j} u \\
       \partial_{x_j}\partial_{x_k} F(u) &=& DF(u) \partial_{x_j}\partial_{x_k}
       u + D^2F(u)(\partial_{x_j}u,\partial_{x_k}u)
     \end{eqnarray*}
     where $DF(u),D^2F(u)$ are uniformly bounded since $u\in C^0(\ol{\Omega})$
     and $u(x) \in \ol{U}$ for all $x\in \ol\Omega$. Therefore
     $\nabla_h F(u) \in L^2(\Omega)$ can be easily estimated. Hence it only remains to consider the second order derivatives. 
      To this end we use that
     \begin{eqnarray*}
       \lefteqn{\left\|D^2F(u)(\partial_{x_j}u,\frac1h\partial_{x_d}u)\right\|_{L^2(\Omega)}\leq C
       \|\partial_{x_j}u\|_{L^{4,\infty}(\Omega)}
       \left\|\frac1h\partial_{x_d}u\right\|_{L^{4,2}(\Omega)}}\\
       &\leq & C\|\partial_{x_j} u\|_{H^1(\Omega)}\left\|\frac1h\partial_{x_d}u\right\|_{H^{1,0}(\Omega)}
\leq C'(R)\|(u, \nabla_h u)\|_{H^{1,0}(\Omega)}
     \end{eqnarray*}
     for all $j=1,\ldots,d-1$ due to Lemma~\ref{lem:Embeddings}. Similarly,
     \begin{eqnarray*}
       \left\|D^2F(u)(\partial_{x_j}u,\partial_{x_k}u)\right\|_{L^2(\Omega)} &\leq& C'(R)\|(u, \nabla_h u)\|_{H^{1,0}(\Omega)}
     \end{eqnarray*}
for all $j,k=1,\ldots, d-1$. From these estimates the statement of the corollary easily follows.
   \end{proof}


For the following let 
$W\colon B_r(I)\subset\R^{d\times d}\to \R$ be a smooth function for some $r>0$ which is frame invariant, i.e., $W(RF)=W(F)$ for every $F\in \R^{d\times d}\to\R^{d\times d}$ and $R\in SO(d)$, and such that $DW(I)=0$ and $D^2W(I)\colon \R^{d\times d}\to\R^{d\times d}$ is positive definite on symmetric matrices. Moreover, we set
$\widetilde{W}(G)= W(I+G)$.
The estimates of derivatives of $D^2 \widetilde{W}(\nabla_h u)$ will be essential for the proof of our main result and will be based on the following lemma:
\begin{lemma}\label{lem:D3G}
  There is some constant $C>0$, $\eps>0$, and $A\in C^\infty(\ol{B_\eps(0)};\mathcal{L}^3(\R^{d\times d}))$ such that for all $G\in
  \R^{d\times d}$ with $|G|\leq \eps$ we have
  \begin{equation*}
    D^3\widetilde{W}(G) = D^3\widetilde{W}(0) + A(G),
  \end{equation*}
  where
  \begin{alignat*}{2}
    |D^3\widetilde{W}(0)|_h&\leq Ch &\quad& \text{for all}\ 0<h\leq 1,\\
    |A(G)| &\leq C|G| &\quad& \text{for all}\ |G|\leq \eps. 
  \end{alignat*}
\end{lemma}
\begin{proof}
  First of all, if $|G|\leq \eps$ for $\eps>0$ sufficiently small, we can use a polar decomposition $I+G= RU$, where $R\in SO(d)$ and $U$ is symmetric and positive definite such that $U^2=
  (I+G)^T(I+G)$. From frame invariance we conclude
  that $W(I+G)=W(U)= \widehat{W}(U^2)=\widehat{W}(I+2\sym G+G^TG)$ for some smooth $\widehat{W}\colon V \subset \R^{d\times d}\to \R$, where $V$ is some open neighborhood of $I$. For this proof we denote $A_s= \sym A$. Straight-forward calculations yield
  \begin{eqnarray*}
    DW(F)(H) &=& D\widehat{W}(U^2)(2 H_s + H^TG+G^TH)\\
    D^2W(F)(H_1,H_2) &=& D^2\widehat{W}(U^2)(2H_{1,s} + H_1^TG+G^TH_1,2
    H_{2,s} + H_2^TG+G^TH_2)\\
    && + D\widehat{W}(U^2)(H_1^TH_2+H_2^TH_1)
\end{eqnarray*}
and
\begin{eqnarray*}
    \lefteqn{D^3W(F)(H_1,H_2,H_3)=}\\
    && D^3\widehat{W}(U^2)(2H_{1,s} + H_1^TG+G^TH_1,2
    H_{2,s} + H_2^TG+G^TH_2,2H_{3,s} + H_3^TG+G^TH_3 )\\
    && + D^2\widehat{W}(U^2)(H_1^TH_2+H_2^TH_1, 2H_{3,s}+ H_3^TG+G^TH_3)\\
&&    + D^2\widehat{W}(U^2)(H_1^TH_3+H_3^TH_1, 2H_{2,s}+ H_2^TG+G^TH_2)\\
&&+ D^2\widehat{W}(U^2)(H_2^TH_3+H_3^TH_2, 2H_{1,s}+ H_1^TG+G^TH_1)
  \end{eqnarray*}
  where $F=I+G$. 
From the latter identities the statements immediately follow.
\end{proof}
For the following we denote
\begin{eqnarray*}
  \|A\|_{H^{m_1,m_2}_h} &:=& \left(\sum_{|\alpha|\leq m_1, j=0,\ldots, m_2} \|\partial_{x'}^\alpha \partial_{x_d}^j A\|_{L^2_h(\Omega)}^2\right)^{\frac12}\\
  \|A\|_{H^{m}_h} &:=& \left(\sum_{|\alpha|\leq m} \|\partial_{x}^\alpha A\|_{L^2_h(\Omega)}^2\right)^{\frac12}
\end{eqnarray*}
where $m,m_1,m_2\in \N_0$ and $A\in H^{m_1,m_2}(\Omega)^{d\times d}$, $A\in H^{m}(\Omega)^{d\times d}$, respectively.
\begin{corollary}\label{cor:D3W} 
  There are some $\eps,C>0$ such that
  \begin{eqnarray}\nonumber
    \lefteqn{\|D^3 \widetilde{W}(Z)(Y_1, Y_2, Y_3)\|_{L^1(\Omega)}}\\\label{eq:D3WEstim}
    &\leq & C h \left(\left\|Y_1 \right\|_{H^{1,1}_h} + \left\|Y_1 \right\|_{H^{2,0}_h} \right)\left\|Y_2\right\|_{L^2_h(\Omega)} \left\|Y_3\right\|_{L^2_h(\Omega)} 
  \end{eqnarray}
  for all $Y_1\in V(\Omega)^{d\times d}, Y_2,Y_3 \in
  L^2(\Omega)^{d\times d} $, $0<h\leq 1$ and $\|Z\|_{L^\infty(\Omega)}\leq \min (\eps,h)$ and 
  \begin{eqnarray}\nonumber
    \lefteqn{\|D^3 \widetilde{W}(Z)(Y_1, Y_2, Y_3)\|_{L^1(\Omega)}}\\\label{eq:D3WEstimb}
    &\leq & C h \left\|Y_1\right\|_{H^1_h(\Omega)}\left\|Y_2\right\|_{H^{1,0}_h(\Omega)} \left\|Y_3\right\|_{L^2_h(\Omega)} 
  \end{eqnarray}
  for all $Y_1\in H^1(\Omega)^{d\times d}, Y_2\in H^{1,0}(\Omega)^{d\times d},Y_3 \in
  L^2(\Omega)^{d\times d} $, $0<h\leq 1$ and $Z\in L^\infty(\Omega)^{d\times d}$ with $\|Z\|_{L^\infty(\Omega)}\leq \min(\eps,h)$.
\end{corollary}
\begin{proof}
  The statement follows directly from Lemma~\ref{lem:D3G}, Korn's
  inequality due to Lemma~\ref{lem:Korn}, and Lemma~\ref{lem:Embeddings}.
\end{proof}


\section{Long-Time Existence for Thin Sticks/Plates}
\subsection{Main Result}
\label{sec:MainResult}
We consider
\begin{equation}\label{eq:Evol1}
  \partial_t^2 u_h - \frac1{h^2} \Div_h D\widetilde{W}(\nabla_h u_h) = f_h h^{1+\theta} \qquad \text{in}\ \Omega\times I
\end{equation}
where $\widetilde{W}(G)= W(I+G)$,  $\Omega= (-L,L)^{d-1}\times
(-\frac12,\frac12)$, $\beta=4+2\theta$, which is equivalent to $\theta= \alpha -3$, and $I=[0,T_\ast]$ for some $T_\ast$ together with the initial and boundary conditions
\begin{eqnarray}\label{eq:BC1}
  \left. D\widetilde{W}(\nabla_h u_h)e_d\right|_{x_d=\pm \frac12} &=&0 \\\label{eq:BC2}
  \left. u_h\right|_{x_j=L} &=& \left. u_h\right|_{x_j=-L},\qquad j=1,\ldots,d-1,\\\label{eq:BC3}
  \left. (u_h,\partial_t u_h)\right|_{t=0} &=& (u_{0,h}, u_{1,h}).
\end{eqnarray}
Here we assume that $W\colon B_r(I)\to \R$ is a smooth function for some $r>0$ which is frame invariant, i.e., $W(RF)=W(F)$ for every $F\in \R^{d\times d}\to\R^{d\times d}$ and $R\in SO(d)$, and such that $DW(I)=0$ and $D^2W(I)\colon \R^{d\times d}\to\R^{d\times d}$ is positive definite on symmetric matrices. -- Note that the latter condition implies that $D^2W(I)$ is elliptic in the sense of Legendre-Hadamard:
\begin{equation}\label{eq:LegendreHadamard}
  (D^2W(I)a\otimes b) :a\otimes b\geq c_0 |a|^2|b|^2\qquad \text{for all}\ a,b\in \R^d
\end{equation}
for some $c_0>0$.
\begin{theorem}\label{thm:main1}
  Let $\theta\geq 0$, $0<T<\infty$, let $f_h\in W^2_1(0,T;H^{1,0})\cap W^1_1(0,T;H^{2,0})$, $0<h\leq 1$, 
and 
let $u_{0,h}\in H^{1,3}(\Omega)^d\cap H^{4,0}(\Omega)^d$, $u_{1,h}\in H^{1,2}(\Omega)^d\cap H^{3,0}(\Omega)^d$
  such that 
  \begin{equation*}
    D\widetilde{W}(\nabla_h u_{0,h})e_d|_{x_d=\pm\frac12} = D^2\widetilde{W}(\nabla_h u_{0,h})\nabla_h u_{1,h} e_d|_{x_d=\pm\frac12} =0, 
  \end{equation*}
 and    
  \begin{eqnarray}\label{eq:AssumInitialData1}
   \max_{k=0,1}\left\|\left(\frac1h \eps_h (u_{1+k,h}),u_{2+k,h}\right)\right\|_{H^{2-k,0}(\Omega)} &\leq& Mh^{1+\theta} \\\label{eq:AssumFData}
   \left\|\left(\frac1h \eps_h (f_h|_{t=0}),\partial_t f_h|_{t=0}\right)\right\|_{H^{1,0}}+ \max_{|\gamma|\leq 1} \| \partial_z^\gamma f_h\|_{W^1_1(0,T;H^{1,0})} &\leq & M 
  \end{eqnarray}
uniformly in $0<h\leq 1$, 
  where 
  \begin{eqnarray}\label{eq:u2h}
    u_{2,h} &=&  h^{1+\theta}f_h|_{t=0} + \frac1{h^2} \Div_h D\widetilde{W}(\nabla_h
  u_{0,h}),\\\label{eq:u3h}
    u_{3,h} &=&  h^{1+\theta}\partial_tf_h|_{t=0} + \frac1{h^2} \Div_h \left(D^2\widetilde{W}(\nabla_h
  u_{0,h})\nabla_h u_{1,h}\right).
% \\
%     u_{4,h} &=&  h^{1+\theta}\partial_t^2f_h|_{t=0} + \frac1{h^2} \Div_h D^2W(\nabla_h
%   u_{0,h})\nabla_h u_{2,h}\\
% && - \frac1{h^2} \Div_h D^3W(\nabla_h
%   u_{0,h})(\nabla_h u_{1,h}, \nabla_h u_{1,h}) \\
  \end{eqnarray}
%and let
%  \begin{equation*}
%    \left\| \div_h (F_nf|_{t=0}
%  \end{equation*}
 If $\theta>0$, then there is some
  $h_0\in (0,1]$ and $C$ depending on $M$ such that for every $0<h\leq h_0$ there is a unique solution $u_h\in C^3([0,T];H^{1,0})\cap C^0([0,T];H^{2,2}\cap H^{4,0})$ of
  (\ref{eq:Evol1})-(\ref{eq:BC3}) satisfying 
 \begin{eqnarray}\label{eq:UnifBdd1} 
   \max_{|\gamma|\leq 1}\left\|\left(\partial_t^2\partial_z^\gamma u_h,
       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h)\right)\right\|_{C([0,T];H^{1,0})}&\leq& Ch^{1+\theta}%\\\label{eq:UnifBdd2} 
%   \max_{|\gamma|= 2}\left\|\left(\partial_t^2\partial_z^\gamma u_h,
%       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h)\right)\right\|_{C([0,T];L^2)}&\leq& Ch^{\frac{d-2}2}
  \end{eqnarray}
%  \begin{equation} 
%    \left\|\left(\partial_t u_h, \frac1h\eps_h(u_h)
%        \right)\right\|_{X_\infty^{m+2}(T)}+ \left\|\left(\nabla_h^2 u,
%          \nabla_h^2 \partial_t u\right)\right\|_{X_\infty^{m}(T)}\leq C h^{1+\theta}
%\end{equation}
uniformly in $0<h\leq h_0$. If $\theta=0$, the same statement holds with $h_0=1$ provided that $M,T>0$ are sufficiently small.
\end{theorem}
\notshow{
\begin{theorem}\label{thm:main2} 
  Let $\theta=0$ and let $f\in X_1^{m+2}(T), 0<h\leq 1$, be bounded for some $T>0$. Then there is some
  $0<T_0\leq T$ and $C=C(f)$ such that for all $0<h\leq 1$ there is a unique solution $u$ of
  (\ref{eq:Evol1})-(\ref{eq:BC3}) satisfying 
  \begin{equation}
    \left\|\left(\partial_t u, \frac1h\eps_h(u)
        \right)\right\|_{X_\infty^{m+2}(T_0)}+ \left\|\left(\nabla_h^2 u,
          \nabla_h^2 \partial_t u\right)\right\|_{X_\infty^{m}(T_0)}\leq C h
\end{equation}
uniformly in $0<h\leq 1$.
\end{theorem}
}
\subsection{Approximate System}\label{sec:Approx}
In order to construct a solution to (\ref{eq:Evol1})-(\ref{eq:BC3}), we first construct
 solutions  for an approximate system. 

To this end, let  $P_n^h\colon L^2(\Omega) \to
  L^2(\Omega)$, $n\in\N$, be the projection on
  the eigenspaces of the eigenvalues not exceeding $n$ of $-\Delta_h$ with domain
  $\mathcal{D}(-\Delta_h)= \{u\in H^2_{per}(\Omega): \partial_{x_d}u|_{x_d=\pm \frac12}=0 \}$.
  Then $P_n^h$ are orthogonal projections  such that 
  \begin{eqnarray}\nonumber
\|\nabla_h P_n^hf\|_{L^2(\Omega)}&=&\|(-\Delta_h)^{\frac12}P_n
  f\|_{L^2(\Omega)}=\|P_n(-\Delta_h)^{\frac12} f\|_{L^2(\Omega)}\\\label{eq:PnH1Estim}
&\leq& 
  \|(-\Delta_h)^{\frac12} f\|_{L^2(\Omega)}= \|\nabla_h
  f\|_{L^2(\Omega)}.
  \end{eqnarray}
 The same is also true for $H^{m,0}(\Omega)$ since
  $-\Delta_h$ and therefore also $P_n^h$ and $(-\Delta_h)^\frac12$ commute with
  tangential derivative $\partial_{x'}^\beta$. Furthermore, the previous estimates imply
  \begin{equation*}
    \|\partial_{x_d} P_n^h f\|_{L^2(\Omega)} \leq h\|\nabla_h f\|_{L^2(\Omega)}\leq \|\nabla f\|_{L^2(\Omega)} 
  \end{equation*}
and therefore
\begin{equation*}
  \|\nabla P_n^h f\|_{L^2(\Omega)}\leq 2\|\nabla f\|_{L^2(\Omega)}.
\end{equation*}
Moreover, by standard elliptic
  theory all
  eigenfunctions are smooth. Therefore $\mathcal{R}(P_n^h)\subseteq
  C^\infty(\ol{\Omega})$.
Finally, there is a constant $C_n>0$ such that
\begin{equation}\label{eq:nEstim}
\|P_n^hf\|_{H^2(\Omega)}\leq C_n\|f\|_{L^2(\Omega)}  
\end{equation}
uniformly in
$0<h\leq 1$ since
\begin{eqnarray*}
  \|P_n^hf\|_{H^2(\Omega)}&\leq& C\left(\|f\|_{L^2(\Omega)}+\|\Delta P_n^hf\|_{L^2(\Omega)} \right)\\
&\leq& C\left(\|f\|_{L^2(\Omega)}+\|\Delta_h P_n^hf\|_{L^2(\Omega)} \right)\\
&\leq& C\left(\|f\|_{L^2(\Omega)}+n\|P_n^hf\|_{L^2(\Omega)} \right)\leq C_n \|f\|_{L^2(\Omega)}
\end{eqnarray*}
where we have used that $\|\Delta_h e_j\|_{L^2(\Omega)}= \lambda_j\|e_j\|_{L^2(\Omega)}\leq n\|e_j\|_{L^2(\Omega)}$ for each eigenfunction $e_j$ to some eigenvalue $\lambda_j\leq n$ and
\begin{equation*}
  c_1\|\Delta_h u\|_{L^2(\Omega)}\leq \|\Delta_{x'} u\|_{L^2(\Omega)}+\frac1{h^2}\left\| \partial_{x_d}^2 u\right\|_{L^2(\Omega)}\leq c_2\|\Delta_h u\|_{L^2(\Omega)}
\end{equation*}
for any $0<h\leq 1$.
 For notational simplicity we will write $P_n$ instead of $P_n^h$ in the following.

 
To motivate the approximation, we note that
\begin{eqnarray}\nonumber
  D\widetilde{W}(\nabla_h u) &=& D^2\widetilde{W}(0)\nabla_h u + 
  \int_0^1D^3\widetilde{W}( \tau\nabla_h u) [\nabla_h u, \nabla_h u](1-\tau)\sd \tau, \\\label{eq:DefnF'}
  &=& D^2\widetilde{W}(0)\nabla_h u + 
  F' (\nabla_h u), 
\end{eqnarray}
by Taylor's expansion since $D\widetilde{W}(0) =DW(I)=0$.
Therefore we define the approximations
\begin{eqnarray}\nonumber
  F_n(\nabla_h u) &:=& D^2\widetilde{W}(0)\nabla_h u + 
  \int_0^1P_nD^3\widetilde{W}(P_n \tau\nabla_h u) [P_n\nabla_h u, P_n\nabla_h u](1-\tau)\sd\tau\\\label{eq:DefnFn}
 &\equiv& D^2\widetilde{W}(0)\nabla_h u + 
  F'_n(P_n \nabla_h u)
\end{eqnarray}
where $n\in\N$.
Replacing $D\widetilde{W}(\nabla_h u)$ by $F_n(\nabla_h u)$ in
(\ref{eq:Evol1}), we obtain the following approximate system.
 \begin{alignat}{2}\label{eq:Approx1}
  \partial_t^2 u_h^n - \frac1{h^2} \Div_h F_n(\nabla_h u_h^n)&= f_h h^{1+\theta} &\quad& \text{in}\ \Omega\times (0,T)\\\label{eq:Approx2}
  \left. F_n(\nabla_h u_h^n) e_d\right|_{x_d=\pm \frac12} &=0 &&
   \\\label{eq:Approx3}
  \left. u_h^n\right|_{x_j=L} &= \left. u_h^n\right|_{x_j=-L},&& j=1,\ldots,d-1,\\\label{eq:Approx4}
  \left. (u_h^n,\partial_t u_h^n)\right|_{t=0} &= (u_{0,h}^n, u_{1,h}^n).
\end{alignat}
Here $u_{0,h}^n, u_{1,h}^n$ will be chosen as solutions of 
  \begin{eqnarray}\label{eq:ApproxCompCond1}
    u_{2,h} &=&  h^{1+\theta}f_h|_{t=0} + \frac1{h^2} \Div_h F_n(\nabla_h
  u_{0,h}^n)\\\label{eq:ApproxCompCond2}
    u_{3,h} &=&  h^{1+\theta}\partial_tf_h|_{t=0} + \frac1{h^2} \Div_h \left(D F_n(\nabla_h
  u_{0,h}^n)\nabla_h u_{1,h}^n\right)
% \\
%     u_{4,h} &=&  h^{1+\theta}\partial_t^2f_h|_{t=0} + \frac1{h^2} \Div_h D^2W(\nabla_h
%   u_{0,h})\nabla_h u_{2,h}\\
% && - \frac1{h^2} \Div_h D^3W(\nabla_h
%   u_{0,h})(\nabla_h u_{1,h}, \nabla_h u_{1,h}) \\
  \end{eqnarray}
together with $\int_{\Omega} u_{j,h}^n \sd x =\int_{\Omega} u_{j,h} \sd x  $, $j=0,1$, and the boundary conditions
  \begin{equation}\label{eq:ApproxCompCond3}
    F_n(\nabla_h u_{0,h}^n)e_d|_{x_d=\pm\frac12} = DF_n(\nabla_h u_{0,h}^n)\nabla_h u_{1,h}^n e_d|_{x_d=\pm\frac12} =0, 
  \end{equation}
where $u_{2,h}, u_{3,h}$ are as in Theorem~\ref{thm:main1}. We will show that (\ref{eq:ApproxCompCond1})-(\ref{eq:ApproxCompCond3}) has a unique solution $u_{0,h}^n, u_{1,h}^n$ satisfying
\begin{equation}\label{eq:ApproxIVEstim}
\max_{k=0,1}\left\|\left(\frac1h \eps_h (u_{k,h}^n), \nabla\frac1h \eps_h (u_{k,h}^n), \nabla_h^2u_{k,h}^n\right)\right\|_{H^{2-k,0}(\Omega)}
   \leq C_0 Mh^{1+\theta} 
\end{equation}
for all $0<h\leq h_0$ 
provided that $h_0\in (0,1]$ is sufficiently small  in the case $\theta >0$ and provided that $M>0$  is sufficiently small (and $h_0=1)$ if $\theta=0$. Here $C_0$ is some universal constant. 
\begin{remark}\label{rem:SolvIV}
 If $k=0$, the statement follows from Proposition~\ref{prop:SolvIV} below. If $k=1$ and $u_{0,h}$ already constructed, the statement follows from Lemma~\ref{lem:LinEstim2} below and the Lemma of Lax-Milgram.  
\end{remark}


The main step now consists in solving (\ref{eq:Approx1})-(\ref{eq:Approx4})
under the same assumptions as in Theorem~\ref{thm:main1}\notshow{,
Theorem~\ref{thm:main2}, respectively,} and showing uniform bounds in $n\in\N$
and $0<h\leq h_0$. More precisely, we show
\begin{theorem}\label{thm:ApproxMain1} 
  Let $\theta\geq 0$, $0<T<\infty$, and let $u_{2,h},u_{3,h},f_h$ be as in Theorem~\ref{thm:main1} . 
 If $\theta>0$, then there are 
  $h_0\in (0,1]$ and $C$ depending only on  $M,T$ such that for
  every $0<h\leq h_0$ and $n\in\N$ there are unique solutions $u_{0,n}^h$ and $u_{1,n}^h$ of (\ref{eq:ApproxCompCond1})-(\ref{eq:ApproxCompCond3}) and
a unique solution $u_h^n$ of
  (\ref{eq:Approx1})-(\ref{eq:Approx4}) satisfying 
 \begin{eqnarray}\label{eq:UnifBddApprox1} 
   \max_{|\gamma|\leq 1}\left\|\left(\partial_t^2\partial_z^\gamma u_h^n,
       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h^n)\right)\right\|_{C([0,T];H^{1,0})}&\leq& Ch^{1+\theta}%\\\label{eq:UnifBddApprox2} 
%   \max_{|\gamma|= 2}\left\|\left(\partial_t^2\partial_z^\gamma u_h^n,
%       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h^n)\right)\right\|_{C([0,T];L^2)}&\leq& Ch^{\frac{d-2}2}
  \end{eqnarray}
uniformly in $0<h\leq h_0$, $n\in\N$. 
If $\theta=0$, the same is true provided $M,T>0$ are sufficiently small.
\end{theorem}
Once Theorem~\ref{thm:ApproxMain1} \notshow{and Theorems~\ref{thm:ApproxMain2} are} is
proved, the main Theorem~\ref{thm:main1}\notshow{-\ref{thm:main2} are} is easily proved by
passing to the limit $n\to\infty$ for a suitable subsequence using the uniform
bounds due to (\ref{eq:UnifBddApprox1}).

For simplicity we will write $u_n$ instead of $u_h^n$ in the following.

In order to construct a solution to (\ref{eq:Approx1})-(\ref{eq:Approx4}), we
differentiate (\ref{eq:Approx1})-(\ref{eq:Approx4}) once with respect to time.
This yields the system
\begin{alignat}{2}\label{eq:Approx1'}
  \partial_t^2 w_n - \frac1{h^2} \Div_h \left(A_n(\nabla_h u_n)\nabla_h w_n\right)
  &= \partial_t f_h h^{1+\theta} &\quad& \text{in}\ \Omega\times (0,T),\\\label{eq:Approx2'}
  \left. A_n(\nabla_h u_n)\nabla_h w_n e_d\right|_{x_d=\pm \frac12} &=0 &&
   \\\label{eq:Approx3'}
  \left. w_n\right|_{x_j=L} &= \left. w_n\right|_{x_j=-L},&& j=1,\ldots,d-1,\\\label{eq:Approx4'}
  \left. (w_n,\partial_t w_n)\right|_{t=0} &= (w_{0,n}, w_{1,n}).
\end{alignat}
for $w_n= \partial_t u_n\equiv \partial_t u_h^n$, where 
\begin{eqnarray}\nonumber
  \lefteqn{A_n(\nabla_hu_n) \nabla_h w}\\\label{eq:DefnAn}
 &=& 
   D^2\widetilde{W}(0)\nabla_h w + 
  \int_0^1 P_nD^3\widetilde{W}(P_n \tau\nabla_h u_n) [P_n\nabla_h u_n, P_n\nabla_h w]\sd\tau, \\\nonumber
&=&D^2\widetilde{W}(0)\nabla_h w + P_n\left(D^2\widetilde{W}(P_n \nabla_h u_n)- D^2\widetilde{W}(0)\right)P_n\nabla_h w 
\end{eqnarray}
since $F_n(\nabla_h u_n)- D^2\widetilde{W}(0) \nabla_h u_n = P_n D\widetilde{W}(P_n \nabla_h u_n) -
P_nD^2\widetilde{W}(0) P_n\nabla_h u_n $ due to (\ref{eq:DefnF'})-(\ref{eq:DefnFn}). -- We note that $A_n(\nabla_h u_n)$ defines a symmetric operator on $L^2(\Omega)^{d\times d}$ since $P_n^\ast =P_n$ and that
\begin{equation*}
  DA_n(\nabla_h u_n)[\nabla_h v, \nabla_h w]= P_n D^3\widetilde{W}(P_n\nabla_h u_n)[P_n\nabla_h v, P_n\nabla_h w].
\end{equation*}
Moreover, we have
$w_{0,n}= u_{1,n}\equiv u_{1,h}^n$
and
\begin{eqnarray*}
w_{1,n}&=& u_{2,h}^n\equiv u_{2,h}= f_h h^{1+\theta} |_{t=0} +\frac1{h^2} \Div_h \left(DW(\nabla_h u_{0,h}) \right),\\
w_{2,n}&=& u_{3,h}^n\equiv u_{3,h}= \partial_t f_h h^{1+\theta} |_{t=0} +\frac1{h^2} \Div_h \left(D^2W(\nabla_h u_{0,h})\nabla_h u_{1,h} \right),
\end{eqnarray*}
provided (\ref{eq:ApproxCompCond1})-(\ref{eq:ApproxCompCond2}) hold. Hence $(w_{1,n},w_{2,n})\equiv (w_1,w_2)$ are independent of $n\in \N_0$ and $u_{0,n},u_{1,n}$. 
First we will solve (\ref{eq:Approx1'})-(\ref{eq:Approx4'}) for $w_n$ and small
times (depending on $n\in\N$) provided that $0<h\leq h_0$ is sufficiently small $h_0\in (0,1]$ if $\theta>0$ and that $M>0$ is sufficiently small if $\theta =0$ (independent of $n\in \N$). Here $u_n$ is determined by $w_n$ via
\begin{equation}\label{eq:LinkUnWn}
u_n(x,t) = u_{0,h}^n(x) + \int_0^t w_n(x,\tau)\sd \tau.  
\end{equation}
Afterwards we will derive uniform bounds in $0<h\leq h_0$, $n\in\N$, and $t\in
(0,T)$.

Finally, we note that, if $w_n$ solves (\ref{eq:Approx1'})-(\ref{eq:Approx4'}) and $u_n$ is
determined by (\ref{eq:LinkUnWn}), then $u_n$ solves
(\ref{eq:Approx1})-(\ref{eq:Approx4}) since (\ref{eq:Approx1'}) implies
\begin{equation*}
    \partial_t^2 u_{n} - \frac1{h^2} \Div_h F_n(\nabla_h u_{n}) = f_h h^{1+\theta} + c,
\end{equation*}
where $c=c(x)$ is independent of $t$, and the initial condition $\partial_t
w_n|_{t=0} = \partial_t^2 u_n|_{t=0}=w_{1,n}$ implies $c\equiv 0$ by the choice of $w_{1,n}$.


\subsection{Estimates for the Linearized Operator}\label{sec:Linear}

Recall that $z=(t,x')$ with the convention that $z_0=t$ and $z_j=x_j$ for $j=1,\ldots, d-1$. Moreover, recall that $\nabla_z=\nabla_{t,x'}= (\partial_t, \nabla_{x'})$. Furthermore, $P_n=P_n^h$, $n\in \N$, $0<h \leq 1$, denotes the smoothing operator defined above and we set $P_\infty=I$.

Let $u_h$ for some $0<h\leq 1$ be given  such that
 \begin{eqnarray}\label{eq:UniformEstim} 
%   \max_{|\gamma|\leq 1}\left\|\left(\partial_t^2\partial_z^\gamma u_h,
%       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h)\right)\right\|_{C([0,T];L^2)}&\leq& Rh\\\label{eq:UniformEstim2} 
   \max_{|\gamma|\leq 1}\left\|\left(\frac1h\eps_h(\partial_{z}^\gamma u_h),
       \nabla_{x,t} \frac1h\eps_h(\partial_{z}^\gamma u_h)\right)\right\|_{C([0,T];H^{1,0})}&\leq& Rh
  \end{eqnarray}
where $R\in (0,R_0]$ for some $0<R_0\leq 1$ to be determined later. 
% -- We note that in (\ref{eq:UniformEstim}) all fourth order derivatives of $u_h$ are estimated except for 
% \begin{equation*}
%   \partial_t^2 u,\ \partial_t^3 \partial_{x_d} u, \  \partial_t^2 \partial_{x_d}^2 u,\ \partial_{x_d}^3 \nabla_{x,t} u. 
% \end{equation*}
We note that (\ref{eq:UniformEstim}) implies that $\frac1h\eps_h(\partial_z^\gamma u_h) \in C([0,T];V(\Omega))$ and $\partial_t^2 \frac1h\eps_h(u_h)\in C([0,T];H^{1,0})$. For the following we denote
\begin{equation*}
      \|f\|_{V_h}= \|(f,\nabla_h f)\|_{H^{1,0}},
\end{equation*}
where $f\in V(\Omega)$. Of course $\|f\|_{V}\leq \|f\|_{V_h}$ for all $0<h\leq 1$.

Because of $\|\nabla_h P_n f\|_{L^2}\leq\|\nabla_h f\|_{L^2}$, Korn's inequality (\ref{eq:Korn}),
and since $P_n$ commutes with derivatives with respect to $z=(t,x')$,
 (\ref{eq:UniformEstim}) implies
\begin{eqnarray}\nonumber
  \lefteqn{\max_{|\gamma|\leq 1}\left\|\left(\partial_z^\gamma P_n \nabla_h u_h, \partial_z^\gamma P_n\frac1h \eps_h (u_h) \right)\right\|_{L^\infty(0,T;V)}}\\\label{eq:LInftyUnifEstim}
&& + \left\|\left(\partial_t^2 P_n \nabla_h u_h, \partial_t^2 P_n\frac1h \eps_h (u_h) \right)\right\|_{L^\infty(0,T;H^{1,0})} \leq C_1 R h
\end{eqnarray}
for some $C_1\geq 1$ depending only on the constant in the Korn inequality.
\begin{remark}
The analysis in the following will be mainly based on (\ref{eq:LInftyUnifEstim}). Therefore we will assume throughout this section that (\ref{eq:LInftyUnifEstim}) holds for some given $\nabla_h u_h$,  $n\in \N_0\cup \{\infty\}$, and $0<h\leq 1$. -- Of course, if $\nabla_h u_h$ satisfies the stronger estimate (\ref{eq:UniformEstim}), we will have (\ref{eq:LInftyUnifEstim}) for any $n\in \N\cup \{\infty\}$. 
\end{remark}
%\begin{eqnarray*}
%  V(\Omega)&:=& H^{1,1}(\Omega)\cap H^{2,0}(\Omega)
%\cong H^1(-\frac12,\frac12;H^1(\Omega'))\cap L^2(-\frac12,\frac12;H^2(\Omega'))\\
%&&\hookrightarrow BUC([-\frac12,\frac12];H^{\frac32}(\Omega'))\hookrightarrow L^\infty(\Omega).
%\end{eqnarray*}
%We note that $\|P_nf\|_{V(\Omega)}\leq \|f\|_{V(\Omega)}$ since the same is true for $H^{1,1}(\Omega)$ and $H^{2,0}(\Omega)$.
 Because of $V(\Omega)\hookrightarrow L^\infty(\Omega)$, cf. Lemma~\ref{lem:Embeddings},  
%(\ref{eq:LInftyd3})-(\ref{eq:LInftyd2}) with $\theta=\frac14$ and (\ref{eq:UniformEstim}) 
(\ref{eq:LInftyUnifEstim}) implies in particular
\begin{eqnarray}
  \label{eq:CoeffLInftyEstim}
  \left\|\left(P_n \nabla_h u_h,P_n \frac1h\eps_h(
      u_h)\right)\right\|_{L^\infty(0,T;L^\infty\cap V)}\leq MRh, 
\end{eqnarray}
where $M$ depends only on $\Omega$.  %with 
Recall that $\widetilde{W}(A)=W(I+A)$ for all $A\in \R^{d\times d}$.
In order to evaluate $D\widetilde{W}(P_n \nabla_hu_h)$,
we will assume that $R_0>0$ is so small that $\widetilde{W}\in C^\infty(\ol{B_{MR_0}(0)})$ and $MR_0 \leq \eps$, where $\eps>0$ is as in Corollary~\ref{cor:D3W}.



Using (\ref{eq:CoeffLInftyEstim}) and (\ref{eq:D3WEstim}), we obtain
\begin{eqnarray}\nonumber
 \lefteqn{\left| \frac1{h^2}\int_0^1\left(D^3\widetilde{W}(\tau P_n\nabla_h u_h(t))[P_n\nabla_h u_h(t),P_n\nabla_h
   v],P_n \nabla_h w\right)_{L^2(\Omega)}\sd \tau\right|}\\\nonumber
%&\leq & \frac{C}{h^2}\|D^3W(\tau \nabla_h u_h(t))\|_{L^\infty(\Omega;\mathcal{L}^3_h)}\|\nabla_h u_h(t)\|_{L^\infty_h(\Omega)}\|\nabla_h v\|_{L^2_h(\Omega)}\|\nabla_h w\|_{L^2_h(\Omega)}\\\nonumber
&\leq& C'_0\frac1h
   \left\|\left(P_n\nabla_h u_h, P_n\frac1h \eps_h (u_h)\right)\right\|_{V(\Omega)}\left\|\frac1h
   \eps_h(v)\right\|_{L^2(\Omega)}\left\|\frac1h
   \eps_h(w)\right\|_{L^2(\Omega)}\\\label{eq:EstimD3W}
 &\leq & C_0R \left\|\frac1h
   \eps_h(v)\right\|_{L^2(\Omega)}\left\|\frac1h
   \eps_h(w)\right\|_{L^2(\Omega)}
\end{eqnarray}
uniformly in $v,w\in H^1_{per}(\Omega)^d$, $0\leq t\leq T$, $0<h\leq 1$.% because of Corollary~\ref{cor:D3W}, (\ref{eq:LInftyUnifEstim}), and (\ref{eq:CoeffLInftyEstim}), where $\delta_0>0$ is chosen sufficiently small.

In particular, we derive
 \begin{eqnarray*}
   \lefteqn{\frac1{h^2}(A_n(\nabla_hu_h(t))\nabla_h v,\nabla_h v)_{L^2(\Omega)}=\frac1{h^2}(D^2\widetilde{W}(0)\nabla_h v,\nabla_h v)_{L^2(\Omega)}}\\
   && +
   \frac1{h^2}\int_0^1\left( D^3\widetilde{W}(\tau P_n \nabla_h u_h(t))[P_n\nabla_h u_h(t),P_n\nabla_h
   v],P_n \nabla_h v\right)_{L^2(\Omega)}\sd \tau\\
 &\geq & % c_0 \left\|\frac1h \eps_h(v)\right\|_{L^2(\Omega)}^2 - C \frac1h
%    \left\|\left(\frac1h \nabla_h
%        u_h\right)\right\|_{H^{m,1}(\Omega)}\left\|\frac1h
%    \eps_h(v)\right\|_{H^{m,0}(\Omega)}^2\\
   \left(c_0 - C_0R_0\right)  \left\|\frac1h
   \eps_h(v)\right\|_{L^2(\Omega)}^2,
 \end{eqnarray*}
uniformly in $v\in H^1_{per}(\Omega)^d$, $t\in [0,T]$, $0<T<\infty$ $0<R\leq R_0$, $0<h\leq 1$, where $c_0>0$ depends only on $D^2\widetilde{W}(0)$ and $\Omega$.
Hence, if $R_0\in (0,1]$ is sufficiently small,  we have
\begin{equation}\label{eq:UniformCoercive'}
  \frac1{h^2}(A_n(\nabla_hu_h(t))\nabla_h v,\nabla_h v)_{L^2(\Omega)} \geq \frac{c_0}2 \left\|\frac1h \eps_h(v)\right\|_{L^2(\Omega)}^2
\end{equation}
for all $v\in H^1_{per}(\Omega)^d$, $t\in[0,T]$, $0<h\leq 1$, $0<R\leq R_0$, and $u_h$ satisfying (\ref{eq:LInftyUnifEstim}), where $c_0$ is as above and depends only on $D^2\widetilde{W}(0)$ and $\Omega$. By the same kind of
expansion for $D^2\widetilde{W}$ and estimates one shows 
\begin{equation}\label{eq:EstimD2W}
  \left|\frac1{h^2}(\partial_{z_j}A_n(\nabla_hu_h(t))\nabla_h v,\nabla_h
    w)_{L^2(\Omega)}\right|
\leq C'R\left\|\frac1h
   \eps_h(v)\right\|_{L^2}\left\|\frac1h
   \eps_h(w)\right\|_{L^2}
\end{equation}
for all $v,w\in H^1_{per}(\Omega)^d$, $j=0,\ldots, d-1$ uniformly in $0<h\leq 1$, $t\in[0,T]$, $0<R\leq R_0$, $0<T<\infty$. Therefore
\begin{eqnarray}\nonumber
  \lefteqn{\frac1{h^2}(A_n(\nabla_hu_h(t))\nabla_h v,\nabla_h v)_{H^{1,0}(\Omega)}}\\\nonumber
 &\geq& \frac{c_0}2 \left\|\frac1h \eps_h(v)\right\|_{L^2(\Omega)}^2+ \sum_{j=1}^{d-1} \frac{c_0}2 \left\|\frac1h \eps_h(\partial_{x_j} v)\right\|_{L^2(\Omega)}^2\\\nonumber
&& - \sum_{j=1}^{d-1} \left|\frac1{h^2}(\partial_{x_j}A_n(\nabla_hu_h(t))\nabla_h v,\nabla_h
    \partial_{x_j}v)_{L^2(\Omega)} \right|\\\label{eq:UniformCoercive}
&\geq & \left(\frac{c_0}2 - CR_0\right) \left\|\frac1h
   \eps_h(v)\right\|_{H^{1,0}(\Omega)}^2\geq \frac{c_0}3 \left\|\frac1h
   \eps_h(v)\right\|_{H^{1,0}(\Omega)}^2
\end{eqnarray}
for all $v\in H^{1,1}(\Omega)^d$  
 if $0<R\leq R_0$ for some $R_0 \in (0,1]$ sufficiently small.

To obtain higher regularity, we will use:
\begin{lemma}\label{lem:LinEstim2} 
  There are constants $C_0>0,R_0\in (0,1]$ independent of $n\in\N\cup\{\infty\}$ (and $R\in (0,R_0]$)  such that, if $w\in
  H^{1,2}(\Omega)^d\cap H^{3,0}(\Omega)^d$  solves
  \begin{eqnarray*}
    - \frac1{h^2}\Div_h (A_n(\nabla_h u_h(t))\nabla_h w) &=& f  \qquad
    \text{in}\ \mathcal{D}'(\Omega)
  \end{eqnarray*}
  for some $f\in H^{1,0}(\Omega)$ and $0<h\leq 1$ and $P_n\nabla_h u$ satisfies (\ref{eq:CoeffLInftyEstim}) for $0<R\leq R_0$,  then
   we have
   \begin{equation}    \left\| \left(
       \nabla \frac1h \eps_h (w),\nabla_h^2 w\right)\right\|_{H^{1,0}(\Omega)}\label{eq:LinEstim2} 
 \leq C_0 \left( \|h^2f\|_{H^{1,0}(\Omega)} + \left\|\frac1h\eps_h (w)\right\|_{H^{2,0}(\Omega)}\right).
   \end{equation} 
   If additionally
  \begin{equation}
    \label{eq:BC}
 e_d\cdot A_n(\nabla_h u_h(t))\nabla_h
  w|_{x_d=\pm\frac12}=0,
  \end{equation}
 then
  \begin{equation}    \max_{j=0,1}\left\|\left(
      \nabla_h^{1+j} w, \nabla^j \frac1h \eps_h(w) \right)\right\|_{H^{1,0}(\Omega)}\label{eq:LinEstim2b} 
\leq C_0 \|f\|_{H^{1,0}(\Omega)}.
  \end{equation} 
\end{lemma}
\begin{proof} 
Let $0<R_0 \leq 1$ be at least as small as above.
  First of all, since $A_n(0)=D^2\widetilde{W}(0)$, we obtain 
  \begin{eqnarray*}
    \lefteqn{\Div_h (A_n(0)\nabla_h w)
= \frac1h\partial_{x_d} (D^2\widetilde{W}(0)\nabla_h w)_d + \Div_{x'}(D^2\widetilde{W}(0)\nabla_h w)'}\\
&=& \frac1{h^2}(D^2\widetilde{W}(0)\partial_{x_d}^2 w\otimes e_d )_d + \frac1h (D^2\widetilde{W}(0)\partial_{x_d}(\nabla_{x'},0) w)_d + \Div_{x'}(D^2\widetilde{W}(0)\nabla_h w)' 
% \\
% (\Div',\frac1h\partial_{x_d}) (\eps_h(w))\\%= (\Delta_h
%   %  w+\nabla_h \Div_h w)\\
%     &=& 
%     \begin{pmatrix}
%       \frac1{h^2}\partial_{x_d}^2 w'\\
%       \frac2{h^2}\partial_{x_d}^2 w_d
%     \end{pmatrix}
%     +
%     \begin{pmatrix}
%       \nabla' \frac1h\partial_{x_d} w_d\\
%       0
%     \end{pmatrix}
%     + (\Div',0) 2\eps_h(w)
  \end{eqnarray*}
  where $A'=(a_{ij})_{i=1,\ldots d,j=1,\ldots d-1}$ for $A\in \R^{d\times d}$. We note that the second and third term consists of terms of
  $\nabla_{x'}\nabla_h w$. Moreover, 
  \begin{equation*}
   (D^2\widetilde{W}(0)\partial_{x_d}^2 w\otimes e_d )_d =M\partial_{x_d}^2 w  
  \end{equation*}
for some symmetric positive definite matrix $M$, which follows from the Legendre-Hadamard condition (\ref{eq:LegendreHadamard}).
  Hence
  \begin{equation*}
    \frac1{h^2} \partial_{x_d}^2 w =M^{-1}\left(\Div_h\left(Q\nabla_h w\right)  -\frac1h (Q\partial_{x_d}(\nabla_{x'},0) w)_d - \Div_{x'}(Q\nabla_h w)'\right)
  \end{equation*}
for $Q= D^2\widetilde{W}(0)$
and therefore
  \begin{eqnarray*}
     \lefteqn{\left\|
      \frac1{h^2} \partial_{x_d}^2 w\right\|_{H^{1,0}(\Omega)}}\\ 
    &\leq& C_0 \left(\left\|\Div_h \left(D^2\widetilde{W}(0)\nabla_h w\right) \right\|_{H^{1,0}(\Omega)}
    + \left\|\nabla_{x'}\nabla_h w\right\|_{H^{1,0}(\Omega)}\right).
  \end{eqnarray*}
  Thus Korn's inequality and $\|\partial_{x_d}\frac1h\eps_h(w)\|_{L^2(\Omega)}\leq \|\nabla_h^2 w\|_{L^2(\Omega)}$ yields
  \begin{eqnarray}\nonumber
      \lefteqn{\left\|\left(\nabla \frac1h \eps_h (w),
      \nabla_h^2  w\right)\right\|_{H^{1,0}(\Omega)}}\\\label{eq:nablah2}
    &\leq& C_0 \left(\left\|\Div_h \left(D^2\widetilde{W}(0)\nabla_h w\right) \right\|_{H^{1,0}(\Omega)}
    + \left\|\nabla_{x'}\frac1h\eps_h(w)\right\|_{H^{1,0}(\Omega)}\right).
  \end{eqnarray}
  Next we use that
  \begin{eqnarray*}
   \lefteqn{\Div_h \left(A_n(\nabla_h u_h)\nabla_h w\right)}\\
 &=&    \Div_h
    \left(D^2\widetilde{W}(0)\nabla_h w\right) +    \int_0^1\Div_h 
    \left(P_nD^3\widetilde{W}(\tau P_n^h\nabla_h u_h)[P_n^h\nabla_hu_h,P_n^h\nabla_h w]\right)\sd \tau \\
 &\equiv &    \Div_h
    \left(D^2\widetilde{W}(0)\nabla_h w\right) +    \Div_h 
    \left(G(P_n^h\nabla_h u_h)[P_n^h\nabla_hu_h,P_n^h\nabla_h w]\right), 
  \end{eqnarray*}
  where $G\in C^\infty(\ol{B_{\eps}(0)};\mathcal{L}^3(\R^{d\times d}))$ for some suitable $\eps>0$. Hence Corollary~\ref{cor:Algebra} implies
  \begin{eqnarray*}
    \lefteqn{\|G(P_n^h\nabla_h u_h)[P_n^h\nabla_hu_h,P_n^h\nabla_h w]\|_{V_h}}\\
      &\leq& C\|G(P_n^h\nabla_h u_h)\|_{V_h}\|P_n^h\nabla_hu_h\|_{V_h}\|P_n^h\nabla_hw\|_{V_h}\\
      &\leq& CR_0\left\|\left(\nabla_hw, \nabla_h^2 w\right)\right\|_{H^{1,0}}.
    \end{eqnarray*}
    where 
$\|f\|_{V_h}= \|(f,\nabla_h f)\|_{H^{1,0}}$
    and we have used (\ref{eq:PnH1Estim}) as well as (\ref{eq:CoeffLInftyEstim}).
  Hence
  \begin{eqnarray}\nonumber
    \lefteqn{\left\|\Div_h \left(D^2\widetilde{W}(0)\nabla_h w\right)
      \right\|_{H^{1,0}(\Omega)}}\\\nonumber
   &\leq & \left\|\Div_h \left(A_n(\nabla u_h)\nabla_h
       w\right)\right\|_{H^{1,0}(\Omega)}\\\nonumber 
&& +\left\|\nabla_h (G(P_n^h\nabla_h u_h)[P_n^h\nabla_hu_h,P_n^h\nabla_h w]\right\|_{H^{1,0}(\Omega)}\\ \label{eq:deltaEstim}
   &\leq & \left\|h^2f\right\|_{H^{1,0}(\Omega)} +  CR_0\left\|\left(\frac1{h}\eps_h(w), \nabla_h^2w\right)\right\|_{H^{1,0}(\Omega)}.
  \end{eqnarray}
Combining the latter estimate with (\ref{eq:nablah2}) for sufficiently small $R_0 \in (0,1]$,  we obtain (\ref{eq:LinEstim2}). 

Now, if additionally (\ref{eq:BC}), then
\begin{equation*}
  \frac1{h^2}(A_n(\nabla_h u_h)\nabla_h  w, \nabla_h \varphi)_{H^{1,0}} = (f,\varphi)_{H^{1,0}}
\end{equation*}
for all $\varphi \in V(\Omega)^d$. Hence, choosing $\varphi = \partial_{x'}^{2\gamma} w_0$ with $w_0= w- \frac1{|\Omega|} \int_{\Omega} w\sd x$ and $|\gamma|\leq 1$ and using integration by parts, we obtain by (\ref{eq:UniformCoercive}), (\ref{eq:LinEstim2}), and (\ref{eq:D1EstimAn}) below
\begin{eqnarray*}
  \lefteqn{\sup_{|\gamma|\leq 1}\left\|\partial_{x'}^\gamma \frac1h \eps_h(w) \right\|_{H^{1,0}(\Omega)}^2}\\
 &\leq& C_0 \|f\|_{H^{1,0}(\Omega)}\max_{|\gamma|\leq 1}\left\|\partial_{x'}^{2\gamma} w_0\right\|_{H^{1,0}(\Omega)}
+ CR \left\|\frac1h \eps_h(w)\right\|_{H^1(\Omega)}\max_{|\gamma|\leq 1}\left\|\partial_{x'}^{\gamma} w_0\right\|_{H^{1,0}(\Omega)}\\
&\leq& C_0 \left(\|f\|_{H^{1,0}(\Omega)}+R\left\|\frac1h \eps_h (w)\right\|_{H^{2,0}(\Omega)}\right)\max_{|\gamma|\leq 1}\left\|\partial_{x'}^{\gamma} \frac1h \eps_h (w)\right\|_{H^{1,0}(\Omega)}.
\end{eqnarray*}
Thus, choosing $R_0$ sufficiently small, we obtain
\begin{equation*}
  \left\| \frac1h \eps_h(w) \right\|_{H^{2,0}(\Omega)} \leq C_0 \|f\|_{H^{1,0}(\Omega)}
\end{equation*}
with $C_0>0$ depending only on $\Omega$. This finishes the proof.
 \end{proof}

 \begin{lemma}\label{lem:DifEstimAn}
Let $P_n\nabla u_h(t)$ satisfy (\ref{eq:LInftyUnifEstim}) for some $n\in\N\cup \{\infty\}$, $0<h\leq 1$, $t\in [0,T]$, and $0<R\leq R_0$, where $R_0\in (0,1]$ is so small that all previous conditions are satisfied.  
Then
   \begin{eqnarray}\label{eq:D1EstimAn}
     \left|\frac1{h^2}\left((\partial_z^\beta A_n(\nabla_hu_h(t)))\nabla_h
         w,\nabla_h v\right)_\Omega\right|
 \leq
     CR\left\|\frac1h\eps_h(w)\right\|_{H^{|\beta|-1}(\Omega)}\left\|\frac1h\eps_h(v)\right\|_{L^2(\Omega)}
    %\label{eq:D2EstimAn}
   \end{eqnarray}
   if $1\leq |\beta|\leq 2$ and 
   \begin{eqnarray}\label{eq:3DiffEstim}
     \left|\frac1{h^2}\left((\partial_z^\beta A_n(\nabla_hu_h(t)))\nabla_h
         w,\nabla_h v\right)_\Omega\right|
 \leq
     CR\left\| \frac1h\eps_h(w)\right\|_{V(\Omega)} \left\|\frac1h\eps_h(v)\right\|_{L^2(\Omega)}
   \end{eqnarray}
if $|\beta|=3$ and $\beta \neq 3 e_0$, i.e., $\partial_z^\beta \neq \partial_t^3$. Moreover, 
 \begin{eqnarray}\label{eq:D1EstimAn'}
     \left|\frac1{h^2}\left((\partial_t\partial_{x_j} A_n(\nabla_hu_h(t)))\nabla_h
         w,\nabla_h v\right)_\Omega\right|
 \leq
     CR\left\|\frac1h\eps_h(w)\right\|_{H^{1,0}(\Omega)}\left\|\frac1h\eps_h(v)\right\|_{L^2(\Omega)}
    %\label{eq:D2EstimAn}
   \end{eqnarray} 
for all $j=1,\ldots, d-1$. The constants $C$ are independent of $\nabla_h u_h(t), w,v, h,n, R$.
 \end{lemma}
 \begin{proof}
If $|\beta|=1$, then (\ref{eq:D1EstimAn}) is just (\ref{eq:EstimD2W}).
Next let $|\beta|=2$.
  Then for $j,k=0,\ldots, d-1$
%   \begin{equation*} 
%     \partial_{z_j} A_n(\nabla_h u_h) = P_nD^3\widetilde{W}(P_n\nabla_h u_h)(P_n\partial_{z_j}\nabla_h u_h),
%   \end{equation*}
%   and 
  \begin{eqnarray*}
    \partial_{z_j}\partial_{z_k} A_n(\nabla_h u_h) &=& P_nD^3\widetilde{W}(P_n\nabla_h
    u_h)(P_n\partial_{z_j}\partial_{z_k}\nabla_h u_h)\\
&& + P_nD^4 \widetilde{W}(P_n\nabla_h
    u_h)(P_n\partial_{z_j}\nabla_h u_h,P_n\partial_{z_k}\nabla_h u_h),
  \end{eqnarray*}
  where 
  \begin{eqnarray*}
   \left\|P_n\partial_{z_j}\partial_{z_k} \frac1h \eps_h (u_h)\right\|_{H^{1,0}(\Omega)}
&\leq& 
    C_0 \left\|P_n \nabla_z \frac1h \eps_h (u_h)\right\|_{V(\Omega)} \leq C_0Rh 
\end{eqnarray*}
due to (\ref{eq:LInftyUnifEstim}). Together with (\ref{eq:D3WEstimb}) the latter estimate implies (\ref{eq:D1EstimAn}) in the case $|\beta|=2$. Moreover, (\ref{eq:D1EstimAn'}) is proved in the same way using that $\partial_t \partial_{x_j} \frac1h\eps_h (u_h) \in H^1(\Omega)$ is uniformly bounded and again (\ref{eq:D3WEstimb}).

  Finally, if $|\beta|=3$ with $\partial_z^\beta\neq \partial_t^3$, we use that
  \begin{eqnarray*}
    \lefteqn{\partial_{z_j}\partial_{z_k}\partial_{z_l} A_n(\nabla_h u_h) = P_nD^3\widetilde{W}(P_n\nabla_h
    u_h)[P_n\partial_{z_j}\partial_{z_k}\partial_{z_l}\nabla_h u_h]}\\
&& + P_nD^4 \widetilde{W}(P_n\nabla_h
    u_h)[P_n\partial_{z_j}\partial_{z_l}\nabla_h u_h,P_n\partial_{z_k}\nabla_h u_h]\\
  &&  +  P_nD^4 \widetilde{W}(P_n\nabla_h
    u_h)[P_n\partial_{z_j}\nabla_h u_h,P_n\partial_{z_k}\partial_{z_l}\nabla_h u_h]\\
&& +  P_nD^4 \widetilde{W}(P_n\nabla_h
    u_h)[P_n\partial_{z_l}\nabla_h u_h,P_n\partial_{z_j}\partial_{z_k}\nabla_h u_h]\\
&&
   + P_nD^5W (P_n\nabla_hu_h) [P_n\partial_{z_j}\nabla_h u_h,P_n\partial_{z_k}\nabla_h u_h,P_n\partial_{z_l}\nabla_h u_h]
  \end{eqnarray*}
  Since $P_n\nabla_z \nabla_h u_h\in L^\infty(\Omega)$  and $P_n\nabla_z^2 \nabla_h u_h\in H^{1,0}(\Omega) \hookrightarrow L^{4,2}(\Omega)$ are of order $CRh$ due to (\ref{eq:LInftyUnifEstim}), the estimates of all parts in
  \begin{equation*}
    \frac1{h^2} \left((\partial_z^\beta A_n(\nabla_h u_h))\nabla_h w, \nabla_h v\right)_\Omega
  \end{equation*}
which come from terms involving $D^4\widetilde{W}$ or $D^5\widetilde{W}$ can be done in a straight forward manner by
\begin{equation*}
  CR \left\| \frac1h \eps_h (w)\right\|_{L^{4,\infty}}\left\| \frac1h \eps_h (v)\right\|_{L^2}\leq   C'R \left\| \frac1h \eps_h (w)\right\|_{H^{1}}\left\| \frac1h \eps_h (v)\right\|_{L^2}
\end{equation*}
uniformly in $0<h\leq 1$ and $n\in \N_0\cup\{\infty\}$.
 It only remains to estimate the part involving the $D^3\widetilde{W}$-term:
To this end we use that (\ref{eq:LInftyUnifEstim}) and (\ref{eq:D3WEstim}) imply
\begin{eqnarray*}
     \lefteqn{\left|\frac1{h^2}\left(( D^3\widetilde{W}(P_n\nabla_hu_h))[P_n\partial_z^\beta \nabla_hu_h,P_n\nabla_h
         w],P_n\nabla_h v\right)_\Omega\right|} \\
   &\leq& \frac{C_0}h \left\|\partial_z^\beta\frac1h\eps_h(u_h)  \right\|_{L^2(\Omega)} \left\| \left( \frac1h \eps_h(w), \nabla_h w \right) \right\|_{V(\Omega)}\left\| \frac1h \eps_h(v)  \right\|_{L^2(\Omega)}\\
   &\leq& CR \left\| \frac1h \eps_h(w) \right\|_{V(\Omega)}\left\| \frac1h \eps_h(v)  \right\|_{L^2(\Omega)}
\end{eqnarray*}
Altogether we obtain (\ref{eq:3DiffEstim}). 
 \end{proof}
 \begin{corollary} 
Let $P_n\nabla_h u_h(t)$, $n\in\N\cup \{\infty\}$, $0<h\leq 1$, $0<R\leq R_0$ be as in Lemma~\ref{lem:DifEstimAn}. Then  we have 
   \label{cor:H10Estims}
   \begin{eqnarray*}
     \left|\frac1{h^2}\left((\partial_t A_n(\nabla_h u_h(t)))\nabla_h w,\nabla_h
         v \right)_{H^{1,0}}\right|
&\leq& CR
\left\|\frac1h\eps_h(w)\right\|_{H^{1,0}}\left\|\frac1h\eps_h(v)\right\|_{H^{1,0}}\\
    \left|\frac1{h^2}\left((\partial_t^2 A_n(\nabla_h u_h))\nabla_h w,\nabla_h
         v \right)_{H^{1,0}}\right|
&\leq& CR \left\|\frac1h\eps_h(w)\right\|_{V(\Omega)}\left\|\frac1h\eps_h(v)\right\|_{H^{1,0}}
   \end{eqnarray*}
uniformly in  $0<h\leq 1$, $0<R\leq R_0$, $n\in \N_0\cup \{\infty\}$,  $t\in [0,T]$, and $0<T<\infty$.
 \end{corollary}
 \begin{proof}
   Because of (\ref{eq:EstimD2W}), we have
   \begin{eqnarray*}
     \left|\frac1{h^2}\left((\partial_t A_n(\nabla_h u_h))\nabla_h w_j,\nabla_h
         v_j \right)_{L^2(\Omega)}\right|
&\leq& CR\left\|\frac1h\eps_h(w_j)\right\|_{L^2}\left\|\frac1h\eps_h(v_j)\right\|_{L^2}
   \end{eqnarray*}
for $j=0,\ldots,d-1$ and $(w_0,v_0)=(w,v)$, $(w_j,v_j)=(\partial_{x_j} w,\partial_{x_j} v)$ if
$j=1,\ldots,d-1$. Moreover, 
\begin{eqnarray*}
     \lefteqn{\left|\frac1{h^2}\left((\partial_t\partial_{x_j} A_n(\nabla_h u_h))\nabla_h w,\nabla_h
         v_j \right)_{L^2(\Omega)}\right|}\\
&\leq&
CR\left\|\frac1h\eps_h(w)\right\|_{H^{1,0}}\left\|\frac1h\eps_h(v_j)\right\|_{L^2}
\leq
C'R\left\|\frac1h\eps_h(w)\right\|_{H^{1,0}}\left\|\frac1h\eps_h(v)\right\|_{H^{1,0}}
\end{eqnarray*}
for $j=1,\ldots,d-1$ due to (\ref{eq:D1EstimAn'}). Altogether this implies the
first estimate.

Similarly, (\ref{eq:D1EstimAn}) yields
   \begin{eqnarray*}
     \lefteqn{\left|\frac1{h^2}\left((\partial_t^2 A_n(\nabla_h u_h))\nabla_h w_j,\nabla_h
         v_j \right)_{L^2(\Omega)}\right|}\\
&\leq&
CR\left\|\frac1h\eps_h(w_j)\right\|_{H^1}\left\|\frac1h\eps_h(v_j)\right\|_{L^2}
\leq C'R\left\|\frac1h\eps_h(w)\right\|_{V(\Omega)}\left\|\frac1h\eps_h(v)\right\|_{H^{1,0}}
   \end{eqnarray*}
where $(w_j,v_j)$ are as above. Finally, (\ref{eq:3DiffEstim}) implies
\begin{eqnarray*}
  \lefteqn{\left|\frac1{h^2}\left((\partial_t^2\partial_{x_j} A_n(\nabla_h u_h))\nabla_h w,\nabla_h
      v_j \right)_{L^2(\Omega)}\right|}\\
  &\leq&
CR\left\|\frac1h\eps_h(w)\right\|_{V(\Omega)}\left\|\frac1h\eps_h(v_j)\right\|_{L^2}
\leq
C'R\left\|\frac1h\eps_h(w)\right\|_{V(\Omega)}\left\|\frac1h\eps_h(v)\right\|_{H^{1,0}}
\end{eqnarray*}
This shows the second estimates.
 \end{proof}

Next we will show solvability of (\ref{eq:ApproxCompCond1}) and the estimate (\ref{eq:ApproxIVEstim}) for $k=0$.
 \begin{proposition}
   \label{prop:SolvIV}
     Let $0<h\leq 1$, $P_n^h$, $n\in\N$, be the smoothing operators from above, let $P_\infty^h=I$, and let $F_n$ be defined as in (\ref{eq:DefnFn}). 
  Then there are constants $C_0>0,M_0\in (0,1]$ independent of $n\in\N\cup\{\infty\}$  such that for any $f\in H^{2,0}(\Omega)^d$ with $\|f\|_{H^{2,0}}\leq M_0h$ and $\int_\Omega f\sd x=0$ there is a solution $w \in H^{2,2}(\Omega)^d\cap H^{4,0}(\Omega)^d $, which is unique up to a constant, such that
  \begin{equation}\label{eq:FnEq}
    \frac1{h^2}\left( F_n(\nabla_h w), \nabla_h\varphi \right)_{L^2(\Omega)} =(f,\varphi)_{L^2(\Omega)}
  \end{equation}
for all $\varphi \in H^1_{per}(\Omega)^d$ and
\begin{equation}\label{eq:FnEstim}
  \left\|\left(\frac1h \eps_h (w), \nabla \frac1h \eps_h (w), \nabla_h^2 w\right)\right\|_{H^{2,0}(\Omega)} \leq C_0\|f\|_{H^{2,0}(\Omega)}.
\end{equation}
for some $C_0>0$ independent of $h,f, n$.
 \end{proposition}
 \begin{proof}

   Because of (\ref{eq:DefnFn}), (\ref{eq:FnEq}) is equivalent to solve
   \begin{eqnarray*}
     \lefteqn{\weight{L_h w, \varphi}_{H^{-1}_h,H^1_h} \equiv \frac1{h^2}\left( D^2\widetilde{W}(0)\nabla_h w, \nabla_h\varphi \right)_{L^2(\Omega)}}\\
 &=&(f,\varphi)_{L^2(\Omega)}- 
\frac1{h^2}\left( F_n'(\nabla_h w), \nabla_h\varphi \right)_{L^2(\Omega)} \equiv (G_h (w), \nabla \varphi)_{L^2(\Omega)}.
   \end{eqnarray*}
We will prove the proposition with the aid of the contraction mapping principle. To this end we note that 
 for every $f\in H^{1,0}(\Omega)^d$ and $F\in H^{1,1}(\Omega)^{d\times d}\cap H^{2,0}(\Omega)^{d\times d}$ there is a unique $w\in H^{2,1}(\Omega)^d\cap H^{3,0}(\Omega)^d$ such that
   \begin{equation}\label{eq:Lh}
     \weight{L_h w, \varphi}_{H^{-1}_h,H^1_h} = (f,\varphi)_{L^2(\Omega)}+ (F,\nabla_h\varphi)_{L^2(\Omega)}
   \end{equation}
for all $\varphi \in H^1_h(\Omega)$ because of the Lemma of Lax-Milgram, Korn's inequality, and since $L_h$ commutes with tangential derivatives.
The solution satisfies
\begin{equation}\label{eq:C0}
  \left\|\frac1h \eps_h(w)\right\|_{H^{2,0}(\Omega)}\leq C_0 \left(\|f\|_{H^{1,0}(\Omega)}+ \|F\|_{H^{2,0}_h(\Omega)}\right)
\end{equation}
for some universal $C_0>0$. Moreover, (\ref{eq:Lh}) implies 
\begin{equation*}
  -\frac1{h^2} \Div_h (D^2\widetilde{W}(0)\nabla_h w)= f-\Div_h F \qquad \text{in}\ \mathcal{D}'(\Omega).
\end{equation*}
Therefore $w\in H^{2,1}(\Omega)^d$ by standard elliptic regularity. Hence Lemma~\ref{lem:LinEstim2} together with the previous estimate imply
\begin{equation*}
  \left\|\left(\frac1h \eps_h(w),\nabla \frac1h \eps_h(w),\nabla_h^2 w\right)\right\|_{H^{1,0}(\Omega)}\leq C_0 \left(\left\|\left(f, h^2\nabla_h F\right)\right\|_{H^{1,0}(\Omega)}+ \|F\|_{H^{2,0}_h(\Omega)}\right)
\end{equation*}
for some universal $C_0>0$. Using (\ref{eq:EstimD3W}) and Corollary~\ref{cor:D3W},  one derives that
\begin{equation*}
  \|G_h(w_1)-G_h(w_2)\|_{H^{2,0}_h(\Omega)} \leq C M_0 \left\|\frac1h\eps_h(w_1-w_2)\right\|_{V(\Omega)}
\end{equation*}
for some $C>0$ provided that 
\begin{equation}\label{eq:Estimwj}
\max_{j=1,2}  \left\|\left(\frac1h \eps_h(w_j),\nabla \frac1h \eps_h(w_j),\nabla_h^2 w_j\right)\right\|_{H^{1,0}(\Omega)}\leq 2C_0M_0 h,
\end{equation}
where $C_0>0$ is as (\ref{eq:C0}) and $M_0\in (0,1]$. Here we note that
\begin{eqnarray*}
  \partial_{x_k} G_h(w_j)&=& -\frac1{h^2}A'_n(\nabla_h w_j)\nabla_h \partial_{x_k}w_j,\\
  \partial_{x_k}\partial_{x_l} G_h(w_j)&=& -\frac1{h^2}A'_n(\nabla_h w_j)\nabla_h \partial_{x_k}\partial_{x_l}w_j\\
&& -\frac1{h^2} P_n D^3W(P_n \nabla_h w_j)[P_n\nabla_h \partial_{x_k} w_j,P_n \nabla_h \partial_{x_l} w_j]
\end{eqnarray*}
for all $k,l=1,\ldots, d-1$, $j=1,2$,
where $A'_n(\nabla_h w_j)=A_n(\nabla_h w_j) - D^2 \widetilde{W}(0)$. To estimate the $A_n'$-terms one uses (\ref{eq:D3WEstim}) or (\ref{eq:EstimD3W}) and to estimate the $D^3 W$-term one uses (\ref{eq:D3WEstimb}).

Furthermore, using Corollary~\ref{cor:Algebra}, one shows in the same way as in the proof of Lemma~\ref{lem:LinEstim2}, that
\begin{equation*}
  h^2\|\nabla_h(G_h(w_1)-G_h(w_2))\|_{H^{1,0}(\Omega)} \leq C M_0 \left\|\left(\nabla_h^2(w_1-w_2), \nabla_h (w_1-w_2)\right)\right\|_{H^{1,0}}
\end{equation*}
for some $C>0$ provided that (\ref{eq:Estimwj}) holds. Hence, if $M_0\in (0,1]$ is sufficiently small, we obtain that $L_h^{-1}G_h\colon X_h\to X_h$ restricted to $\ol{B_{2C_0M_0h}(0)}$ is a contraction, where $X_h$ is normed by
\begin{equation*}
   \|w\|_{X_h}:= \left\|\left(\frac1h \eps_h(w),\nabla \frac1h \eps_h(w),\nabla_h^2 w\right)\right\|_{H^{1,0}(\Omega)}.
\end{equation*}
Therefore we obtain a unique solution $w$ solving (\ref{eq:FnEq}) and satisfying (\ref{eq:FnEstim}) with $H^{2,0}(\Omega)$ replaced by $H^{1,0}(\Omega)$. In order to obtain (\ref{eq:FnEstim}), one can simply use that $w_j:= \partial_{x_j}w$, $j=1,\ldots, d-1$, solves
\begin{equation*}
    \frac1{h^2}\left( A_n(\nabla_h w)\nabla_h w_j, \nabla_h\varphi \right)_{L^2(\Omega)} =(\partial_{x_j}f,\varphi)_{L^2(\Omega)} \qquad \text{for all}\ \varphi \in H^1_{per}(\Omega)
\end{equation*}
 and apply Lemma~\ref{lem:LinEstim2}.
 \end{proof} 




The following lemma contains the essential estimate for the linearized system to (\ref{eq:Evol1})-(\ref{eq:BC3}):
\begin{lemma}\label{lem:LinEstim3} 
  Let $0<T<\infty$, $0<h\leq 1$, $0<R\leq R_0$, $n\in \N\cup \{\infty\}$ be given, and let $R_0$ be as in Lemma~\ref{lem:LinEstim2}. 
  Assume that $u_h$ satisfies (\ref{eq:LInftyUnifEstim}) and that $f\in W^1_1(0,T;H^{1,0})^d$, $w_0\in H^{1,2}(\Omega)^d\cap H^{3,0}(\Omega)^d, w_1 \in V(\Omega)^d$.
%   and let $\tilde{R}:= R^{\frac14}(R')^{\frac34}+R$. % Alt: $\tilde{R}:= h^{\frac12} +(RR')^{\frac12}+R$.
  Then there is a unique $w\in C^0([0,T];H^{2,1}(\Omega)\cap H^{3,0}(\Omega))^d\cap C^2([0,T];H^{1,0}(\Omega))^d$  that solves
\begin{eqnarray}\label{eq:stEq}
  \partial_t^2 w - \frac1{h^2} \Div_h (A_n (\nabla_h
  u_h)\nabla_h w) &=& f\\\label{eq:Neumann1}
   A_n (\nabla_h
  u_h)\nabla_h w\, e_d|_{x_d=\pm\frac12} &=&0 \\\label{eq:Periodic1}
  \left. w\right|_{x_j=L} &=& \left. w\right|_{x_j=-L}, j=1,\ldots, d-1\\
  \left. (w,\partial_t w)\right|_{t=0} &=& (w_0, w_1).
\end{eqnarray}
Moreover, there are some constants $C_L, C'\geq 1$ depending only on $\Omega$ and $W$  such that
  \begin{eqnarray}\label{eq:LinEstim}
    \lefteqn{\left\|\left(\partial_t^2 
          w,\frac1h \eps_h(w), \nabla_{x,t}\frac1h \eps_h(w)\right)\right\|_{C([0,T];H^{1,0})}}\\\nonumber
&\leq& C_Le^{C'RT} \left( 
      \| f\|_{W^1_1(0,T;H^{1,0})}
 + \left\|\left(\frac1h \eps_h(w_1), w_2, f|_{t=0}\right)\right\|_{H^{1,0}}  \right)
  \end{eqnarray}
  where
  \begin{eqnarray}\label{eq:LinEstimw2}
    w_{2}&=& \frac1{h^2}\Div_h (A_n(\nabla_h
u_h|_{t=0})\nabla_h w_0)+   f|_{t=0}.
  \end{eqnarray}
\end{lemma}
\begin{proof} 
%In the following, constants, which are independent of $R'$ will be denoted by $C_0,C_0',\ldots$, while possibly $R'$-dependent constant will be denoted by $C,C',\ldots$.
Existence of a solution $w\in C^1([0,T];H^{1,0}(\Omega))\cap C^0([0,T];V(\Omega))$ can be obtained by the standard energy method, cf. e.g. \cite[Theorem~10.8]{RenardyRogers} with $H=H^{1,0}(\Omega)^d, V=V(\Omega)^d$ and 
 \begin{equation*}
   a(t;v,w) = \frac1{h^2}(A_n(\nabla_h u_h) \nabla_h v,\nabla_h w)_{H^{1,0}(\Omega)}, \qquad v,w\in V(\Omega)^d,
  \end{equation*}
  which is bounded and coercive on $V(\Omega)^d$ because of (\ref{eq:UniformCoercive}). If $n\neq \infty$, then $w\in C^2([0,T];H^{1,0}(\Omega))\cap C^1([0,T];V(\Omega))$ can be obtained by the same technique as in \cite[Section~30.1]{WlokaPDE}, where we note that
  \begin{equation*}
    \frac1{h^2} \div_h ((\partial_t A_n(\nabla_h u_h))\nabla_h \cdot) \colon V(\Omega)\to H^{1,0}(\Omega)
  \end{equation*}
is a bounded linear operator with operator norm bounded uniformly in $t\in [0,T]$
  because of the smoothing operator $P_n$ in the definition of $A_n$ and $\partial_t \nabla_h u_h \in C([0,T];L^2(\Omega))$. Moreover, $w\in C^0([0,T];H^{1,2}(\Omega)\cap H^{3,0}(\Omega)))$ follows from (\ref{eq:stEq}) with $\partial_t^2 w,f\in C^{0}([0,T];H^{1,0}(\Omega))$, (\ref{eq:DefnAn}) with $n<\infty$ and standard elliptic theory. Finally, if $n=\infty$, then existence of a solution $w\in C^2([0,T];H^{1,0}(\Omega))\cap C^0([0,T];H^{1,2}(\Omega)\cap H^{3,0}(\Omega))$ can be obtained from the case $n\in \N$ by using the uniform bounds due to (\ref{eq:LinEstim}) proved below and passing to the limit $n\to \infty$.

Hence the main task is to  establish (\ref{eq:LinEstim}).
%   If $0<T\leq 1$, we assume without loss of generality that $f(0)=0$. Otherwise we replace $f$ by 
% $\tilde{f}=f-f|_{t=0}$ and $w$ by $\tilde{w}=w+v_0$, where $v_0$  solves
%   \begin{equation*}
%     \frac1{h^2}(A_n(\nabla_h u_h|_{t=0})\nabla_h v_0,\nabla_h \varphi)_{H^{1,0}} = (f|_{t=0}, \varphi)_{H^{1,0}} 
%   \end{equation*}
%   for all $\varphi \in V(\Omega)$. Then the estimate for $\tilde{w}$ in terms of $\tilde{f}$ implies the original estimate because of Lemma~\ref{lem:LinEstim2}. 
% since
%   \begin{equation*}
%     \|\tilde{f}\|_{W^1_1(0,T;L^2)} \leq \|\partial_t f\|_{L^1(0,T;L^2)} + T \|\tilde{f}\|_{L^\infty(0,T;L^2)} \leq 2 \|\partial_t f\|_{L^1(0,T;L^2)}
%   \end{equation*}
%   for all $0<T\leq 1$
%   due to $\partial_t \tilde{f}= \partial_t f$ and $\tilde{f}(0)=0$. (Note that $w_2$ does not change during this transformation.)
%  This reduction will ensure that
%  \begin{equation}\label{eq:UnifTimeBdd}
%    \|f\|_{L^\infty(0,T;H)} \leq C_0\|f\|_{W^1_1(0,T;H)}
%  \end{equation}
%  with $H=H^{1,0}$
%  uniformly for all $0<T<\infty$. (For $T\geq 1$ the estimate holds for general $f\in W^1_1(0,T;H)$ due to (\ref{eq:StrongBUCConv}) and $\|f(0)\|_{H}\leq \|f\|_{W^1_1(0,1;H)}$. For $0<T\leq 1$ one needs $f(0)=0$ in order to have a uniform constant.) 
% In particular this implies
% \begin{eqnarray}\nonumber
%   |(f, \partial_t g)_{L^2(0,t;H)}| &\leq&|(\partial_t f, g)|_{L^2(0,t;H)} + \left|(f(\tau),g(\tau))_H|_{\tau=0}^t\right|\\\label{eq:DualEstim}
% &\leq& C_0\|f\|_{W^1_1(0,T;H)}\|g\|_{L^\infty(0,t;H)}
% \end{eqnarray}
% uniformly in $g\in W^1_1(0,T;H)$ and $0\leq t\leq T<\infty$.
 First of all, we note that (\ref{eq:stEq})-(\ref{eq:Periodic1}) imply 
 \begin{equation*}
   a(t):=\int_\Omega w(t)\sd x= \int_\Omega w_0 \sd x + t \int_\Omega w_1 \sd x + \int_0^t (t-\tau)\int_\Omega  f(\tau,x) \sd x \sd \tau.
 \end{equation*}
 Hence, replacing $w(t)$ by $w(t)-a(t)$ and subtracting from $(w_0,w_1,f)$ their mean values with respect to $\Omega$, we can reduce to the case
 \begin{equation*}
    \int_\Omega w_0 \sd x= \int_\Omega w_1 \sd x = \int_\Omega f(t) \sd x = \int_\Omega w(t)\sd x= 0
 \end{equation*}
 for all $0\leq t\leq T$.

Now we
differentiate (\ref{eq:stEq}) with respect to $t$ and multiply with $\partial_t^2 w$ in
  $H:=H^{1,0}(\Omega)$. 
Then we obtain
  \begin{eqnarray*}
    \lefteqn{\frac12\frac{d}{dt}\left( \left\|\partial_t^2 w\right\|_{H}^2+\frac1{h^2}\left(A_n(\nabla_hu_h)\nabla_h\partial_t w,
        \nabla_h \partial_t w\right)_{H}  \right)}\\
&\leq &  \left|\left(\partial_t f,\partial_t^2 w\right)_{H}\right|+ \frac32 \left|\frac1{h^2}\left( (\partial_t A_n(\nabla_h
  u_h))\nabla_h \partial_t w,\nabla_h \partial_t w\right)_{H}\right|\\
&&
+  \left|\frac1{h^2}\left( (\partial_t^2 A_n(\nabla_h
  u_h))\nabla_h w,\nabla_h \partial_t w\right)_{H}\right|
 -  \frac1{h^2}\frac{d}{dt}\left((\partial_t A_n(\nabla_hu_h))\nabla_h w,
        \nabla_h \partial_t w\right)_{H}
  \end{eqnarray*}
in the sense of distributions,
  where we have used 
  \begin{eqnarray}\nonumber
    \lefteqn{\frac{d}{dt}\left( \frac1{2h^2}\left(A_n(\nabla_hu_h)\nabla_h\partial_t w,
        \nabla_h \partial_t w\right)_{H} + \frac1{h^2}\left((\partial_t A_n(\nabla_hu_h))\nabla_h w,
        \nabla_h \partial_t w\right)_{H} \right) }\\\nonumber
    &=& -\frac12\frac{d}{dt} \|\partial_t^2 w\|_{H}^2 + \left(\partial_t f, \partial_t^2 w\right)_{H} + \frac3{2h^2} \left((\partial_t A_n(\nabla_hu_h))\nabla_h\partial_t w,
        \nabla_h \partial_t w\right)_{H}\\\label{eq:EnergyDiff}
&&+  \frac1{h^2}\left((\partial_t^2 A_n(\nabla_hu_h))\nabla_h w,
        \nabla_h \partial_t w\right)_{H}  
  \end{eqnarray}
  and (\ref{eq:Neumann1})-(\ref{eq:Periodic1}). Due to Corollary~\ref{cor:H10Estims} we have
  \begin{eqnarray*} 
   \frac1{h^2} \left|\left((\partial_t A_n(\nabla_hu_h))\nabla_h\partial_t w,
        \nabla_h \partial_t w\right)_{H}\right|
    &\leq& CR\left\|\frac1{h}\eps_h(\partial_t w)\right\|_{H}^2,\\
\frac1{h^2}\left|\left((\partial_t^2 A_n(\nabla_hu_h))\nabla_h w,
        \nabla_h \partial_t w\right)_{H}\right|
    &\leq& CR\left\|\frac1{h}\eps_h(w)\right\|_{V(\Omega)}\left\|\frac1{h}\eps_h(\partial_t w)\right\|_{H}
  \end{eqnarray*} 
for every $t\in [0,T]$.
  Moreover, because of Corollary~\ref{cor:H10Estims} again,
  \begin{eqnarray*}
    \lefteqn{\sup_{0\leq \tau \leq t}\left|\frac1{h^2}\left((\partial_t A_n(\nabla_hu_h(\tau)))\nabla_h w(\tau),
        \nabla_h \partial_t w(\tau)\right)_{H}\right|}\\
  &\leq & C R\left\| \frac1h \eps_h (w)\right\|_{L^\infty(0,t;H)}
\left\| \frac1h \eps_h (\partial_t w)\right\|_{L^\infty(0,t;H)}
  \end{eqnarray*}
  Therefore the previous estimates, (\ref{eq:UniformCoercive}), and Young's inequality imply
  \begin{eqnarray}\nonumber
    \lefteqn{\sup_{0\leq \tau \leq t}\left\|\left(\partial_t^2 w(\tau),\frac1h \eps_h(\partial_t w(\tau))
        \right)\right\|_{H}^2  }\\
\nonumber
    &\leq & CR \left\|\left( \partial_t^2 w, \frac1h \eps_h(w),\nabla_{x,t}\frac1h\eps_h( w)\right)\right\|_{L^2(0,t;H)}^2+
  C_0\|\partial_t f\|_{L^1(0,T;H)}^2   \\\nonumber %\label{eq:FirstEstim}
&& + C_0\left\|\left(\frac1h \eps_h(w_1),
        w_2\right)\right\|_{H}^2 + C R\left\| \frac1h \eps_h (w)\right\|_{L^\infty(0,t;H)}^2.
  \end{eqnarray}
  Now
  \begin{eqnarray*}
    \left\| \frac1h \eps_h (w)\right\|_{L^\infty(0,t;H)}^2 &\leq& C_0\left(\max_{j=0,1}\left\| \frac1h \eps_h (\partial_t^j w)\right\|_{L^2(0,t;H)}^2 + \left\|\frac1h \eps_h (w_0)\right\|_{H}^2 \right), 
    \end{eqnarray*}
due to 
\begin{equation}\label{eq:UnivLinftyEb}
\|f\|_{L^\infty(0,t;H)}\leq C_0 \left(\|f\|_{W^1_2(0,t;H)} + \|f|_{t=0}\|_{H}\right)  
\end{equation}
with some $C_0>0$ independent of $t>0$, cf. (\ref{eq:StrongBUCConv}), and
    \begin{eqnarray*}
   \lefteqn{\left\|\left(\frac1h \eps_h (w), \nabla \frac1h \eps_h (w)\right)\right\|_{L^\infty(0,t;H)}}\\
    &\leq& C_0 \left( \|f\|_{L^\infty(0,t;H)} + \|\partial_t^2 w\|_{L^\infty(0,t;H)}\right) \\
 &\leq& C_0 \left( \|f\|_{W^1_1(0,t;H)}+\|f|_{t=0}\|_{H} + \|\partial_t^2 w\|_{L^\infty(0,t;H)}\right) 
  \end{eqnarray*}
due to (\ref{eq:LinEstim2b}) and (\ref{eq:UnivLinftyEb})
  uniformly in $0\leq t\leq T$.
Hence we conclude
  \begin{eqnarray}\nonumber
    \lefteqn{\sup_{0\leq \tau \leq t}\left\|\left(\partial_t^2 w(\tau), \frac1h \eps_h(w),\nabla_{x,t}\frac1h \eps_h( w(\tau))
        \right)\right\|_{H}^2  }\\
\nonumber
    &\leq & CR \left\|\left( \partial_t^2 w,\frac1h \eps_h(w),\nabla_{x,t}\frac1h\eps_h(
        w)\right)\right\|_{L^2(0,t;H)}^2   \\\nonumber %\label{eq:FirstEstim}
&& +
  C_0\| f\|_{W^1_1(0,T;H)}^2+ C_0\left\|\left(\frac1h \eps_h(w_1),
        w_2, f|_{t=0}\right)\right\|_{H}^2,
  \end{eqnarray}
  where we have used $R\leq 1$ and (\ref{eq:LinEstimw2}).
%  and 
%   \begin{eqnarray*} 
%     \left\|\frac1h \eps_h (w)\right\|_{L^2(\Omega)} &\leq& C_0 \left(\left\|\nabla_{x'}\frac1h \eps_h (w)\right\|_{L^2(\Omega)} + \left\|\partial_{x_d}\frac1h \eps_h (w)\right\|_{L^2(\Omega)}\right)\\
% &\leq& C_0 \left(\left\|\nabla_{x'}\frac1h \eps_h (w)\right\|_{L^2(\Omega)} + \left\|\nabla_h^2 w\right\|_{L^2(\Omega)}\right)
%   \end{eqnarray*}
% due to Poincar\'e's inequality.
  Therefore the Lemma of Gronwall yields
  \begin{eqnarray*}
    \lefteqn{\left\|\left(\partial_t^2 w,\frac1h \eps_h(w), \nabla_{x,t}\frac1h \eps_h(w)
        \right)\right\|_{L^\infty(0,T;H)}^2}\\
    &\leq& C_Le^{C'RT}\left(\left\|\left(\frac1h \eps_h(w_1),
        w_2, f|_{t=0}\right)\right\|_{H}^2 + \|f\|_{W^1_1(0,T;H)}^2 \right).
  \end{eqnarray*}
  This shows (\ref{eq:LinEstim}).
\end{proof}

Finally, we consider (\ref{eq:Approx1'})-(\ref{eq:Approx4'}) with $f$ replaced
by $-\Div_h f$ in its weak form,
namely:
\begin{eqnarray}\nonumber
  -(\partial_t w,\partial_t \varphi)_{Q_T} &+& \frac1{h^2} (A_n(\nabla_h
  u_h)\nabla_h w,\nabla_h \varphi)_{Q_T}\\\label{eq:wEq1}
 &=& (f,\nabla_h \varphi)_{Q_T} + \weight{w_1,\varphi|_{t=0}}_{H^{-1}_h,H^1_h} \\
  \left. w\right|_{x_j=L} &=& \left. w\right|_{x_j=-L}\quad \text{for all}\ j=1,\ldots, d-1\\\label{eq:wEq3}
  w|_{t=0} &=& w_0.
\end{eqnarray}
for all $\varphi \in C^1([0,T];H^1_{per}(\Omega)^d)$ with $\varphi|_{t=T}=0$.
\begin{lemma}\label{lem:LinEstimWeak} 
Assume that $u_h$ satisfies
  (\ref{eq:LInftyUnifEstim}) with $R\in (0,R_0]$ and some given $0<h\leq 1$, $n\in \N\cup \{\infty\}$ and   let $R_0\in (0,1]$ be so small that \eqref{eq:EstimD2W} and \eqref{eq:UniformCoercive'} hold.
  Let $w\in H^1(Q_T)^d$ be a solution of (\ref{eq:wEq1})-(\ref{eq:wEq3}) for some $f\in L^1(0,T;L^2(\Omega)^{d\times d})$, $w_0\in L^2(\Omega)^d$, and $w_1\in H^1_{per}(\Omega)^d$ and let $u(t)= \int_0^t w(\tau)\sd \tau$. Then there are some $C_0,C>0$  independent of $w$
  and $0<T<\infty$
  such that
  \begin{eqnarray}\nonumber
    \lefteqn{\left\|\left(w, \frac1h \eps_h (u)\right)\right\|_{L^\infty(0,T;L^2)}}\\\label{eq:WeakLinEstim}
    &\leq& C_0e^{CRT}\left(\|f\|_{L^1(0,T;L^2_h)} +
      \left\|w_0\right\|_{L^2} + \|w_1\|_{H^{-1}_h(\Omega)}\right).
  \end{eqnarray} 
\end{lemma}
\begin{proof}
  Let $0\leq T'\leq T$ and define $\tilde{u}_{T'}(t)= -\int_t^{T'}w(\tau) \sd \tau$.
  We choose $\varphi= \tilde{u}_{T'}\chi_{[0,T']}$ in (\ref{eq:wEq1}) (after a standard approximation). Then
  \begin{eqnarray*}
    \lefteqn{\frac12\|w(T')\|_{L^2(\Omega)}^2  +\frac1{2h^2} (A_n(\nabla_h
  u_h)\nabla_h \tilde{u}_{T'}(0),\nabla_h \tilde{u}_{T'}(0))_{\Omega}}\\
&=& -\frac1{2h^2} ((\partial_t A_n(\nabla_h
  u_h))\nabla_h \tilde{u}_{T'},\nabla_h \tilde{u}_{T'})_{Q_{T'}} - (f,\nabla_h \tilde{u}_{T'})_{Q_{T'}}\\
&& - \weight{w_1, \tilde{u}_{T'}(0)}_{H^{-1}_h,H^1_h}  
  + \frac12\|w_0\|_{L^2(\Omega)}^2
  \end{eqnarray*}
Hence (\ref{eq:UniformCoercive'}), (\ref{eq:EstimD2W}), and $\tilde{u}_{T'}(0)= -u(T')$ imply
\begin{eqnarray*}
  \lefteqn{\|w(T')\|_{L^2(\Omega)}^2 + \left\|\frac1h\eps_h(u(T'))\right\|_{L^2}^2 \leq CR \left\|\frac1h\eps_h(\tilde{u}_{T'})\right\|_{L^2(Q_{T'})}^2}\\
    && + C \|w_0\|_{L^2(\Omega)}^2
+ C\left(\|f\|_{L^1(0,T;L^2_h)}+\|w_1\|_{H^{-1}_h}\right)\left\|\frac1h \eps_h (u) \right\|_{L^\infty(0,T';L^2)}
\end{eqnarray*}
for all $0\leq T'\leq T$. Since $\tilde{u}_{T'}(t)= -u(T')+u(t)$, we obtain
\begin{equation*}
  \left\|\frac1h\eps_h(\tilde{u}_{T'})\right\|_{L^2(Q_{T'})}^2 \leq
\left\|\frac1h\eps_h(u)\right\|_{L^2(Q_{T'})}^2 + T'\left\| \frac1h\eps_h(u(T'))\right\|_{L^2(\Omega)}^2.
\end{equation*}
Hence there is some $\kappa>0$ independent of $R\in (0,R_0]$, $h\in (0,1]$, such that
\begin{eqnarray*}
  \lefteqn{\|w\|_{L^\infty(0,T';L^2)}^2 + \left\|\frac1h\eps_h(u)\right\|_{L^\infty(0,T';L^2)}^2}\\
    &\leq& CR \left\|\frac1h\eps_h(u)\right\|_{L^2(Q_{T'})}^2 + C_0\left( \|w_0\|_{L^2(\Omega)}^2+\|w_1\|_{H^{-1}_h(\Omega)}^2 + \|f\|_{L^1(0,T;L^2_h)}^2\right)
\end{eqnarray*}
provided that $RT'\leq \kappa$. By the lemma of Gronwall we obtain (\ref{eq:WeakLinEstim}) for all $0<T<\infty$ such that $RT\leq \kappa$. Now, if $0<T<\infty$ with $RT> \kappa$, we apply the latter estimate successively for some $0=T_0<T_1<\ldots< T_N=T$ such that $R (T_{j+1}-T_j) \leq \kappa$, $j=0,\ldots,N-1$, and $N \leq 2 R\kappa^{-1}T$.  Hence we obtain
  \begin{eqnarray*}
    \lefteqn{\left\|\left(w, \frac1h \eps_h (u)\right)\right\|_{L^\infty(0,T;L^2)}}\\
    &\leq& (C_0)^Ne^{CRT}\left(\|f\|_{L^1(0,T;L^2_h)} +
      \left\|w_0\right\|_{L^2} + \|w_1\|_{H^{-1}_h(\Omega)}\right),
  \end{eqnarray*}
  where 
  \begin{equation*}
    (C_0)^N\leq \exp \left(2\kappa^{-1}\ln C_0 RT \right)\leq \exp(C'_0RT)
  \end{equation*}
  since $N \leq 2 R\kappa^{-1}T$. This implies (\ref{eq:WeakLinEstim}) for some modified $C_0,C$ independent of $R\in (0,R_0]$, $h\in (0,1]$, $0<T<\infty$.
\end{proof}
\subsection{Local in Time Existence }\label{sec:LocalInTime}


For the following we assume that $\theta\geq 0$, $0<T\leq 1$, and  $(u_{2,h},u_{3,h}), f_h$ are as in Theorem~\ref{thm:ApproxMain1} and set $w_{1}=u_{2,h}, w_2= u_{3,h}$. Moreover, we assume that $R_0\in (0,1]$ is so small that all the statements in Section~\ref{sec:Linear} are applicable. -- Note that $T\leq 1$ is not a restriction for the proof of Theorem~\ref{thm:ApproxMain1} and Theorem~\ref{thm:main1}. By a simple scaling with $T^{-1}$ in time $t$ and $h$ we can always reduce to this case changing $M>0$ by a  certain factor depending on $T$ if necessary. (Of course this finally influence the smallness assumption of $h_0>0$ in the case $\theta>0$ and the starting smallness assumption on $M$ if $\theta=0$.) %%% Nicht sehr schoen!!!

Moreover,
let $C_L\geq 1$ be the constant in Lemma~\ref{lem:LinEstim3} and let $C_0\geq 1$ be as in (\ref{eq:ApproxIVEstim}). Then (\ref{eq:AssumInitialData1})-(\ref{eq:AssumFData}), (\ref{eq:ApproxIVEstim}) imply
\begin{eqnarray}\nonumber
 \lefteqn{ \left\|h^{1+\theta}\partial_t f_h\right\|_{W^1_1([0,T];H^{1,0})} + \max_{k=0,1}\left\|\left(\frac1h \eps_h (u_{k,h}^n),\nabla\frac1h \eps_h (u_{k,h}^n), \nabla_h^2u_{k,h}^n\right)\right\|_{H^{2-k,0}(\Omega)}}\\\label{eq:DataEstim}
 &&  + \max_{|\gamma|\leq 1}\left\|\left(\frac1h \eps_h(w_1),w_2, h^{1+\theta}\partial_z^\gamma f_h|_{t=0}\right)\right\|_{H^{1,0}(\Omega)}\leq \tilde{M} h^{1+\theta}\qquad \qquad\quad
\end{eqnarray}
where $\tilde{M}= (2+C_0)M$.
If $\theta>0$, we can find some $h_0\in (0,1]$ (depending on $M$) such that $R:=6C_L \tilde{M} h_0^{\theta}\leq R_0$. 
%  If $\theta>0$, we can find some $0<h_0\leq 1$ such that (\ref{eq:AssumInitialData1Approx}) with $M$ replaced by $3C_LM$ 
%   implies
%   \begin{equation}\label{eq:LInftyAssump}
%     \left\|\frac1h\eps_h(u_{0,n})\right\|_{V(\Omega)}< \delta_0h 
%                                 
%   \end{equation}
% for all $0<h\leq h_0$, where $\delta_0$ is as in Section~\ref{sec:Linear} and
% $V(\Omega)= H^{1,1}(\Omega)\cap H^{2,0}(\Omega)$. 
If
$\theta=0$, we assume that $M>0$ is so small that $R:=6C_L \tilde{M}\leq R_0$. 
%Then (\ref{eq:LInftyAssump}) follows 
% from (\ref{eq:AssumInitialData1Approx}) with $M$ replaced by $3C_LM$   for all $0<h\leq 1$. 
In this case we set $h_0=1$.

Under the latter assumptions we will prove:
\begin{theorem}\label{thm:LocalInTime} 
%   Let $\theta\geq 0$, $0<T<\infty$,
%   let $f_h\in H^4(Q_T)$, $0<h\leq 1$, be such that
%   \begin{equation*}
%     \|f_h\|_{H^4(Q_T)} \leq M_f\qquad \text{for all}\ h\in (0,1],
%   \end{equation*}
%   and let $u_{0,n},u_{1,n}$ and $h_0,M$ be as above.
For every $n\in\N$ and $0<h\leq h_0$ there is some $T_0>0$ and a unique
  $u^n_h\in C^3([0,T'];H^{1,0})\cap C([0,T];H^{2,2}\cap H^{4,0})$ solving
  (\ref{eq:Approx1})-(\ref{eq:Approx4}) on  $(0,T')$ with $T'=\min (T,T_0)$, satisfying 
 \begin{equation} \label{eq:SoluEstimLocal}
   \max_{|\gamma|\leq 1}\left\|\left(\partial_t^2\partial_z^\gamma u_h^n,\tfrac1h \eps_h(\partial_z^\gamma u_h^n),
       \nabla_{x,t}\partial_{z}^\gamma\tfrac1h\eps_h(u_h^n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 3C_L\tilde{M}h^{1+\theta}%\\\label{eq:UnifBddApprox2} 
%   \max_{|\gamma|= 2}\left\|\left(\partial_t^2\partial_z^\gamma u_h^n,
%       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_h^n)\right)\right\|_{C([0,T];L^2)}&\leq& Ch^{\frac{d-2}2}
  \end{equation}
uniformly in $0<h\leq h_0$. 
Here  $T_0>0$ depends only on $M,n$.
\end{theorem}
We solve (\ref{eq:Approx1})-(\ref{eq:Approx4}) by solving
(\ref{eq:Approx1'})-(\ref{eq:Approx4'}). To this end, let $w_n^0= 0$ and let $w_n^{k+1}$,
$k\in\N_0$, be defined recursively by 
\begin{alignat}{2}\label{eq:Iteration1}
  \partial_t^2 w_n^{k+1} - \frac1{h^2} \Div_h \left(A_n(\nabla_h u_n^k)\nabla_h w_n^{k+1}\right)
  &= \partial_t f_h h^{1+\theta} &\ & \text{in}\ \Omega\times (0,T'),\\\label{eq:Iteration2}
  \left. A_n(\nabla_h u_n^k)\nabla_h w_n^{k+1} e_d\right|_{x_d=\pm \frac12} &=0, &&
   \\\label{eq:Iteration3}
  \left. w_n^{k+1}\right|_{x_j=-L} &= \left. w_n^{k+1}\right|_{x_j=L},&&
  j=1,\ldots,d-1,\\\label{eq:Iteration4} 
  \left. (w_n^{k+1},\partial_t w_n^{k+1})\right|_{t=0} &= (w_{0,n}, w_{1}),
\end{alignat}
where $w_{0,n}=u_{1,h}^n$, 
\begin{equation}\label{eq:DefnUnk}
  u_n^{k}(x,t)= u_{0,h}^n(x)+ \int_0^t
w_n^{k}(x,\tau)\sd \tau\qquad \text{for all}\ k\in\N_0,
 \end{equation}
and $u_{0,h}^n,u_{1,h}^n$ solve (\ref{eq:ApproxCompCond1}),(\ref{eq:ApproxCompCond2}), respectively, cf. Remark~\ref{rem:SolvIV}. The existence of a unique solution $w_h^{k+1}$ follows from Lemma~\ref{lem:LinEstim3}. 
% and $u_{0,n}^{k+1}, w_{0,n}^{k+1}$ are determined as solutions of
% \begin{eqnarray}\label{eq:ApproxCompCond1'}
%   u_{2,h} &=&  h^{1+\theta}f_h|_{t=0} + \frac1{h^2} \Div_h D\widetilde{W}(\nabla_h
%   u_{0,n}^{k+1})+ \frac1{h^2} \Div_h F_n'(\nabla_h
%   u_{0,h}^k)\\\label{eq:ApproxCompCond2'}
%     u_{3,h} &=&  h^{1+\theta}\partial_tf_h|_{t=0} + \frac1{h^2} \Div_h \left(D F_n(\nabla_h
%   u_{0,n}^{k})\nabla_h w_{1,n}^{k+1}\right)
% \end{eqnarray} 

As usual for short time existence of hyperbolic equations, we first show
boundedness of $(u_n^k)_{k\in\N}$ in some suitable ``high norms'' and then
convergence of $(u_n^k)_{k\in\N}$ in some ``low norms'' provided that $T_0=T_0(n)>0$
is sufficiently small and $n\in\N$ is fixed.

\noindent
In order to get ``Boundedness in High Norms'',
we show that $(w_n^k)_{k\in\N}$ satisfies for sufficiently small $T'=T'(n)\in (0,T]$
 \begin{equation}\label{eq:BddUk1b} 
 \left\|\left(\partial_t^2 w_n^k, \frac1h\eps_h(w_n^k),
       \nabla_{x,t}\frac1h\eps_h(w_n^k)
     \right)\right\|_{C([0,T'];H^{1,0})}\leq 2C_L\tilde{M}h^{1+\theta}
%\\\label{eq:BddUk2} 
% \max_{j=0,1}  \left\|\left(\partial_t^{2+j}\nabla_z u_n^k,
%       \partial_t^{j}\nabla_{x,t}\nabla_z\frac1h \eps_h(u_n^k)\right)\right\|_{C([0,T'];L^2)}&\leq& 5C_LMh^{\frac{d-2}2}
  \end{equation}
uniformly in $0<h\leq h_0$, where $C_L$ is the
constant in Lemma~\ref{lem:LinEstim3}. To this end we use:
\begin{lemma}\label{lem:CnBdds}
  There is some $0<T'(n)\leq \min(1,T)$ depending only on $M,n$ and $\min(1,T)$ such that,
  if $u_n^k,w_n^k$ satisfy (\ref{eq:DefnUnk}) and
 \begin{equation}\label{eq:BddUk1} 
 \left\|\left(\partial_t^2 w_n^k, \frac1h\eps_h(w_n^k),
       \nabla_{x,t}\frac1h\eps_h(w_n^k)
     \right)\right\|_{C([0,T'];H^{1,0})}\leq 4C_L\tilde{M}h^{1+\theta} 
 \end{equation}
then the solution $w_n^{k+1}$ of (\ref{eq:Iteration1})-(\ref{eq:Iteration4}) satisfies
 \begin{equation*}
 \left\|\left(\partial_t^2 w_n^{k+1}, \frac1h\eps_h(w_n^{k+1}),
       \nabla_{x,t}\frac1h\eps_h(w_n^{k+1})
     \right)\right\|_{C([0,T'];H^{1,0})}\leq 2C_L\tilde{M}h^{1+\theta}
  \end{equation*}
  and $u_n^k$ satisfies
 \begin{eqnarray}\label{eq:BddUk1''} 
  \lefteqn{\max_{|\gamma|\leq 1}\left\|\left(\partial_z^\gamma P_n \nabla_h u_n^k, \partial_z^\gamma P_n\frac1h \eps_h (u_n^k) \right)\right\|_{L^\infty(0,T';V)}}\\\nonumber
&& + \left\|\left(\partial_t^2 P_n \nabla_h u_n^k, \partial_t^2 P_n\frac1h \eps_h (u_n^k) \right)\right\|_{L^\infty(0,T';H^{1,0})} \leq 6C_1C_L\tilde{M} h^{1+\theta}\leq C_1R_0 h 
  \end{eqnarray}
uniformly in $0<h\leq h_0$,
\end{lemma}
%  -- Note that, according to the
% assumptions above, (\ref{eq:BddUk1}) implies
%   \begin{equation*}
%     \left\|\frac1h\eps_h(u_n^k)\right\|_{C([0,T'];V(\Omega))}< \delta_0h.
%   \end{equation*}
%  We prove these bounds by mathematical induction. Obviously (\ref{eq:BddUk1})-(\ref{eq:BddUk2}) holds true for $k=0$
% since $u_n^0=u_{0,h}$.
\begin{proof}
 First of all, if (\ref{eq:BddUk1}) holds for some $k\in\N$ and some $T'>0$,
 then 
 $ P_n \nabla_h u_n^k$ satisfies
  \begin{eqnarray}\nonumber
 \lefteqn{  \left\|\left(\partial_tP_n\nabla_hu_n^k,
        \partial_tP_n\frac1h\eps_h(u_n^k)\right)\right\|_{C([0,T'];V)}}\\\nonumber%\label{eq:BddUk1} 
 &+&\left\|\left(\partial_t^2P_n\nabla_hu_n^k,
        \partial_t^2P_n\frac1h\eps_h(u_n^k)\right)\right\|_{C([0,T'];H^{1,0})} \leq 4C_1C_L\tilde{M}h^{1+\theta}%\\\label{eq:BddUk2'} 
 %    \max_{|\gamma|= 2}\left\|\left(\partial_t^2\partial_z^\gamma \tilde{u}_n^k,
 %        \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(\tilde{u}_n^k)\right)\right\|_{C([0,T'];L^2)}&\leq&
 %    C_n Mh^{\frac{d-2}2}
   \end{eqnarray}
 uniformly in $0< h\leq h_0$ where $C_1\geq 1$ can be chosen as the same constant as in (\ref{eq:LInftyUnifEstim}). 
Moreover, there is some $C_n>0$ depending only on $n\in\N$
 such that
$\|P_n^hf\|_{H^2(\Omega)}\leq C_n\|f\|_{L^2(\Omega)}$ uniformly in
$0<h\leq 1$, cf. (\ref{eq:nEstim}).
%  -- Note that  
% \begin{eqnarray*}
% \|(-\Delta)^{\frac12}u\|_{L^2}&=& \|\nabla u\|_{L^2}\leq \|\nabla_h u\|_{L^2}=\|(-\Delta_h)^{\frac12}u\|_{L^2}\\
%  \Rightarrow\|P_n^h u\|_{H^2(\Omega)}&\leq& C\left(\|u\|_{L^2}+ \|(-\Delta)P_n^hu\|_{L^2}\right)\\
% & \leq& C\left(\|u\|_{L^2}+ \|(-\Delta_h)P_n^hu\|_{L^2}\right)\leq C_n \|u\|_{L^2}. 
%\end{eqnarray*}
Now, using
\begin{equation}\label{eq:Simple1}
  \nabla_hu_n^k =  \nabla_h u_{0,h}^n + \int_0^t \nabla_h w_n^k(\tau) \sd \tau,  
\end{equation}
(\ref{eq:ApproxIVEstim}), (\ref{eq:BddUk1}), and the previous estimate, we conclude 
 \begin{eqnarray}\label{eq:BddUk1'} 
  \lefteqn{\max_{|\gamma|\leq 1}\left\|\left(\partial_z^\gamma P_n \nabla_h u_n^k, \partial_z^\gamma P_n\frac1h \eps_h (u_n^k) \right)\right\|_{C([0,T'];V)}}\\\nonumber
&& + \left\|\left(\partial_t^2 P_n \nabla_h u_n^k, \partial_t^2 P_n\frac1h \eps_h (u_n^k) \right)\right\|_{C([0,T'];H^{1,0})} \leq (4C_1C_L+ 1+ C_n' T')\tilde{M}h^{1+\theta} 
  \end{eqnarray}
uniformly in $0<h\leq h_0$, $0<T'\leq T$, $j=1,\ldots,d-1$ for some $C_n'$ depending on $n\in\N$.
Hence, if $T'=T'(n)>0$ is so small that $1+C_n'T'\leq 2C_1C_L$, we obtain
(\ref{eq:BddUk1''})
%  \begin{eqnarray}\label{eq:BddUk1''} 
%   \lefteqn{\max_{|\gamma|\leq 1}\left\|\left(\partial_z^\gamma P_n \nabla_h u_n^k, \partial_z^\gamma P_n\frac1h \eps_h (u_n^k) \right)\right\|_{L^\infty(0,T';V)}}\\\nonumber
% && + \left\|\left(\partial_t^2 P_n \nabla_h u_n^k, \partial_t^2 P_n\frac1h \eps_h (u_n^k) \right)\right\|_{L^\infty(0,T';H^{1,0})} \leq 6C_1C_L\tilde{M} h^{1+\theta}\leq C_1R_0 h 
%   \end{eqnarray}
uniformly in $0<h\leq h_0$, where $6C_L\tilde{M}h_0^\theta \leq R_0$ by the choice of $h_0, M$.

% Since $\|\nabla_h P_n^hf\|_{H^{j,0}(\Omega)}\leq \|\nabla_h
% f\|_{H^{j,0}(\Omega)}$ for all $j\in \N_0$, $\tilde{u_n}^k|_{t=0}=P_n
% u_{0,h}$, and is bounded in $\partial_t \nabla_h \tilde{u}_n^k \in
% C([0,T'];V(\Omega))$,  
% we have
% \begin{equation*}
%   \left\|\left(\nabla_h \tilde{u}_n^k, \frac1h
%       \eps_h(\tilde{u}_n^k)\right)\right\|_{C([0,T'], V(\Omega))}< \delta_0 h
% \end{equation*} 
% for all $0<h\leq h_0$ if $T'(n)>0$ is sufficiently small. 

Because of (\ref{eq:BddUk1''}), $P_n \nabla_h u_n^k$ satisfies (\ref{eq:LInftyUnifEstim}) for $T$ replaced by $T'=T'(n)$ and given $n\in \N$. Hence we can apply
Lemma~\ref{lem:LinEstim3} to conclude 
  \begin{eqnarray}\nonumber
    \lefteqn{\left\|\left(\partial_t^2 
          w_n^{k+1},\frac1h\eps_h(w_n^{k+1}), \nabla_{x,t} \frac1h\eps_h(w_n^{k+1})\right)\right\|_{C([0,T'],H^{1,0})}}\\\nonumber
&\leq& C_Le^{6C'M h_0^\theta T'}\left( 
   h^{1+\theta}\|
      f_h\|_{W^2_1(0,T';H^{1,0})}\right.\\\nonumber
&& \left. + \left\|\left(\frac1h \eps_h(w_{1}),
      w_{2}, h^{1+\theta}\partial_t f_h|_{t=0}\right)\right\|_{H^{1,0}}\right)\\
&\leq& C_L e^{C'R_0 T'}\tilde{M}h^{1+\theta}
  \end{eqnarray}
for all $T'>0$ such that $1+C_n' T' \leq 2C_1C_L$.
  Hence, if $T'(n)>0$ is chosen sufficiently small, we obtain $C_L e^{C'R_0T'}\leq
  2C_L$.
This shows the estimate for $w_n^{k+1}$.
\end{proof}

\noindent
\begin{proof*}{of Theorem~\ref{thm:LocalInTime}}
Let $w_k^n,u_k^n$ be defined as above.
Because of the latter lemma (\ref{eq:BddUk1b}) holds for all $k\in\N$ provided that
$T'=T'(n)>0$ is chosen as in the lemma.  

In order to show
``contraction in low norms'',
let $z_n^{k+1}:= w_n^{k+1}-w_n^k$ and $Z_n^{k+1}:= u_n^{k+1}-u_n^k$. Because of (\ref{eq:Iteration1})-(\ref{eq:Iteration2}), 
 $z_n^{k+1}$ solves
\begin{eqnarray*}%\label{eq:Diff1}
  \lefteqn{-(\partial_t z_n^{k+1},\partial_t \varphi)_{Q_{T'}}+
  \frac1{h^2}(A_n(\nabla_h u_n^k)\nabla_h z_n^{k+1},\nabla_h
  \varphi)_{Q_{T'}}}\\
 &=& -\frac1{h^2}(A_n(\nabla_h u_n^k)-A_n(\nabla_h u_n^{k-1}))\nabla_h w_n^k,\nabla_h
  \varphi)_{Q_{T'}} 
\end{eqnarray*}
for all $\varphi \in C^1([0,T'];L^2(\Omega))^d\cap C^0([0,T'];H^1(\Omega))^d$ with $\varphi|_{t=T}=0$ and $
\left.(z_h^{k+1}, \partial_t z_h^{k+1})\right|_{t=0} = (0, 0)$. Here
\begin{eqnarray*}
  \lefteqn{\left|\frac1{h^2}((A_n(\nabla_h u_n^k)-A_n(\nabla_h u_n^{k-1}))\nabla_h w_n^k,\nabla_h
  \varphi)_{Q_{T'}}\right|}\\
&\leq& \frac{CT'}h \left\|\left(P_n\nabla_h w_n^k,P_n\frac1h
    \eps_h(w_n^k)\right)\right\|_{L^\infty(0,T';V)}\left\|\frac1h\eps_h(Z_n^k)\right\|_{L^\infty(0,T';L^2)}
\left\|\frac1h\eps_h(\varphi)\right\|_{L^\infty(0,T';L^2)}\\
&\leq&  C'T'\tilde{M}h_0^\theta \left\|\frac1h\eps_h(Z_n^k)\right\|_{L^\infty(0,T';L^2)}
\left\|\frac1h\eps_h(\varphi)\right\|_{L^\infty(0,T';L^2)}
\end{eqnarray*}
because of Corollary~\ref{cor:D3W}, (\ref{eq:BddUk1}),
and 
\begin{eqnarray*}
  \lefteqn{\frac1{h^2}((A_n(\nabla_h u_n^k)-A_n(\nabla_h u_n^{k-1}))\nabla_h w_n^k,\nabla_h
  \varphi)_{\Omega}}\\
 &=& \int_0^1 \left(D^3\widetilde{W}((1-\tau)\nabla_h u_n^{k-1}+\tau\nabla_h u_n^{k})[P_n \nabla_hZ_n^k,P_n\nabla_h w_n^k],P_n \nabla_h
  \varphi\right)_{\Omega} \sd \tau.
\end{eqnarray*}
Hence 
\begin{equation*}
  \left\|\left(\partial_tZ_n^{k+1},
      \frac1h\eps_h (Z_n^{k+1})\right)\right\|_{C([0,T'];L^2)} \leq 
  Ce^{C'\tilde{M}h_0^\theta T'}T' \left\|
      \frac1h\eps_h(Z_n^k)\right\|_{C([0,T'];L^2)}
\end{equation*}
due to Lemma~\ref{lem:LinEstimWeak}.
 Therefore there is some $T'_0>0$ depending only on $\tilde{M}$ such that
\begin{equation*}
  \left\|\left(\partial_tZ_n^{k+1},
      \frac1h\eps_h (Z_n^{k+1})\right)\right\|_{L^\infty(0,T';L^2)} \leq 
  \frac12 \left\|\left(\partial_tZ_n^k,
      \frac1h\eps_h(Z_n^k)\right)\right\|_{L^\infty(0,T';L^2)}
\end{equation*}
provided that $0<T'(n) \leq T'_0$.
This shows that $(u_n^k)_{k\in\N}$ converges to some $u_n\in
C([0,T'];H^1(\Omega))$ as $k\to\infty$ and $\partial_t
u_n^k\to \partial_t u_n$ in $C([0,T'];L^2(\Omega))$ for some sufficiently small $T'(n)$. Because of
(\ref{eq:BddUk1b}) and (\ref{eq:BddUk1''}), $u_n$ and $w_n:=\partial_t u_n$  satisfy the same estimates as $u_k^n$, $w_k^n$, respectively. By interpolation $\nabla_h
u_n^k\to_{k\to\infty} \nabla_h u_n$ in $L^\infty(Q_{T'})$ strongly. Therefore
$u_n$ and $w_n$ solve
(\ref{eq:Approx1'})-(\ref{eq:Approx4'}).
Consequently $u_n$ solves (\ref{eq:Approx1})-(\ref{eq:Approx4}).

Finally, we have to prove (\ref{eq:SoluEstimLocal}).
We know that $u_n$ constructed above on $(0,T'(n))$  satisfies
 \begin{equation*}
  \left\|\left(\partial_t^{3} u_n,\partial_t \frac1h\eps_h(u_n),
       \partial_t \nabla_{x,t}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 2C_L\tilde{M}h^{1+\theta}
%\\\label{eq:BddUn2} 
%   \left\|\left(\partial_t^{2+j}\nabla_z u_n,
%       \partial_t^{1+j}\nabla_{x,t}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];L^2)}&\leq& M'h^{\frac{d-2}2}
  \end{equation*}
  due to (\ref{eq:BddUk1b}). Moreover, using (\ref{eq:Simple1}) and (\ref{eq:DataEstim}), we conclude 
 \begin{equation}\label{eq:BddUnb} 
  \max_{j=0,1}\left\|\left(\partial_t^{2+j} u_n,\partial_t^j \frac1h\eps_h(u_n),
       \partial_t^j \nabla_{x,t}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 3C_L\tilde{M}h^{1+\theta}.
  \end{equation}
Hence it remains to show that we can replace one time derivative  in (\ref{eq:BddUnb}) by $\partial_{x_j}$, $j=1,\ldots, d-1$.
To this end let $w_n^j= \partial_{x_j}u_n$, $j=1,\ldots, d-1$. Then $w_n^j$ solves
\begin{alignat}{2}\label{eq:Approx1j}
  \partial_t^2 w_n^j - \frac1{h^2} \Div_h A_n(\nabla_h u_n)\nabla_h w_n^j
  &= \partial_{x_j} f_h h^{1+\theta} &\quad& \text{in}\ \Omega\times (0,T')\\\label{eq:Approx2j}
  \left. A_n(\nabla_h u_n)\nabla_h w_n^j e_d\right|_{x_d=\pm \frac12} &=0, &&
   \\\label{eq:Approx3j}
  \left. w_n^j\right|_{x_j=L} &= \left. w_n^j\right|_{x_j=-L},&& j=1,\ldots,d-1,\\\label{eq:Approx4j}
  \left. (w_n^j,\partial_t w_n^j)\right|_{t=0} &= (w_{0,h}^j, w_{1,h}^j)
\end{alignat}
with $w_{k,h}^j= \partial_{x_j} u^n_{k,h}$, $k=0,1$. Hence Lemma~\ref{lem:CnBdds} with $w_n^k=w_n, u_n^k=u_n$, $w_n^{k+1}= w_n^j$, and $\partial_t f$ replaced by $\partial_{x_j} f$ implies the estimates of the remaining terms in (\ref{eq:SoluEstimLocal}).
\end{proof*}

\subsection{Uniform bounds and Proof of Theorem~\ref{thm:main1}}\label{sec:uniform}

%If $\theta>0$, we choose $R'=2CM(1+T^\frac12)$ and $\delta_0\in (0,1]$ as in Section~\ref{sec:Linear}, Lemma~\ref{lem:LinEstim2}, resp., and set $M':= 2C(M+MT^\frac12)$. If $\theta=0$, we choose $M$ and $MT^\frac12$ so small that $2C(M+MT^\frac12)\leq \delta_0$. 

We know that $u_n$ constructed above on $(0,T'(n))$ satisfies
 \begin{equation}\label{eq:BddUn} 
 \max_{|\gamma|\leq 1} \left\|\left(\partial_t^2\partial_z^\gamma u_n,\partial_z^\gamma \frac1h\eps_h(u_n),
       \partial_z^\gamma \nabla_{x,t}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 3C_L\tilde{M}h^{1+\theta}
%\\\label{eq:BddUn2} 
%   \left\|\left(\partial_t^{2+j}\nabla_z u_n,
%       \partial_t^{1+j}\nabla_{x,t}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];L^2)}&\leq& M'h^{\frac{d-2}2}
  \end{equation}
where $\tilde{M}=C_0M+2M$ due to Theorem~\ref{thm:LocalInTime}.
In the case $\theta>0$, it remains to show that we can
replace $T'(n)$ by an arbitrary $0<T\leq 1$ if $0<h\leq h_0$ for some sufficiently small $0<h_0\leq 1$ independent of
$n\in\N$. If $\theta=0$, we will show that $T'(n)$ can be replaced by $0<T\leq 1$ if
$M$ is sufficiently small
(independent of $n\in\N_0$).

To this end we apply Lemma~\ref{lem:LinEstim3}
for $w^j_n=\partial_{x_j}u_n$ and $w_n= \partial_t u_n$ using the equations for $w^j_n,w_n$ and that (\ref{eq:UniformEstim}) is valid for $u=u_n$ and $R=6C_L\widetilde{M}h^\theta\leq R_0$, which implies (\ref{eq:LInftyUnifEstim}) uniformly in $n\in \N_0\cup \{\infty\}$. Thus we obtain
 \begin{equation*}
  \max_{|\gamma|= 1} \left\|\left(\partial_t^2\partial_{z}^\gamma u_n,\partial_{z}^\gamma\frac1h\eps_h(u_n),
       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq C_Le^{C'\tilde{M}h^\theta}\tilde{M}h^{1+\theta}%\\
%   \max_{|\gamma|= 1}\left\|\left(\partial_t^2\partial_{z}^\gamma\partial_{x_j} u_n,
%       \nabla_{x,t}\partial_{z}^\gamma\partial_{x_j}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];L^2)}&\leq&
%   4C_Le^{C'\tilde{R}(h) T}(M+MT^\frac12+ C'T^2h^\theta)h^{\frac{d-2}2}
  \end{equation*}
  uniformly in $n\in\N$ and $0<h\leq h_0$.  Due to (\ref{eq:ApproxIVEstim}) and (\ref{eq:Simple1}), we conclude
 \begin{equation*}
  \max_{|\gamma|\leq 1} \left\|\left(\partial_t^2\partial_{z}^\gamma u_n,\partial_{z}^\gamma\frac1h\eps_h(u_n),
       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 2C_Le^{C'\tilde{M}h^\theta}\tilde{M}h^{1+\theta}%\\
%   \max_{|\gamma|= 1}\left\|\left(\partial_t^2\partial_{z}^\gamma\partial_{x_j} u_n,
%       \nabla_{x,t}\partial_{z}^\gamma\partial_{x_j}\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];L^2)}&\leq&
%   4C_Le^{C'\tilde{R}(h) T}(M+MT^\frac12+ C'T^2h^\theta)h^{\frac{d-2}2}
  \end{equation*}
If $\theta>0$, we can now choose $0<h_0\leq 1$ so small that
 \begin{equation*}
  \max_{|\gamma|\leq 1} \left\|\left(\partial_t^2\partial_{z}^\gamma u_n,\partial_{z}^\gamma\frac1h\eps_h(u_n),
       \nabla_{x,t}\partial_z^\gamma\frac1h\eps_h(u_n)\right)\right\|_{C([0,T'];H^{1,0})}\leq 3C_L\tilde{M}h^{1+\theta}
  \end{equation*}
  uniformly in $n\in\N$ and $0<h\leq h_0$. If $\theta=0$, then we choose $\tilde{M}=(2+C_0)M$
  sufficiently small to obtain the same estimates.
  Hence we get a uniform bound as long as the solutions exists. 
Applying the results of Section~\ref{sec:LocalInTime} with the only difference that the bound $M$ of the norm of the initial values is replaced by
$3C_L\tilde{M}$ and a shift in time by $T'(n)$, we can continue the solution $u_h$ on $[0,\widetilde{T}]$, $\widetilde{T}=\min(T'(n)+T'',T)$, where $T''>0$ depends only on $n$ and $3C_L\tilde{M}$. This extended solution can be constructed such that (\ref{eq:BddUk1}) is valid. But the arguments of this section show that then even
 \begin{equation*}
  \max_{|\gamma|\leq 1} \left\|\left(\partial_t^2\partial_{z}^\gamma u_n,\partial_{z}^\gamma\frac1h\eps_h(u_n),
       \nabla_{x,t}\partial_{z}^\gamma\frac1h\eps_h(u_n)\right)\right\|_{C([0,\widetilde{T}];H^{1,0})}\leq 3C_L\tilde{M}h^{1+\theta}. 
  \end{equation*}
 Therefore we can repeat this argument finitely many times and can continue the solution for all $t\in [0,T]$.

Using the uniform bounds, it is easy to pass to the limit $n\to \infty$ for a suitable subsequence. 
Uniqueness of the solution can be shown by similar estimates as in
the proof of Theorem~\ref{thm:LocalInTime}. This finishes the proof of Theorem~\ref{thm:main1}.

\section{First Order Asymptotics}\label{sec:Asymptotics}

Throughout this section we assume that
$$
f_h(x,t) = 
\begin{pmatrix}
  0 \\ g(x',t)
\end{pmatrix},
$$
for some given  $g\in W^2_1(0,T;H^3_{per}((-L,L)^{d-1}))\cap W^1_1(0,T;H^5_{per}((-L,L)^{d-1}))$. As seen in the proof of Lemma~\ref{lem:LinEstim2}, 
we can assume without loss of generality that $\int_{(-L,L)^{d-1}} g(x') dx'=0$. Moreover, we assume that $0<\theta \leq 1$.

For simplicity let $W(F)= \dist(F,SO(d))^2$, which implies $D^2W(0)F= \sym F$. %%%Richtig? Oder 2\eps_h(u)?
Let $L_2v=\Div'\eps'(v)$ and let $v$ be the solution of the $(d-1)$-dimensional wave equation
\begin{alignat*}{2}
  \partial_t^2 v + \frac1{12}\Delta^2_{x'} v &= g &\qquad & \text{in}\  (-L,L)^{d-1}\times (0,T),\\
v|_{x_j=-L} &=v|_{x_j=L} && \text{for}\ j=1,\ldots,d-1,\\
(v, \partial_t v)|_{t=0}&= (v_0, v_1) &&\text{in}\  (-L,L)^{d-1},
\end{alignat*}
where $v_0 \in H^7_{per}((-L,L)^{d-1}), v_1 \in H^5_{per}((-L,L)^{d-1})$.
By standard methods the latter system possesses a unique solution 
\begin{alignat*}{1}
v \in &\, C^2([0,T];H^5_{per}((-L,L)^{d-1}))\cap C^1([0,T];H^7_{per}((-L,L)^{d-1}))\\
&\cap C^0([0,T];H^9_{per}((-L,L)^{d-1})).
\end{alignat*}
Using $v$, we define an approximate solution
$\tilde{u}^h$ of (\ref{eq:Evol1})-(\ref{eq:BC3}) by
\begin{eqnarray*}
  \tilde{u}^h(x,t) &=&
  h^{\theta}\begin{pmatrix}
    0\\ hv
  \end{pmatrix}
  +
 h^{2+\theta}\begin{pmatrix}
    -x_d \nabla_{x'} v\\ 0
  \end{pmatrix}
  +
 h^{4+\theta}\begin{pmatrix}
   (\frac13x_d^3-\frac14x_d) \nabla_{x'} \Delta_{x'}v\\ 0
  \end{pmatrix}\\
&&  +
 h^{5+\theta}\begin{pmatrix}
   0\\(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16}) \Delta_{x'}^2v
  \end{pmatrix}.
\end{eqnarray*}
Then
\begin{eqnarray*}
  \nabla_h  \tilde{u}^h &=& h^{1+\theta}
  \begin{pmatrix}
    0 &-\nabla_{x'} v\\
    \nabla_{x'}v^T & 0
  \end{pmatrix}
  +
h^{2+\theta}
  \begin{pmatrix}
    -x_d\nabla_{x'}^2 v &h(x_d^2-\frac14)\nabla_{x'} \Delta_{x'}v\\
    0 & 0
  \end{pmatrix}\\
&&+h^{4+\theta}
  \begin{pmatrix}
     (\frac13x_d^3-\frac14x_d) \nabla_{x'}^2 \Delta_{x'}v&0\\
    0 & (\frac1{24}x_d-\frac1{6}x_d^3) \Delta_{x'}^2v
  \end{pmatrix}\\
&&+h^{5+\theta}
  \begin{pmatrix}
     0 &0\\
    (\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16}) \nabla_{x'}^T\Delta_{x'}^2v &0 
  \end{pmatrix}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
 \lefteqn{\eps_h  (\tilde{u}^h) = 
h^{2+\theta}
  \begin{pmatrix}
    -x_d\nabla_{x'}^2 v &\frac{h}2(x_d^2-\frac14)\nabla_{x'} \Delta_{x'}v\\
    \frac{h}2(x_d^2-\frac14)\nabla_{x'}^T \Delta_{x'}v & h^2 (\frac1{24}x_d-\frac1{6}x_d^3) \Delta_{x'}^2v
  \end{pmatrix}}\\
&&+h^{4+\theta}
  \begin{pmatrix}
     (\frac13x_d^3-\frac14x_d) \nabla_{x'}^2 \Delta_{x'}v&\frac{h}2(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16}) \nabla_{x'}\Delta_{x'}^2v\\
    \frac{h}2(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16}) \nabla_{x'}^T\Delta_{x'}^2v & 0
  \end{pmatrix}
\end{eqnarray*}
and therefore
\begin{equation}\label{eq:BCApproxSol}
  \frac1h \eps_h(\tilde{u}^h)e_d|_{x_d=\pm\frac12} =0.
\end{equation}
Moreover,
\begin{eqnarray*}
  \frac1{h^2}\Div_h \eps_h  (\tilde{u}^h) &=& 
h^{\theta}
  \begin{pmatrix}
    -x_d\nabla_{x'}\Delta_{x'} v +x_d\nabla_{x'} \Delta_{x'}v\\
    \frac{h}2(x_d^2-\frac14)\Delta_{x'}^2v + h (\frac1{24}-\frac1{2}x_d^2) \Delta_{x'}^2v
  \end{pmatrix}\\
&&+h^{2+\theta}
  \begin{pmatrix}
     (\frac13x_d^3-\frac14x_d) \nabla_{x'} \Delta_{x'}^2v+\frac{1}2(\frac1{24}x_d-\frac1{6}x_d^3) \nabla_{x'}\Delta_{x'}^2v\\
    \frac{h}2(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16})
    \Delta_{x'}^3v 
  \end{pmatrix}
\\
&\equiv&
h^{1+\theta}
  \begin{pmatrix}
    0\\
    -\frac1{12}\Delta_{x'}^2v 
  \end{pmatrix}+r_h,
%&&+h^{2+\theta}
%  \begin{pmatrix}
%     (\frac13x_d^3-\frac14x_d) \nabla_{x'} \Delta_{x'}^2v+\frac{1}2(\frac1{24}x_d-\frac1{6}x_d^3) \nabla_{x'}\Delta_{x'}^2v\\
%    \frac{h}2(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16})
%    \Delta_{x'}^3v 
%  \end{pmatrix}.
\end{eqnarray*}
where 
\begin{equation}\label{eq:rhEstim}
  \|r_h\|_{C([0,T];L^2(\Omega))}\leq Ch^{2+\theta}.
\end{equation}
Thus $\tilde{u}_h$ is a solution of 
\begin{alignat}{2}\label{eq:ApproxEvol1}
  \partial_t^2 \tilde{u}_h - \frac1{h^2} \Div_h\left( D^2\widetilde{W}(0)\nabla_h \tilde{u}_h\right) &= f_h h^{1+\theta} -r_h &\qquad& \text{in}\ \Omega\times (0,T)\\\nonumber
  \left. \left(D^2\widetilde{W}(0)\nabla_h \tilde{u}_h\right) e_d\right|_{x_d=\pm \frac12} &= 0 \\\nonumber
  \left. \tilde{u}_h\right|_{x_j=L} &= \left. \tilde{u}_h\right|_{x_j=-L},&\qquad& j=1,\ldots,d-1,\\\nonumber
  \left. (\tilde{u}_h,\partial_t \tilde{u}_h)\right|_{t=0} &= (\tilde{u}_{0,h}, \tilde{u}_{1,h}),
\end{alignat}
where
\begin{eqnarray*}
  \tilde{u}_{j,h}(x) &=&
  h^{1+\theta}\begin{pmatrix}
    0\\ hv_j
  \end{pmatrix}
  +
 h^{2+\theta}\begin{pmatrix}
    -x_d \nabla_{x'} v_j\\ 0
  \end{pmatrix}
  +
 h^{4+\theta}\begin{pmatrix}
   (\frac13x_d^3-\frac14x_d) \nabla_{x'} \Delta_{x'}v_j\\ 0
  \end{pmatrix}\\
&&  +
 h^{5+\theta}\begin{pmatrix}
   0\\(\frac1{48}x_d^2-\frac1{24}x_d^4-\frac1{24\cdot 16}) \Delta_{x'}^2v_j
  \end{pmatrix}, \quad j=0,1.
\end{eqnarray*}

We will compare this approximate solution  with an exact solution of the $d$-dimensional system (\ref{eq:Evol1})-(\ref{eq:BC3}) for an appropriate choice of initial values. 
\begin{theorem}\label{thm:FirstOrderAsymp}
  Let $0<\theta \leq 1$, let $v_0,v_1,f$, $\tilde{u}_{0,h}, \tilde{u}_{1,h}$ and $\tilde{u}_h$ be defined as above. 
Then for some sufficiently small $h_0\in (0,1]$ there are initial values $(u_{0,h},u_{1,h})$ satisfying (\ref{eq:AssumInitialData1}) and such that
\begin{equation*}
  \max_{j=0,1}\left\|\frac1h \eps_h(u_{j,h}) -\frac1h \eps_h(\tilde{u}_{j,h})\right\|_{L^2(\Omega)} \leq C h^{1+2\theta}.
\end{equation*}
Moreover, if $u_h$ is the solution of (\ref{eq:Evol1})-(\ref{eq:BC3}) due to Theorem~\ref{thm:main1}, then
\begin{eqnarray*}
 \left\|\left(\partial_t (u_h - \tilde{u}_h), \frac1h \eps_h(u_h-\tilde{u}_h) \right)\right\|_{L^\infty(0,T;L^2)}\leq Ch^{1+2\theta}
\end{eqnarray*}
for all $0<h\leq h_0$ and some $C>0$ independent of $h$.
\end{theorem}
\begin{proof}
We construct the initial values 
$u_{0,h}, u_{1,h}$ as solution of 
\begin{eqnarray*}
  \frac1{h^2}(DW(\nabla_h u_{0,h}),\nabla_h \varphi)_\Omega &=&  (u_{2,h},\varphi)_\Omega - (h^{1+\theta}f|_{t=0},\varphi)_\Omega,\\
  \frac1{h^2}(D^2W(\nabla_h u_{0,h})\nabla_h u_{1,h},\nabla_h \varphi)_\Omega &=&  (u_{3,h},\varphi)_\Omega - (h^{1+\theta}\partial_t f|_{t=0},\varphi)_\Omega 
\end{eqnarray*}
for all $\varphi\in H^1_{per}(\Omega)^d$, where
\begin{equation*}
  u_{2+j,h} = h^{1+\theta}
  \begin{pmatrix}
    0 \\ -\frac{h}{12} \Delta_{x'}^2 \partial_t^jv|_{t=0}  
  \end{pmatrix}
  +h^{2+\theta}
  \begin{pmatrix}
    - \frac{x_d}{12} \nabla_{x'}\Delta_{x'}^2 \partial_t^jv|_{t=0}\\ 0  
  \end{pmatrix}, \quad j=0,1.
\end{equation*}
We note that $\int_\Omega u_{2+j,h} \sd x=0$ and
\begin{equation*}
  \frac1h \eps_h(u_{2+j,h}) = h^{1+\theta}
  \begin{pmatrix}
    -x_d\nabla_{x'}^2 \Delta_{x'}^2 v_j & 0 \\
    0 & 0
  \end{pmatrix}
\end{equation*}
by a similar calculation as above. In particular, this implies
\begin{equation*}
  \left\|\left(\frac1h \eps_h(u_{2+j,h}),\frac1h \eps_h(\partial_t^j f|_{t=0})\right)\right\|_{H^{2-j,0}}\leq Ch^{1+\theta},\qquad j=0,1,
\end{equation*}
where we note that $f$ is independent of $x_d$.
 Because of Proposition~\ref{prop:SolvIV}, the Lemma of Lax-Milgram, and Lemma~\ref{lem:LinEstim2}, $(u_{0,h},u_{1,h})$ exist for all $0<h\leq h_0$ if $h_0\in (0,1]$ is sufficiently small and satisfy 
 \begin{eqnarray*}
     \left\|\left(\frac1h \eps_h (u_{1,h}),\frac1h \eps_h (u_{0,h}), \nabla \frac1h \eps_h (u_{0,h}), \nabla_h^2 u_{0,h}\right)\right\|_{H^{2,0}(\Omega)} \leq Ch^{1+\theta}.
 \end{eqnarray*}
Altogether, $(u_{0,h},u_{1,h}, u_{2,h}, u_{3,h})$ satisfy
(\ref{eq:AssumInitialData1}) and (\ref{eq:u2h})-(\ref{eq:u3h}).
Moreover, we note that 
\begin{equation*}
  \max_{j=0,1}\left\|\frac1h \eps_h(u_{j,h}) -\frac1h \eps_h(\tilde{u}_{j,h})\right\|_{L^2(\Omega)} \leq C h^{1+2\theta}
\end{equation*}
since
\begin{eqnarray*}
  \frac1{h^2}(\eps_h (u_{0,h}-\tilde{u}_{0,h}),\eps_h (\varphi))_\Omega &=&  \frac1{h^2}(F'(\nabla_h u_{0,h}),\nabla_h \varphi)_\Omega + (r_{0,h},\varphi)_{\Omega},\\
  \frac1{h^2}(\eps_h (u_{1,h}-\tilde{u}_{1,h}),\eps_h (\varphi))_\Omega &=&  \frac1{h^2}((DW^2(\nabla_h u_{0,h})-DW^2(0))\nabla_h u_{1,h},\nabla_h \varphi)_\Omega \\
&& + (r_{1,h},\varphi)_{\Omega}
\end{eqnarray*}
due to (\ref{eq:ApproxEvol1}),
where $F'$ is as in (\ref{eq:DefnF'}) and $\|(r_{0,h}, r_{1,h})\|_{L^2(\Omega)}\leq Ch^{2+\theta}$. Here we have used that
\begin{eqnarray*}
  \left|\frac1{h^2}\left(F'(\nabla_h u_{0,h}),\nabla_h \varphi\right)_\Omega\right| \leq Ch^{1+2\theta}\left\|\frac1h\eps_h(\varphi)\right\|_{L^2(\Omega)}\\
  \left|\frac1{h^2}\left((DW^2(\nabla_h u_{0,h})-DW^2(0))\nabla_h u_{1,h},\nabla_h \varphi\right)_\Omega\right| \leq Ch^{1+2\theta}\left\|\frac1h\eps_h(\varphi)\right\|_{L^2(\Omega)}
\end{eqnarray*}
for all $\varphi \in H^1_{per}(\Omega)^d$
because of (\ref{eq:EstimD3W}) and the estimates for $u_{0,h}, u_{1,h}$.

Now let $u_h$ be the solution of (\ref{eq:Evol1})-(\ref{eq:BC3}) due to Theorem~\ref{thm:main1} and consider $w_h= \partial_t u_h - \partial_t \tilde{u}_h$. 
Then $w_h$ solves
\begin{eqnarray*}
  -(\partial_t w_h,\partial_t \varphi)_{Q_T} &+& \frac1{h^2} (DW(\nabla_h
  u_h)\nabla_h w_h,\nabla_h \varphi)_{Q_T} - (w_{1,h},\varphi|_{t=0})_\Omega\\
 &=& -\frac1{h^2}\left((DW(\nabla_h u_h)-D^2W(0))\nabla_h \partial_t \tilde{u}_h, \nabla_h \varphi \right)_{Q_T}- (r_h,\varphi)_{Q_T}  \\
  \left. w\right|_{x_j=L} &=& \left. w\right|_{x_j=-L}, \quad j=1,\ldots, d-1,\\
  w|_{t=0} &=& w_{0,h}.
\end{eqnarray*}
for all $\varphi \in C^1([0,T];H^1_{per}(\Omega)^d)$ with $\varphi|_{t=T}=0$, where $w_{j,h}= \partial_t^{1+j} u_h|_{t=0} - \partial_t^{1+j} \tilde{u}_h|_{t=0}$, $j=0,1$ and $r_h$ satisfies (\ref{eq:rhEstim}).
Moreover, 
\begin{eqnarray*}
  \lefteqn{\left|\frac1{h^2}\left((DW(\nabla_h u_h)-D^2W(0))\nabla_h \partial_t \tilde{u}_h, \nabla_h \varphi \right)_{\Omega}\right|}\\
 &\leq& \frac{C}h \left\|\frac1h \eps_h (u_h)\right\|_{L^\infty(0,T;L^2)}\left\|\frac1h \eps_h (\partial_t \tilde{u}_h)\right\|_{L^\infty(0,T;V)} \left\|\frac1h \eps_h (\varphi)\right\|_{L^2(\Omega)}\\
&\leq & Ch^{1+2\theta}\left\|\frac1h \eps_h (\varphi)\right\|_{L^2(\Omega)}
\end{eqnarray*}
due to (\ref{eq:EstimD3W}). Hence Lemma~\ref{lem:LinEstimWeak} implies
\begin{eqnarray*}
 \left\|\left(\partial_t (u_h - \tilde{u}_h), \frac1h \eps_h(u_h-\tilde{u}_h) \right)\right\|_{L^\infty(0,T;L^2)}\leq Ch^{1+2\theta},
\end{eqnarray*}
which proves the theorem.
\end{proof}

\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{AbelsMoreMuellerGammaConv}
H.~Abels, M.~G. Mora, and S.~M{\"u}ller.
\newblock The time-dependent von {K}{\'a}rm{\'a}n plate equation as a limit of
  3d nonlinear elasticity.
\newblock {\em In preparation}.

\bibitem{Amann}
H.~Amann.
\newblock {\em Linear and Quasilinear Parabolic Problems, Volume 1: Abstract
  Linear Theory}.
\newblock Birkh{\"{a}}user, Basel - Boston - Berlin, 1995.

\bibitem{Antman}
S.~S. Antman.
\newblock {\em Nonlinear problems of elasticity}, volume 107 of {\em Applied
  Mathematical Sciences}.
\newblock Springer, New York, second edition, 2005.

\bibitem{BallOpenProblems}
J.~M. Ball.
\newblock Some open problems in elasticity.
\newblock In {\em Geometry, mechanics, and dynamics, Eds. P. Newton, P. Holmes
  and A. Weinstein}, pages 3--59. Springer, New York, 2002.

\bibitem{ChenWahl}
C.~P. Chen and W.~von Wahl.
\newblock Das {R}and-{A}nfangswertproblem f\"ur quasilineare
  {W}ellengleichungen in {S}obolevr\"aumen niedriger {O}rdnung.
\newblock {\em J. Reine Angew. Math.}, 337:77--112, 1982.

\bibitem{Dafermos}
C.~M. Dafermos.
\newblock {\em Hyperbolic conservation laws in continuum physics}.
\newblock Springer-Verlag, Berlin, second edition, 2005.

\bibitem{FrieseckeJamesMueller}
G.~Friesecke, R.~D. James, and S.~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{GeKruseMarsden}
Z.~Ge, H.~P. Kruse, and J.~E. Marsden.
\newblock The limits of {H}amiltonian structures in three-dimensional
  elasticity, shells, and rods.
\newblock {\em J. Nonlinear Sci.}, 6(1):19--57, 1996.

\bibitem{Hornung6}
P.~Hornung.
\newblock Regularity results for flat minimizers of the {Willmore} functional.
\newblock {\em preprint}, 2008.

\bibitem{Hornung7}
P.~Hornung.
\newblock Approximation of {$W^{2,2}$} isometric immersions by smooth isometric
  immersions {I}: Geometry of countably developable mappings.
\newblock {\em preprint}, 2009.

\bibitem{MarsdenEtAl}
T.~J.~R. Hughes, T.~Kato, and J.~E. Marsden.
\newblock Well-posed quasi-linear second-order hyperbolic systems with
  applications to nonlinear elastodynamics and general relativity.
\newblock {\em Arch. Rational Mech. Anal.}, 63(3):273--294 (1977), 1976.

\bibitem{KochLasiecka}
H.~Koch and I.~Lasiecka.
\newblock Hadamard well-posedness of weak solutions in nonlinear dynamic
  elasticity-full von {K}arman systems.
\newblock In {\em Evolution equations, semigroups and functional analysis
  ({M}ilano, 2000)}, Progr. Nonlinear Differential Equations Appl., pages
  197--216. Birkh\"auser, Basel, 2002.

\bibitem{KochStahel}
H.~Koch and A.~Stahel.
\newblock Global existence of classical solutions to the dynamical von
  {K}\'arm\'an equations.
\newblock {\em Math. Methods Appl. Sci.}, 16(8):581--586, 1993.

\bibitem{KohnVogeliusII}
Robert~V. Kohn and Michael Vogelius.
\newblock A new model for thin plates with rapidly varying thickness. {II}. {A}
  convergence proof.
\newblock {\em Quart. Appl. Math.}, 43(1):1--22, 1985.

\bibitem{LasieckaStabilization}
I.~Lasiecka.
\newblock Uniform stabilizability of a full von {K}arman system with nonlinear
  boundary feedback.
\newblock {\em SIAM J. Control Optim.}, 36(4):1376--1422 (electronic), 1998.

\bibitem{MajdaCompFluidFlow}
A.~Majda.
\newblock {\em Compressible fluid flow and systems of conservation laws in
  several space variables}.
\newblock Applied Mathematical Sciences. Springer-Verlag, New York, 1984.

\bibitem{MielkeWeakConvMethods}
A.~Mielke.
\newblock Weak-convergence methods for {H}amiltonian multiscale problems.
\newblock {\em Discrete Contin. Dyn. Syst.}, 20(1):53--79, 2008.

\bibitem{MonneauJustification}
R.~Monneau.
\newblock Justification of the nonlinear {K}irchhoff-{L}ove theory of plates as
  the application of a new singular inverse method.
\newblock {\em Arch. Ration. Mech. Anal.}, 169(1):1--34, 2003.

\bibitem{RenardyRogers}
M.~Renardy and R.~C. Rogers.
\newblock {\em An Introduction to Partial Differential Equations}.
\newblock Springer-Verlag, Berlin-Heidelberg-New York, 1993.

\bibitem{Stein:SingInt}
E.~M. Stein.
\newblock {\em Singular Integrals and Differentiability Properties of
  Functions}.
\newblock Princeton Hall Press, Princeton, New Jersey, 1970.

\bibitem{Triebel1}
H.~Triebel.
\newblock {\em Interpolation Theory, Function Spaces, Differential Operators}.
\newblock North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.

\bibitem{WlokaPDE}
J.~Wloka.
\newblock {\em Partielle {D}ifferentialgleichungen}.
\newblock B. G. Teubner, Stuttgart, 1982.

\bibitem{ZeidlerIIa}
E.~Zeidler.
\newblock {\em Nonlinear functional analysis and its applications. {II}/{A}}.
\newblock Springer-Verlag, New York, 1990.

\end{thebibliography}

\noindent
{\bf Addresses:}\\ Helmut Abels\\
NWF I - Mathematik\\
Universit\"at Regensburg\\
93040 Regensburg, Germany\\
e-mail: helmut.abels@mathematik.uni-regensburg.de\\[1ex]
Maria Giovanna Mora\\
Scuola Internazionale Superiore di Studi Avanzati\\
via Beirut 2\\
34014 Trieste, Italy\\
e-mail: mora@sissa.it\\[1ex]
Stefan M\"uller\\
Hausdorff Center for Mathematics\\
 Institute for Applied Mathematics\\
Universit\"at Bonn\\
Endenicher Allee 60\\
53115 Bonn, Germany\\
e-mail: stefan.mueller@hcm.uni-bonn.de 

 

%\bibliography{../../Bibliography}
%\bibliographystyle{plain}

\end{document}






0909.2796.extracted_bib
\begin{thebibliography}{10}

\bibitem{AbelsMoreMuellerGammaConv}
H.~Abels, M.~G. Mora, and S.~M{\"u}ller.
\newblock The time-dependent von {K}{\'a}rm{\'a}n plate equation as a limit of
  3d nonlinear elasticity.
\newblock {\em In preparation}.

\bibitem{Amann}
H.~Amann.
\newblock {\em Linear and Quasilinear Parabolic Problems, Volume 1: Abstract
  Linear Theory}.
\newblock Birkh{\"{a}}user, Basel - Boston - Berlin, 1995.

\bibitem{Antman}
S.~S. Antman.
\newblock {\em Nonlinear problems of elasticity}, volume 107 of {\em Applied
  Mathematical Sciences}.
\newblock Springer, New York, second edition, 2005.

\bibitem{BallOpenProblems}
J.~M. Ball.
\newblock Some open problems in elasticity.
\newblock In {\em Geometry, mechanics, and dynamics, Eds. P. Newton, P. Holmes
  and A. Weinstein}, pages 3--59. Springer, New York, 2002.

\bibitem{ChenWahl}
C.~P. Chen and W.~von Wahl.
\newblock Das {R}and-{A}nfangswertproblem f\"ur quasilineare
  {W}ellengleichungen in {S}obolevr\"aumen niedriger {O}rdnung.
\newblock {\em J. Reine Angew. Math.}, 337:77--112, 1982.

\bibitem{Dafermos}
C.~M. Dafermos.
\newblock {\em Hyperbolic conservation laws in continuum physics}.
\newblock Springer-Verlag, Berlin, second edition, 2005.

\bibitem{FrieseckeJamesMueller}
G.~Friesecke, R.~D. James, and S.~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{GeKruseMarsden}
Z.~Ge, H.~P. Kruse, and J.~E. Marsden.
\newblock The limits of {H}amiltonian structures in three-dimensional
  elasticity, shells, and rods.
\newblock {\em J. Nonlinear Sci.}, 6(1):19--57, 1996.

\bibitem{Hornung6}
P.~Hornung.
\newblock Regularity results for flat minimizers of the {Willmore} functional.
\newblock {\em preprint}, 2008.

\bibitem{Hornung7}
P.~Hornung.
\newblock Approximation of {$W^{2,2}$} isometric immersions by smooth isometric
  immersions {I}: Geometry of countably developable mappings.
\newblock {\em preprint}, 2009.

\bibitem{MarsdenEtAl}
T.~J.~R. Hughes, T.~Kato, and J.~E. Marsden.
\newblock Well-posed quasi-linear second-order hyperbolic systems with
  applications to nonlinear elastodynamics and general relativity.
\newblock {\em Arch. Rational Mech. Anal.}, 63(3):273--294 (1977), 1976.

\bibitem{KochLasiecka}
H.~Koch and I.~Lasiecka.
\newblock Hadamard well-posedness of weak solutions in nonlinear dynamic
  elasticity-full von {K}arman systems.
\newblock In {\em Evolution equations, semigroups and functional analysis
  ({M}ilano, 2000)}, Progr. Nonlinear Differential Equations Appl., pages
  197--216. Birkh\"auser, Basel, 2002.

\bibitem{KochStahel}
H.~Koch and A.~Stahel.
\newblock Global existence of classical solutions to the dynamical von
  {K}\'arm\'an equations.
\newblock {\em Math. Methods Appl. Sci.}, 16(8):581--586, 1993.

\bibitem{KohnVogeliusII}
Robert~V. Kohn and Michael Vogelius.
\newblock A new model for thin plates with rapidly varying thickness. {II}. {A}
  convergence proof.
\newblock {\em Quart. Appl. Math.}, 43(1):1--22, 1985.

\bibitem{LasieckaStabilization}
I.~Lasiecka.
\newblock Uniform stabilizability of a full von {K}arman system with nonlinear
  boundary feedback.
\newblock {\em SIAM J. Control Optim.}, 36(4):1376--1422 (electronic), 1998.

\bibitem{MajdaCompFluidFlow}
A.~Majda.
\newblock {\em Compressible fluid flow and systems of conservation laws in
  several space variables}.
\newblock Applied Mathematical Sciences. Springer-Verlag, New York, 1984.

\bibitem{MielkeWeakConvMethods}
A.~Mielke.
\newblock Weak-convergence methods for {H}amiltonian multiscale problems.
\newblock {\em Discrete Contin. Dyn. Syst.}, 20(1):53--79, 2008.

\bibitem{MonneauJustification}
R.~Monneau.
\newblock Justification of the nonlinear {K}irchhoff-{L}ove theory of plates as
  the application of a new singular inverse method.
\newblock {\em Arch. Ration. Mech. Anal.}, 169(1):1--34, 2003.

\bibitem{RenardyRogers}
M.~Renardy and R.~C. Rogers.
\newblock {\em An Introduction to Partial Differential Equations}.
\newblock Springer-Verlag, Berlin-Heidelberg-New York, 1993.

\bibitem{Stein:SingInt}
E.~M. Stein.
\newblock {\em Singular Integrals and Differentiability Properties of
  Functions}.
\newblock Princeton Hall Press, Princeton, New Jersey, 1970.

\bibitem{Triebel1}
H.~Triebel.
\newblock {\em Interpolation Theory, Function Spaces, Differential Operators}.
\newblock North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978.

\bibitem{WlokaPDE}
J.~Wloka.
\newblock {\em Partielle {D}ifferentialgleichungen}.
\newblock B. G. Teubner, Stuttgart, 1982.

\bibitem{ZeidlerIIa}
E.~Zeidler.
\newblock {\em Nonlinear functional analysis and its applications. {II}/{A}}.
\newblock Springer-Verlag, New York, 1990.

\end{thebibliography}