# Convergence of equilibria
for incompressible elastic plates
in the von Kármán regime

Marta Lewicka  Marta Lewicka, University of Pittsburgh, Department of
Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
<EMAIL_ADDRESS>and  Hui Li  Hui Li, Penn State University, 2030 Mary
Ellen Lane, State College, PA 16803, USA<EMAIL_ADDRESS>
###### Abstract.

We prove convergence of critical points $u^{h}$ of the nonlinear elastic
energies $E^{h}$ of thin incompressible plates
$\Omega^{h}=\Omega\times(-h/2,h/2)$, which satisfy the von Kármán scaling:
$E^{h}(u^{h})\leq Ch^{4}$, to critical points of the appropriate limiting
(incompressible von Kármán) functional.

##  1\. Introduction and the main result

In this paper we prove convergence of critical points of the nonlinear elastic
energies on thin incompressible plates in the von Kármán scaling regime, to
critical points of the appropriate limiting (incompressible von Kármán)
functional.

###  1.1. Elastic energy of thin incompressible plates

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, simply connected
domain. For $h>0$, define $\Omega^{h}$ to be the $3$d plate with the midplate
$\Omega$ and thickness $h$:

$\Omega^{h}=\left\\{x=(x^{\prime},x_{3});~{}x^{\prime}\in\Omega,~{}x_{3}\in\left(-\frac{h}{2},\frac{h}{2}\right)\right\\}.$

The elastic energy of a deformation $u^{h}\in
W^{1.2}(\Omega^{h},\mathbb{R}^{3})$ of the homogeneous plate $\Omega^{h}$,
scaled by its unit thickness, is given by:

(1.1) $I^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W_{in}(\nabla
u^{h})~{}\mbox{d}x,$

while the total energy, relative to the external force with the density
$f^{h}\in L^{2}(\Omega^{h},\mathbb{R}^{3})$, is:

(1.2) $J^{h}(u^{h})=\frac{1}{h}\int_{\Omega^{h}}W_{in}(\nabla
u^{h})~{}\mbox{d}x-\frac{1}{h}\int_{\Omega^{h}}f^{h}\cdot u^{h}~{}\mbox{d}x.$

The elastic energy density $W_{in}:\mathbb{R}^{3\times
3}\rightarrow[0,\infty]$ in (1.1) is assumed to be infinite at compressible
deformations:

$W_{in}(F)=\left\\{\begin{array}[]{ll}W(F)&\mbox{ if}~{}~{}\det F=1,\\\
+\infty&\mbox{ otherwise}.\end{array}\right.$

The effective density $W:\mathbb{R}^{3\times 3}\rightarrow[0,\infty)$ above,
which acts when $\det F=1$, is required to satisfy the following conditions:

  * (i)

(frame invariance) $W(RF)=W(F)$, for each proper rotation $R\in SO(3)$, and
each $F\in\mathbb{R}^{3\times 3}$.

  * (ii)

(normalisation) $W(F)=0$ for all $F\in SO(3)$.

  * (iii)

(non-interpenetration) $W(F)=+\infty$ if $\det F\leq 0$, and $W(F)\to+\infty$
as $\det F\to 0+$.

  * (iv)

(bound from below) $W(F)\geq c~{}\mbox{dist}^{2}(F,SO(3))$ with a constant
$c>0$ independent of $F$.

  * (v)

(bound from above) There exists a constant $C>0$ such that for each $F$ with
$\det F>0$, i.e. for each $F\in\mathbb{R}_{+}^{3\times 3}$ there holds:

(1.3) $|DW(F)F^{T}|\leq C(W(F)+1).$

  * (vi)

(regularity) $W$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{3\times
3}$.

  * (vii)

(local regularity) $W$ is of class $\mathcal{C}^{2}$ in a small neighborhood
of $SO(3)$.

The growth conditions in (iv) and (v) will be crucial in the present analysis.
Condition (iv) has been introduced in the context of [6] and it allows to use
the nonlinear version of Korn’s inequality [5], ultimately serving to control
the local deviations of the deformation $u^{h}$ from rigid motions, by the
elastic energy $I^{h}(u^{h})$. Condition (v) has been introduced in [1] (see
also [2]) in the context of inner variations, in order to control the related
strain in terms of the energy. Both conditions are compatible with other
requirements above. Indeed, examples of $W$ satisfying (i) – (vii) are:

$\begin{split}W_{1}(F)&=|(F^{T}F)^{1/2}-\mbox{Id}|^{2}+|\log\det F|^{q},\\\
W_{2}(F)&=|(F^{T}F)^{1/2}-\mbox{Id}|^{2}+\left|\frac{1}{\det
F}-1\right|^{q}\mbox{ for }\det F>0,\end{split}$

where $q>1$ and $W$ equals $+\infty$ if $\det F\leq 0$ [11].

###  1.2. Notation

Given a matrix $F\in\mathbb{R}^{n\times n}$, we denote its trace by
$\mbox{Tr}~{}F$ and its transpose by $F^{T}$. The symmetric part of $F$ is
given by $\mbox{sym}~{}F=\frac{1}{2}(F+F^{T})$. The cofactor of $F$ is the
matrix: $\mbox{cof}~{}F$, where $[\mbox{cof
}F]_{ij}=(-1)^{i+j}\det\hat{F}_{ij}$ and each
$\hat{F}_{ij}\in\mathbb{R}^{(n-1)\times(n-1)}$ is obtained from $F$ by
deleting its $i$th row and $j$th column. The identity matrix is denoted by
$\mbox{Id}_{n}$.

In what follows, we shall use the matrix norm $|F|=(\mbox{Tr}(F^{T}F))^{1/2}$,
which is induced by the inner product:
$F_{1}:F_{2}=\mbox{Tr}(F_{1}^{T}F_{2})$. To avoid notational confusion, we
will often write $\langle F_{1}:F_{2}\rangle$ instead of $F_{1}:F_{2}$. In
general, $3\times 3$ matrices will be denoted by $F$ and $2\times 2$ matrices
will be denoted by $F^{\prime\prime}$. Unless noted otherwise,
$F^{\prime\prime}$ is the principal $2\times 2$ minor of $F$.

Finally, by $\mathcal{C}^{k}_{b}(\mathbb{R}^{n},\mathbb{R}^{s})$ we denote the
space of continuous functions whose derivatives up to the order $k$ are
continuous and bounded in $\mathbb{R}^{n}$.

###  1.3. The limiting energy

The following $2$d energy functional has been rigorously derived in [10] as
the $\Gamma$-limit of the scaled incompressible energies $h^{-4}I^{h}$ in
(1.1), when $h\to 0$:

(1.4)
$\mathcal{I}(w,v)=\frac{1}{2}\int_{\Omega}\mathcal{Q}_{2}^{in}\left(\mbox{sym}\nabla
u+\frac{1}{2}\nabla v\otimes\nabla
v\right)~{}\mbox{d}x+\frac{1}{24}\int_{\Omega}\mathcal{Q}_{2}^{in}\left(\nabla^{2}v\right)~{}\mbox{d}x,$

acting on couples $w\in W^{1,2}(\Omega,\mathbb{R}^{2}),v\in
W^{2,2}(\Omega,\mathbb{R})$. The fields $(w,v)$ may be identified as the in-
plane and the out-of-plane displacements, respectively. Roughly speaking, any
minimizing sequence of $h^{-4}J^{h}$, where $f^{h}(x)\approx
h^{3}f(x^{\prime})e_{3}$ and $\int_{\Omega}f=0$, will have the structure:

$u^{h}_{|\Omega}\approx(\bar{R})^{T}\left(\mbox{id}+hve_{3}+h^{2}w\right)-c^{h}$

asymptotically as $h\to 0$, with $(w,v)$ as above and $\bar{R}\in SO(3)$
maximizing $\int_{\Omega}f(x^{\prime})e_{3}\cdot
Rx^{\prime}~{}\mbox{d}x^{\prime}$ among all rotations $R$, while
$c^{h}\in\mathbb{R}^{3}$ are constant translation vectors. Moreover,
$(w,v,\bar{R})$ minimize the following total limiting energy:

$\mathcal{J}(w,v,\bar{R})=\mathcal{I}(w,v)-\bar{R}_{33}\int_{\Omega}fv.$

A precise formulation of the statements above can be found in [9].

The energy in (1.4) is the incompressible version of the von Kármán
functional, which has been derived (for compressible case, i.e. without the
assumption that $\det\nabla u^{h}=1$) by means of $\Gamma$-convergence in [6].
The quadratic forms $\mathcal{Q}_{2}^{in}$ differ from the standard
$\mathcal{Q}_{2}$ in [6] in as much as minimization in (1.5) below is taken
over the out-of-plane stretches which preserve the incompressibility
constraint. Namely, $\mathcal{Q}_{2}^{in}$ in (1.4) are given as:

(1.5) $\begin{split}&\forall F^{\prime\prime}\in\mathbb{R}^{2\times
2}\quad\mathcal{Q}_{2}^{in}(F^{\prime\prime})=\min_{d\in\mathbb{R}^{3}}\Big{\\{}Q_{3}(F^{\prime\prime}+d\otimes
e_{3}+e_{3}\otimes d);~{}\mbox{Tr}(F^{\prime\prime}+d\otimes
e_{3}+e_{3}\otimes d)=0\Big{\\}},\\\ &\forall F\in\mathbb{R}^{3\times
3}\quad\mathcal{Q}_{3}(F)=D^{2}W(\mbox{Id})(F,F).\end{split}$

Both forms $\mathcal{Q}$ above are positive semidefinite, and strictly
positive definite on symmetric matrices. We also introduce the linear
operators $\mathcal{L}_{2}^{in}:\mathbb{R}^{2\times
2}\rightarrow\mathbb{R}^{2\times 2}$ and $\mathcal{L}_{3}:\mathbb{R}^{3\times
3}\rightarrow\mathbb{R}^{3\times 3}$ such that:

(1.6) $\begin{split}&\forall F^{\prime\prime}\in\mathbb{R}^{2\times
2}\quad\langle\mathcal{L}_{2}^{in}(F^{\prime\prime}):F^{\prime\prime}\rangle=\mathcal{Q}_{2}^{in}(F^{\prime\prime}),\\\
&\forall F\in\mathbb{R}^{3\times
3}\quad\langle\mathcal{L}_{3}(F):F\rangle=\mathcal{Q}_{3}(F).\end{split}$

Note that symmetric operators $\mathcal{L}$ are uniquely given by:
$\langle\mathcal{L}(F_{1}):F_{2}\rangle=\frac{1}{4}\left(\mathcal{Q}(F_{1}+F_{2})-\mathcal{Q}(F_{1}-F_{2})\right)$.

###  1.4. Critical points and the incompressible inner variations

Following [2], we now define the critical points $u^{h}$ of the functionals
$J^{h}$ in (1.2) with respect to inner variations, that is requesting that the
derivative of $J^{h}$ at an incompressible equilibrium $u^{h}$ be zero:

$\frac{\mathrm{d}}{\mathrm{d}\epsilon}_{|\epsilon=0}J^{h}(u^{h}_{\epsilon})=0,$

along all curves $\epsilon\mapsto u^{h}_{\epsilon}$ of incompressible
deformations of $\Omega^{h}$ having the form:
$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))$, with $u^{h}_{0}=u^{h}$ at
$\epsilon=0$. This requirement is translated into the following condition:

(1.7) $\int_{\Omega^{h}}\left\langle DW(\nabla u^{h})(\nabla
u^{h})^{T}:\nabla\phi(u^{h}(x))\right\rangle~{}\mbox{d}x=\int_{\Omega^{h}}f^{h}\cdot\phi(u^{h})~{}\mbox{d}x,\quad\forall\phi\in\mathcal{C}_{b}^{1}(\mathbb{R}^{3},\mathbb{R}^{3})\mbox{
with }\mbox{div}~{}\phi=0.$

We refer to section 2 for the derivation and discussion of (1.7). Let us only
note now that the incompressible inner variations:

$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))=u^{h}(x)+\epsilon\phi(u^{h}(x))+\mathcal{O}(\epsilon^{2}).$

replace the classical variations $u^{h}_{\epsilon}(x)=u^{h}(x)+\epsilon
w^{h}(x)$ used in definition of minimizers of $J^{h}$, and also they replace
the inner variations $u^{h}_{\epsilon}(x)=u^{h}(x)+\epsilon\phi(u^{h}(x))$
considered in [2] and [11] in the compressible case.

###  1.5. The main result

The following is our main result:

######  Theorem 1.1.

For each $h<<1$, let $u^{h}\in W^{1,2}(\Omega^{h},\mathbb{R}^{3})$ be a
critical point of $J^{h}$, i.e. it satisfies (1.7) subject to the external
forces $f^{h}(x)=h^{3}f(x^{\prime})e_{3}$. Assume that:

(1.8) $I^{h}(u^{h})\leq Ch^{4},$

for a constant $C>0$ independent of $h$. Then there exists a sequence of
proper rotations $\bar{R}^{h}\in SO(3)$, and translations
$c^{h}\in\mathbb{R}^{3}$, such that for the renormalized deformations:

(1.9)
$y^{h}(x^{\prime},x_{3})=(\bar{R}^{h})^{T}u^{h}(x^{\prime},hx_{3})-c^{h}\in
W^{1,2}(\Omega^{1},\mathbb{R}^{3}),$

the following convergences hold, up to a subsequence in $h$, as $h\to 0$:

  * (i)

$\bar{R}^{h}\to\bar{R}=[\bar{R}_{ij}]_{i,j:1..3}\in SO(3)$.

  * (ii)

$y^{h}\to x^{\prime}$ in $W^{1,2}(\Omega^{1})$.

  * (iii)

For the scaled out-of-plane displacements:

(1.10)
$v^{h}(x^{\prime})=\frac{1}{h}\int_{-1/2}^{1/2}y_{3}^{h}(x^{\prime},x_{3})~{}\mathrm{d}x_{3},$

there exists $v\in W^{2,2}(\Omega,\mathbb{R})$ such that $v^{h}\to v$ strongly
in $W^{1,2}(\Omega)$.

  * (iv)

For the scaled in-plane displacements:

(1.11)
$w^{h}(x^{\prime})=\frac{1}{h^{2}}\int_{-1/2}^{1/2}\left((y^{h})^{\prime}(x^{\prime},x_{3})-x^{\prime}\right)~{}\mathrm{d}x_{3}$

there exists $w\in W^{1,2}(\Omega,\mathbb{R}^{2})$ such that
$w^{h}\rightharpoonup w$ weakly in $W^{1,2}(\Omega,\mathbb{R}^{2})$.

  * (v)

The limiting displacements $(w,v)$ solve the following Euler-Lagrange
equations of the functional (1.4), expressed in the variational form:

(1.12) $\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla
v\right):\nabla\tilde{w}\right\rangle~{}\mathrm{d}x^{\prime}=0$ (1.13)
$\begin{split}&\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla v\right):(\nabla
v\otimes\nabla\tilde{v})\right\rangle~{}\mathrm{d}x^{\prime}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{12}\int_{\Omega}\left\langle\mathcal{L}_{2}^{in}(\nabla^{2}v):\nabla^{2}\tilde{v}\right\rangle~{}\mathrm{d}x^{\prime}=\bar{R}_{33}\int_{\Omega}f\tilde{v}~{}\mathrm{d}x^{\prime},\end{split}$

for every $\tilde{w}\in W^{1,2}(\Omega,\mathbb{R}^{2})$ and every
$\tilde{v}\in W^{2,2}(\Omega,\mathbb{R})$.

We note that (1.8) are automatically satisfied by any minimizing sequence of
$u^{h}$ of the total energy $J^{h}$, under the assumption that
$f^{h}(x)=h^{3}f(x^{\prime})e_{3}$ [6]. Also, (1.7) holds for every minimum of
$J^{h}$ (see Theorem 2.3), and the assertions (i) - (v) are then a direct
consequence [10] of the fact that $\frac{1}{h^{4}}J^{h}$ $\Gamma$-converges to
$\mathcal{J}$. In general, $\Gamma$-convergence does not assure that a limit
of a sequence of equlibria is an equilibrium of the $\Gamma$-limit. In the
present situation, this turns out to be the case.

###  1.6. Relation to other works

Our work is largely inspired by [11] and [10]. To put it in a larger
perspective, recall that one of the fundamental questions in the mathematical
theory of elasticity has been to rigorously justify various 2d plate models
present in the engineering literature, in relation to the three-dimensional
theory. This goal has been largely accomplished in [6], where a hierarchy of
limiting 2d energies has been derived; the distinct theories are
differentiated by their validity in the corresponding scaling regimes
$h^{\beta},$ $\beta\geq 2$, i.e. in presence of assumption (1.8) where $h^{4}$
is replaced by $h^{\beta}$.

Under the additional incompressibility constraint, the works [3, 4] proved
compactness properties and the $\Gamma$-convergence of the functionals
$\frac{1}{h^{\beta}}I^{h}$ as in (1.1), for the so-called Kirchhoff scaling
$\beta=2$, while [10] treated the case $\beta=4$ including as well a more
complex case of shells when the midsurface $\Omega$ is a generic 2d
hypersurface in $\mathbb{R}^{3}$. In view of the fundamental property of
$\Gamma$-convergence, it follows that the global almost-minimizers of the
energies (1.2) converge to the minimizers of the limiting energy (given by
(1.4) in the von Kármán regime).

Regarding convergence of stationary points for thin plates, the first result
has been obtained in [12] under the von Kármán scaling $\beta=4$ (see also [7]
for an extension to thin shells). These results relied on the crucial
assumption that the elastic energy density $W$ is differentiable everywhere
and its derivative satisfies a linear growth condition: $|DW(F)|\leq
C(|F|+1)$. This assumption is contradictory with the physically expected non-
interpenetration condition, and subsequently it has been removed in [11] and
exchanged with Ball’s condition (1.3), while the equilibrium equations have
been rephrased in terms of the inner variations. In the present paper we
follow the same approach; indeed the concept of inner variations comes up
naturally in the context of incompressible elasticity.

To conclude, we now comment on the isotropic case. For an isotropic energy
density $W$ with the Lamé constants $\lambda$ and $\mu$, the Euler-Lagrange
equations (1.12) – (1.13) of (1.4) are:

(1.14)
$\frac{\mu}{3}\Delta^{2}v=[v,\Phi],\qquad\Delta^{2}\Phi=-\frac{3\mu}{2}[v,v],$

where $v$ is the out-of-plane displacement, while the in-plane displacement
$w$ can be recovered through the Airy stress potential $\Phi$, by means of:

$\mbox{cof}\nabla^{2}\Phi=2\mu\Big{[}\mbox{sym}\nabla w+\frac{1}{2}\nabla
v\otimes\nabla v+\Big{(}\mbox{div}w+\frac{1}{2}|\nabla
v|^{2}\Big{)}\mbox{Id}\Big{]}.$

The Airy’s bracket $[\cdot,\cdot]$ is defined as:
$[v,\Phi]=\nabla^{2}v:(\mbox{cof}\nabla^{2}\Phi)$. As expected, the system
(1.14) can be now obtained as the incompressible limit, i.e. when passing with
the Poisson ratio $\nu\to\frac{1}{2}$, of the classical (compressible) von
Kármán system:

$B\Delta^{2}v=[v,\Phi],\qquad\Delta^{2}\Phi=-\frac{S}{2}[v,v],$

where $S=2\mu(1+\nu)$ is Young’s modulus, $\nu=\frac{\lambda}{2(\mu+\lambda)}$
is the Poisson ratio, and $B=\frac{S}{12(1-\nu^{2})}$ is bending stiffness. By
the change of variable $\Phi=2\mu\Phi_{1}$ one can eliminate the parameter
$\mu$ entirely and write (1.14) in its equivalent form:

$\Delta^{2}v=6[v,\Phi_{1}],\qquad\Delta^{2}\Phi_{1}=-\frac{3}{4}[v,v].$

Acknowledgments. M.L. was partially supported by the NSF Career grant
DMS-0846996 and by the Polish MN grant N N201 547438.

##  2\. Incompressible inner variations and critical points

Following [2], we want to define the critical points $u^{h}$ of the
functionals $J^{h}$ in (1.2) by taking inner variations. That is, we request
that the derivative of $J^{h}$ at an incompressible equilibrium $u^{h}$ be
zero along all curves $\epsilon\mapsto u^{h}_{\epsilon}$ of incompressible
deformations of $\Omega^{h}$ having the form:
$u^{h}_{\epsilon}(x)=\Phi(\epsilon,u^{h}(x))$, with $u^{h}_{0}=u^{h}$ at
$\epsilon=0$. This requirement imposes the following conditions on the flow
$\Phi:[0,\epsilon_{0})\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$:

(2.1) $\begin{split}&\forall\epsilon\quad\Phi(\epsilon,\cdot)\mbox{ is
incompressible, i.e }\quad\forall
y\in\mathbb{R}^{3}\quad\det\nabla\Phi(\epsilon,y)=1,\\\ &\forall
y\in\mathbb{R}^{3}\quad\Phi(0,y)=y.\end{split}$

Assuming sufficient smoothness of $\Phi$, the above immediately implies:

$\begin{split}0&=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\det\nabla\Phi(0,y)=\left\langle\mbox{cof}\nabla\Phi(0,y):\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right\rangle=\left\langle\mbox{Id}:\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right\rangle\\\
&=\mbox{Tr}\left(\frac{\mathrm{d}}{\mathrm{d}\epsilon}\nabla\Phi(0,y)\right)=\mbox{div}\left(\frac{\mathrm{d}}{\mathrm{d}\epsilon}\Phi(0,y)\right)=:\mbox{div}~{}\phi(y).\end{split}$

On the other hand, any divergence-free vector field $\phi$ generates a path of
incompressible deformations. We recall this standard fact below, for the sake
of completeness.

######  Lemma 2.1.

Let $\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{n},\mathbb{R}^{n})$ such that
$\mathrm{div}~{}\phi=0$. Consider the ODE:

(2.2) $\left\\{\begin{array}[]{ll}u^{\prime}(\epsilon)=\phi(u(\epsilon)),\\\
u(0)=y.\end{array}\right.$

and denote its flow by $\Phi(\epsilon,y)=u(\epsilon)$ solving (2.2). Then
$\Phi$ satisfies (2.1).

###### Proof.

Let $\epsilon,\delta>0$ and note that:
$\Phi(\epsilon+\delta,y)=\Phi(\delta,\Phi(\epsilon,y))=\Phi(\delta,y_{1})$
where we put $y_{1}=\Phi(\epsilon,y)$. Hence, denoting the spacial gradient by
$\nabla$, we obtain:

$\det\nabla\Phi(\epsilon+\delta,y)=\det\nabla\Phi(\delta,y_{1})\det\nabla\Phi(\epsilon,y),$

Consequently:

(2.3)
$\begin{split}\frac{\mbox{d}}{\mbox{d}\epsilon}\left(\det\nabla\Phi(\epsilon+\delta,y)\right)&=\frac{\mbox{d}}{\mbox{d}\delta}\left(\det\nabla\Phi(\epsilon+\delta,y)\right)=\frac{\mbox{d}}{\mbox{d}\delta}\left(\det\nabla\Phi(\delta,y_{1})\right)\left(\det\nabla\Phi(\epsilon,y)\right)\\\
&=\left\langle\mbox{cof}~{}\nabla\Phi(\delta,y_{1}):\frac{\mbox{d}}{\mbox{d}\delta}\nabla\Phi(\delta,y_{1})\right\rangle\det\nabla\Phi(\epsilon,y).\end{split}$

Above, we used the formula for the derivative of the determinant of a matrix
function $A(t)$, namely: $(\det A(t))^{\prime}=\mbox{cof}A(t):A(t)^{\prime}.$
For $\delta=0$, (2.3) implies:

$\frac{\mbox{d}}{\mbox{d}\epsilon}\left(\det\nabla\Phi(\epsilon,y)\right)=\langle\mbox{cof}\nabla\Phi(0,y_{1}):\nabla\phi(y_{1})\rangle=\langle\mbox{Id}:\nabla\phi(y_{1})\rangle=\mbox{Tr}\nabla\phi=\mbox{div}~{}\phi=0.$

But $\det\nabla\Phi(0,y)=\det\mbox{Id}_{n}=1$, which achieves the claim.

We are now ready to derive the equilibrium equations (1.7). The result is
essentially similar to Theorem 2.4 [2], which dealt with the compressible
inner variations $u^{h}_{\epsilon}=u^{h}(x)+\epsilon\phi\circ u^{h}$ of a
deformation $u^{h}$ with clamped boundary conditions. The growth condition
(1.3) will be crucial in passing to the limit in the nonlinear term in
$J^{h}$, to which end we are going to use the following Lemma from [2]:

######  Lemma 2.2.

(Lemma 2.5 (i) [2]) Assume that $W$ satisfies (1.3). Then there exists
$\gamma>0$ such that if $A\in\mathbb{R}^{3\times 3}_{+}$ and
$|A-\mathrm{Id}|<\gamma$, then:

$|DW(AF)F^{T}|\leq 3C(W(F)+1)\qquad\forall F\in\mathbb{R}^{3\times 3}_{+},$

where $C$ is the constant in condition (1.3).

######  Theorem 2.3.

Let $\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$ be such that
$\mathrm{div}~{}\phi=0$. Given a deformation $u^{h}\in
W^{1,2}(\Omega^{h},\mathbb{R}^{3})$ with $\det\nabla u^{h}=1$, and such that
$\int_{\Omega^{h}}W(\nabla u^{h})~{}\mathrm{d}x<+\infty$, define
$u_{\epsilon}^{h}(x)=\Phi(\epsilon,u^{h}(x))$. Then:

$\frac{\mathrm{d}}{\mathrm{d}\epsilon}_{|\epsilon=0}J^{h}(u^{h}_{\epsilon})=0$

is equivalent to:

$\int_{\Omega^{h}}\left\langle DW(\nabla u^{h})(\nabla
u^{h})^{T}:\nabla\phi(u^{h}(x))\right\rangle~{}\mathrm{d}x=\int_{\Omega^{h}}f^{h}\cdot\phi(u^{h})~{}\mathrm{d}x.$

###### Proof.

For the notational convenience, in what follows we drop the index $h$ and
write $U$ instead of $\Omega^{h}$, which stands now for a fixed open bounded
domain in $\mathbb{R}^{3}$. It is easy to notice that:

(2.4) $\lim_{\epsilon\to
0}\frac{1}{\epsilon}(\Phi(\epsilon,y)-y)=\phi(y)\quad\mbox{ uniformly in
}\mathbb{R}^{3}.$

It directly implies that:

$\lim_{\epsilon\to
0}\frac{1}{\epsilon}\int_{U}f\cdot(\Phi(\epsilon,u(x))-u(x))~{}\mbox{d}x=\int_{U}f\cdot\phi(u(x))~{}\mbox{d}x.$

To treat the nonlinear term, consider:

(2.5) $\begin{split}\frac{1}{\epsilon}&\int_{U}\big{(}W(\nabla
u_{\epsilon})-W(\nabla u)\big{)}~{}\mbox{d}x-\int_{U}\big{\langle}DW(\nabla
u)(\nabla u)^{T}:\nabla\phi(u)\big{\rangle}~{}\mbox{d}x\\\
&=\int_{U}\fint_{0}^{\epsilon}\Big{\langle}DW\big{(}\nabla\Phi(s,u)\nabla
u\big{)}(\nabla
u)^{T}:\nabla\phi(\Phi(s,u))\Big{\rangle}-\Big{\langle}DW(\nabla u)(\nabla
u)^{T}:\nabla\phi(u)\Big{\rangle}~{}\mbox{d}s~{}\mbox{d}x.\end{split}$

Since the integrand below converges to $0$ pointwise by (2.4), and it is
bounded by the function $2\|\nabla\phi\|_{L^{\infty}}|DW(\nabla u)(\nabla
u)^{T}|$ which is integrable in view of (1.3), we obtain:

$\lim_{\epsilon\to 0}\int_{U}\Big{\langle}DW(\nabla u)(\nabla
u)^{T}:\fint_{0}^{\epsilon}\nabla\phi(\Phi(s,u))-\nabla\phi(u)~{}\mbox{d}s\Big{\rangle}~{}\mbox{d}x=0,$

by the dominated convergence theorem. Similarly:

$\lim_{\epsilon\to
0}\int_{U}\fint_{0}^{\epsilon}\Big{\langle}\Big{(}DW(\nabla\Phi(s,u)\nabla
u)-DW(\nabla u)\Big{)}(\nabla
u)^{T}:\nabla\phi(\Phi(s,u))\Big{\rangle}~{}\mbox{d}s~{}\mbox{d}x=0,$

where the pointwise convergence follows by the formula (2.4), its counterpart
for $\nabla\Phi$, and the continuity of $DW$ on $\mathbb{R}^{3\times 3}_{+}$.
The integrands, for small $\epsilon$, are dominated by the $L^{1}(U)$ function
$4C\|\nabla\phi\|_{L^{\infty}}(W(\nabla u)+1)$ in view of Lemma 2.2 and the
growth condition (1.3).

Therefore, the left hand side in (2.5) converges to $0$ as well. This
completes the proof.

##  3\. The equilibrium equation (1.7)

In this section, we review several facts from [6] and [11], to set the stage
for a proof of Theorem 1.1 and to rewrite the equation (1.7) using the change
of variables (1.9).

The first crucial step in the dimension reduction argument of [6] is finding
the appropriate approximations of the deformations gradients $u^{h}$. Under
the sole assumption:

(3.1) $\frac{1}{h}\int_{\Omega^{h}}W(\nabla u^{h})~{}\mbox{d}x\leq Ch^{4},$

an application of a nonlinear verion of Korn’s inequality [5], yields
existence of rotation fields $R^{h}\in W^{1,2}(\Omega,\mathbb{R}^{3\times 3})$
with $R^{h}(x)\in SO(3)$ a.e. in $\Omega$, so that:

(3.2) $\|\nabla u^{h}(x^{\prime},hx_{3})-R^{h}\|_{L^{2}(\Omega^{1})}\leq
Ch^{2}\qquad\mbox{and}\qquad\|\nabla R^{h}\|_{L^{2}(\Omega)}\leq Ch.$

Recall that $\Omega^{1}=\Omega\times(-\frac{1}{2},\frac{1}{2})$ is the common
domain of the rescaled deformations
$y^{h}(x^{\prime},x_{3})=(\bar{R}^{h})^{T}u^{h}(x^{\prime},hx_{3})-c^{h}$, and
the typical point in $\Omega^{1}$ is denoted by $x=(x^{\prime},x_{3})$. Then,
the detailed analysis in [6] shows that convergences in (i) – (iv) of Theorem
1.1 hold, as a consequence of (1.8) implying (3.1). The constant rotations
$\bar{R}^{h}\in SO(3)$ are given by:

$\bar{R}^{h}=\mathbb{P}_{SO(3)}\left(\fint_{\Omega^{h}}\nabla
u^{h}~{}\mbox{d}x\right),$

where the orthogonal projection $\mathbb{P}_{SO(3)}$ onto $SO(3)$ above is
well defined; see also [8] for detailed calculations. Further, there holds:

(3.3) $\|R^{h}-\bar{R}^{h}\|_{L^{2}(\Omega)}\leq
Ch\qquad\mbox{and}\qquad\lim_{h\to
0}(\bar{R}^{h})^{T}R^{h}=\mbox{Id}\quad\mbox{in
}W^{1,2}(\Omega,\mathbb{R}^{3\times 3}),$

and upon defining the matrix fields $A^{h}\in
W^{1,2}(\Omega,\mathbb{R}^{3\times 3})$:

(3.4)
$A^{h}(x^{\prime})=\frac{1}{h}\left((\bar{R}^{h})^{T}R^{h}(x^{\prime})-\mbox{Id}\right),$

it also follows that:

(3.5) $A^{h}\rightharpoonup
A=\left[\begin{array}[]{c|c}0&\begin{minipage}{22.76228pt}$$\vspace{3mm}-\nabla
v$$\end{minipage}\\\ \hline\cr\nabla
v&\begin{minipage}{22.76228pt}$$\vspace{3mm}0$$\end{minipage}\end{array}\right]\qquad\mbox{weakly
in }W^{1,2}(\Omega,\mathbb{R}^{3\times 3}).$

The same convergence holds strongly in $L^{q}(\Omega,\mathbb{R}^{3\times 3})$
for each $q\geq 1$.

######  Lemma 3.1.

We have:

(3.6) $\lim_{h\to 0}y^{h}=(x^{\prime},0)\quad\mbox{ and }\quad\lim_{h\to
0}\frac{y^{h}_{3}}{h}=x_{3}+v(x^{\prime})\qquad\mbox{in
}~{}W^{1,2}(\Omega^{1}).$

Consequently, for every $\omega_{h}>0$ and $p\in[1,5]$:

(3.7)
$\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\leq\frac{C}{\omega_{h}^{2}}\quad\mbox{
and
}\quad\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\left|\frac{y^{h}_{3}(x)}{h}\right|^{p}~{}\mathrm{d}x\leq\frac{C}{\omega_{h}^{\frac{2}{p+1}}}.$

###### Proof.

By (3.2), (3.3), and applying the Poincaré-Wirtinger inequality on segments
$\\{x^{\prime}\\}\times(-\frac{1}{2},\frac{1}{2})$, we see that:

$\begin{split}\left\|\frac{y_{3}^{h}}{h}-x_{3}-v^{h}(x^{\prime})\right\|_{L^{2}(\Omega^{1})}&\leq
C\Big{\|}\frac{\partial_{3}y_{3}^{h}}{h}-1\Big{\|}_{L^{2}(\Omega^{1})}=C\Big{\|}[(\bar{R}^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})]_{33}-1\Big{\|}_{L^{2}(\Omega^{1})}\\\ &\leq
C\|(\bar{R}^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})-\mbox{Id}\|_{L^{2}(\Omega^{1})}\\\ &\leq C\|\nabla
u^{h}(x^{\prime},hx_{3})-R^{h}\|_{L^{2}(\Omega^{1})}+C\|R^{h}-\bar{R}^{h}\|_{L^{2}(\Omega^{1})}\leq
Ch.\end{split}$

Together with (1.10), the above inequality implies the second assertion in
(3.6). The first assertion follows then directly in view of (1.11).

To prove (3.7), note that for every $p\in[1,5]$:

(3.8)
$\begin{split}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\left|\frac{y^{h}_{3}(x)}{h}\right|^{p}~{}\mathrm{d}x&\leq\left\|\frac{y^{h}_{3}}{h}\right\|^{p}_{L^{p+1}}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{\frac{1}{p+1}}\\\
&\leq
C\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{\frac{1}{p+1}},\end{split}$

by the Hölder inequality and the Sobolev embedding
$W^{1,2}(\Omega^{1})\hookrightarrow L^{6}(\Omega^{1})$ combined with (3.6).
When $p=1$, it implies:

$\left|\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\leq\frac{1}{\omega_{h}}\int_{\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\frac{|y^{h}_{3}(x)|}{h}~{}\mathrm{d}x\leq\frac{C}{\omega_{h}}\left|\left\\{x\in\Omega;~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}$

Hence, the first assertion in (3.7) follows, as well as the second one, in
view of (3.8).

Define the strain $G^{h}\in L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$ and the
scaled stress $E^{h}\in L^{1}(\Omega^{1},\mathbb{R}^{3\times 3})$ as:

$\begin{split}&G^{h}(x^{\prime},x_{3})=\frac{1}{h^{2}}\left((R^{h})^{T}\nabla
u^{h}(x^{\prime},hx_{3})-\mbox{Id}\right),\\\
&E^{h}(x^{\prime},x_{3})=\frac{1}{h^{2}}DW(\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3}))(\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3}))^{T}.\end{split}$

We now gather the fundamental properties of $E^{h}$ and $G^{h}$ from [11],
that will be used in the sequel.

######  Lemma 3.2.

(Section 4, [11])

  * (i)

Up to a subsequence, $G^{h}\rightharpoonup G$ weakly in
$L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$, where $G$ is the limiting strain
whose principal $2\times 2$ minor $G^{\prime\prime}$ satisfies:

(3.9)
$\begin{split}&G^{\prime\prime}(x^{\prime},x_{3})=G_{0}(x^{\prime})-x_{3}G_{1}(x^{\prime}),\quad\mbox{
with: }\\\ &\mathrm{sym}~{}G_{0}=\mathrm{sym}\nabla w+\frac{1}{2}\nabla
v\otimes\nabla v,\qquad G_{1}=\nabla^{2}v.\end{split}$

  * (ii)

Each $E^{h}(x)$ is symmetric, and there holds:

(3.10) $|E^{h}|\leq
C\left(\frac{1}{h^{2}}W(\mathrm{Id}+h^{2}G^{h})+|G^{h}|\right).$

  * (iii)

Up to a subsequence, $E^{h}\rightharpoonup E$ weakly in
$L^{1}(\Omega^{1},\mathbb{R}^{3\times 3})$, and $E=\mathcal{L}_{3}(G)\in
L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$.

  * (iv)

For a given, fixed $\gamma\in(0,2)$, define
$B_{h}=\\{x\in\Omega^{1};~{}h^{2-\gamma}|G^{h}(x)|\leq 1\\}$. Then:

(3.11) $|\Omega^{1}\setminus B_{h}|\leq
Ch^{2(2-\gamma)}\qquad\mbox{and}\qquad\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mathrm{d}x\leq Ch^{2-\gamma}.$

Moreover, calling $\chi_{h}$ the characteristic function of $B_{h}$, we have:

(3.12) $\chi_{h}E^{h}\rightharpoonup E\quad\mbox{weakly in
}L^{2}(\Omega^{1},\mathbb{R}^{3\times 3}).$

The below more convenient form of the equilibrium condition will be repeatedly
used in the proof of Theorem 1.1.

######  Lemma 3.3.

Condition (1.7) is equivalent to:

(3.13)
$\begin{split}&\int_{\Omega^{1}}\left\langle(\bar{R}^{h})^{T}R^{h}E^{h}(x^{\prime},x_{3})(R^{h})^{T}\bar{R}^{h}:\nabla\phi(y^{h}(x^{\prime},x_{3}))\right\rangle~{}\mathrm{d}x_{3}~{}\mathrm{d}x^{\prime}\\\
&\qquad\qquad\qquad\qquad\qquad=h\int_{\Omega^{1}}\left\langle
f(x^{\prime})e_{3},\bar{R}^{h}\phi(y^{h}(x^{\prime},x_{3}))\right\rangle~{}\mathrm{d}x_{3}~{}\mathrm{d}x^{\prime},\end{split}$

for each $\phi\in\mathcal{C}_{b}^{1}(\mathbb{R}^{3},\mathbb{R}^{3})$ with
$\mathrm{div}\phi=0$.

###### Proof.

For a given divergence free
$\phi\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$, consider:

$\psi(y)=\bar{R}^{h}\phi\left((\bar{R}^{h})^{T}y-c^{h}\right),$

which satisfies $\psi\in\mathcal{C}^{1}_{b}$ and $\mbox{div}\psi=0$, and
moreover:

$\nabla\psi\left(u^{h}(x^{\prime},hx_{3})\right)=\bar{R}^{h}\nabla\phi\left(y^{h}(x^{\prime},x_{3})\right)(\bar{R}^{h})^{T}.$

Use now (1.7) with the divergence-free test function $\psi$:

$\begin{split}&\int_{\Omega}\int_{-1/2}^{1/2}\left\langle DW\left(\nabla
u^{h}(x^{\prime},hx_{3})\right)\left(\nabla
u^{h}(x^{\prime},hx_{3})\right)^{T}:\bar{R}^{h}\nabla\phi(y^{h}(x^{\prime},x_{3}))(\bar{R}^{h})^{T}\right\rangle~{}\mbox{d}x_{3}~{}\mbox{d}x^{\prime}\\\
&\hskip
170.71652pt=h^{3}\int_{\Omega}\int_{-1/2}^{1/2}f(x^{\prime})e_{3}\cdot\bar{R}^{h}\phi(y^{h}(x^{\prime},x_{3}))~{}\mbox{d}x_{3}~{}\mbox{d}x^{\prime}.\end{split}$

The formula (3.13) now follows directly, in view of:

$\begin{split}DW(\nabla u^{h}(x^{\prime},hx_{3}))(\nabla
u^{h}(x^{\prime},hx_{3}))^{T}&=R^{h}DW(\mbox{Id}+h^{2}G^{h}(x))(\mbox{Id}+h^{2}G^{h}(x))^{T}(R^{h})^{T}\\\
&=h^{2}R^{h}E^{h}(x^{\prime},x_{3})(R^{h})^{T}.\end{split}$

##  4\. Identification of the operators in (1.12) – (1.13)

######  Lemma 4.1.

Let $G\in\mathbb{R}^{3\times 3}$ and a symmetric matrix
$E\in\mathbb{R}^{3\times 3}$ satisfy:

$\mathcal{L}_{3}(G)=E,\quad\mathrm{Tr}~{}G=0\quad\mbox{and}\quad
E_{13}=E_{23}=0.$

Then:

(4.1)
$\mathcal{L}_{2}^{in}(G^{\prime\prime})=E^{\prime\prime}-E_{33}\mathrm{Id}_{2}.$

###### Proof.

Since $\mathcal{L}$ and $\mathcal{Q}$ depend only on the symmetric parts of
their arguments, we may without loss of generality assume that $G$ is
symmetric.

Firstly, by definitions in (1.5), (1.6), it follows that for every
$F^{\prime\prime}\in\mathbb{R}^{2\times 2}$ there is a unique tangential
minimizer $d=d(F^{\prime\prime})\in\mathbb{R}^{2}$, in the sense that:

(4.2)
$\mathcal{Q}_{2}^{in}(F^{\prime\prime})=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}F^{\prime\prime}&d\\\
d&-\mbox{Tr }F^{\prime\prime}\end{array}\right])\quad\mbox{ and
}\quad\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}F^{\prime\prime}&d\\\
d&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}0&c\\\
c&0\end{array}\right]\Big{\rangle}=0\quad\forall c\in\mathbb{R}^{2}.$

The second identity above is just the Euler-Lagrange equation for the
minimization in (1.5). By convexity of this minimization problem, it also
follows that $d$ is linear:

(4.3)
$d(F^{\prime\prime}+G^{\prime\prime})=d(F^{\prime\prime})+d(G^{\prime\prime})$

Observe now that:

$\begin{split}\mathcal{Q}_{2}(G^{\prime\prime})&=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right])=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]):\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\Big{(}E+\mathcal{L}_{3}(\left[\begin{array}[]{cc}0&d(G^{\prime\prime})-G_{13,23}\\\
d(G^{\prime\prime})-G_{13,23}&0\end{array}\right])\Big{)}:\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}:G^{\prime\prime}\rangle+E_{33}G_{33}\\\
&\qquad\qquad\quad+\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&G_{33}\end{array}\right]):\left[\begin{array}[]{cc}0&d(G^{\prime\prime})-G_{13,23}\\\
d(G^{\prime\prime})-G_{13,23}&0\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}:G^{\prime\prime}\rangle+E_{33}G_{33}=\langle
E:G\rangle=\mathcal{Q}_{3}(G),\end{split}$

where we repeatedly used the assumptions on $G$ and $E$, and (4.2).
Consequently, by uniqueness of the minimizer $d$, it follows that:

(4.4) $d(G^{\prime\prime})=G_{13,23}.$

Take any $F^{\prime\prime}\in\mathbb{R}^{2\times 2}$. By (4.2) and (4.3), we
see that:

$\mathcal{Q}_{2}(G^{\prime\prime}+F^{\prime\prime})=\mathcal{Q}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}+F^{\prime\prime}&d(G^{\prime\prime})+d(F^{\prime\prime})\\\
d(G^{\prime\prime})+d(F^{\prime\prime})&G_{33}-\mbox{Tr
}F^{\prime\prime}\end{array}\right]).$

Expanding the above and removing $\mathcal{Q}_{2}(G^{\prime\prime})$ and
$\mathcal{Q}_{2}(F^{\prime\prime})$ from both sides, we obtain:

$\begin{split}\langle\mathcal{L}_{2}(G^{\prime\prime}):F^{\prime\prime}\rangle&=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&-\mbox{Tr
}G^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\mathcal{L}_{3}(\left[\begin{array}[]{cc}G^{\prime\prime}&d(G^{\prime\prime})\\\
d(G^{\prime\prime})&-\mbox{Tr
}G^{\prime\prime}\end{array}\right]):\left[\begin{array}[]{cc}F^{\prime\prime}&0\\\
0&-\mbox{Tr }F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\
&=\Big{\langle}\mathcal{L}_{3}(G):\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}=\Big{\langle}E:\left[\begin{array}[]{cc}F^{\prime\prime}&d(F^{\prime\prime})\\\
d(F^{\prime\prime})&-\mbox{Tr
}F^{\prime\prime}\end{array}\right]\Big{\rangle}\\\ &=\langle
E^{\prime\prime}-E_{33}\mbox{Id}_{2}:F^{\prime\prime}\rangle,\end{split}$

by (4.4) and assumptions on $E$ and $G$. The expression (4.1) follows now
directly.

In section 5 below we shall prove that for almost every $x\in\Omega^{1}$ there
holds:

(4.5) $\mbox{Tr }G(x)=0\quad\mbox{and}\quad E_{13}(x)=E_{23}(x)=0.$

Therefore, recalling Lemma 3.2 (iii), we observe that the limiting stress and
strain satisfy the assumptions of Lemma 4.1 pointwise almost everywhere. We
now record the following simple conclusion which will be used in deriving the
Euler-Lagrange equations (1.12), (1.13).

######  Lemma 4.2.

Let $E,G\in L^{2}(\Omega^{1},\mathbb{R}^{3\times 3})$ be the limiting strain
and stress as in Lemma 3.2, which are related to $(w,u)$ by (3.9). Then, for
almost every $x^{\prime}\in\Omega$, there holds:

(4.6)
$\begin{split}&\int_{-1/2}^{1/2}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\mathcal{L}_{2}^{in}\left(\mathrm{sym}\nabla
w+\frac{1}{2}\nabla v\otimes\nabla v\right),\\\
&\int_{-1/2}^{1/2}x_{3}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=-\frac{1}{12}\mathcal{L}_{2}^{in}\left(\nabla^{2}v\right).\end{split}$

###### Proof.

By Lemma 5.1, Lemma 5.3, Lemma 4.1 and (3.9) we see that:

$\begin{split}&\int_{-1/2}^{1/2}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\int_{-1/2}^{1/2}\mathcal{L}_{2}^{in}(G^{\prime\prime})~{}\mathrm{d}x_{3}\\\
&\qquad\qquad=\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}G^{\prime\prime}(x^{\prime},x_{3})~{}\mathrm{d}x_{3}\right)=\mathcal{L}_{2}^{in}(G_{0}(x^{\prime}))=\mathcal{L}_{2}^{in}(\mbox{sym
}G_{0}(x^{\prime}))\\\
&\int_{-1/2}^{1/2}x_{3}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2})~{}\mathrm{d}x_{3}=\int_{-1/2}^{1/2}x_{3}\mathcal{L}_{2}^{in}(G^{\prime\prime})~{}\mathrm{d}x_{3}\\\
&\qquad\qquad=\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}x_{3}G^{\prime\prime}(x^{\prime},x_{3})~{}\mathrm{d}x_{3}\right)=-\mathcal{L}_{2}^{in}\left(\int_{-1/2}^{1/2}x_{3}^{2}G_{1}(x^{\prime})~{}\mathrm{d}x_{3}\right)=-\frac{1}{12}\mathcal{L}_{2}^{in}(G_{1}(x^{\prime})).\end{split}$

This concludes the proof, in view of (3.9).

##  5\. Two further properties of $G$ and $E$

In this section we derive the two fundamental properties of the incompressible
stress and strain, allowing for pointwise application of Lemma 4.1, and
ultimately leading to formulas in (4.6).

######  Lemma 5.1.

The limiting strain $G(x)$ is traceless, for almost every $x\in\Omega^{1}$.

###### Proof.

Recall that $\nabla
u^{h}(x^{\prime},hx_{3})=R^{h}(x^{\prime})\big{(}\mbox{Id}+h^{2}G^{h}(x^{\prime},x_{3})\big{)}$.
Therefore:

$1=\det\nabla u^{h}=\det(\mbox{Id}+h^{2}G^{h})=1+h^{2}\mbox{Tr
}G^{h}+h^{4}\mbox{Tr }\mbox{cof }G^{h}+h^{6}\det G^{h},$

and consequently:

(5.1) $\mbox{Tr }G^{h}+h^{2}\mbox{Tr }\mbox{cof }G^{h}+h^{4}\det G^{h}=0.$

Fix an exponent $\gamma\in(\frac{2}{3},2)$ and define
$B_{h}=\\{x\in\Omega^{1};~{}h^{2-\gamma}|G^{h}(x)\leq 1\\}$ as in Lemma 3.2
(iv). Then:

$\begin{split}\int_{\Omega^{1}\setminus B_{h}}|h^{4}\det
G^{h}|&=\int_{\Omega^{1}\setminus B_{h}}|\mbox{Tr }G^{h}+h^{2}\mbox{Tr
}\mbox{cof }G^{h}|\\\ &\leq|\Omega^{1}\setminus
B_{h}|^{1/2}\left(\int_{\Omega^{1}\setminus B_{h}}|\mbox{Tr
}G^{h}|^{2}\right)^{1/2}+h^{2}\int_{\Omega^{1}}|\mbox{Tr }\mbox{cof
}G^{h}|\leq C(h^{2-\gamma}+h^{2}),\end{split}$

where we used (3.11) and the boundedness of $G^{h}$ in $L^{2}(\Omega^{1})$. On
the other hand, we have:

$\int_{B_{h}}|h^{4}\det
G^{h}|=\frac{h^{4}}{h^{6-3\gamma}}\int_{B_{h}}|\det(h^{2-\gamma}G^{h})|\leq
Ch^{3\gamma-2}.$

Hence, by (5.1) and, again the boundedness of $\mbox{Tr }\mbox{cof }G^{h}$ in
$L^{1}(\Omega^{1})$, it follows that:

$\int_{\Omega^{1}}|\mbox{Tr }G^{h}|\leq\int_{\Omega^{1}}|h^{2}\mbox{Tr
}\mbox{cof }G^{h}|+\int_{\Omega^{1}}|h^{4}\det G^{h}|\to 0,\quad\mbox{as }h\to
0.$

Observing that $\mbox{Tr }G^{h}\rightharpoonup\mbox{Tr }G$ weakly in
$L^{2}(\Omega^{1})$, we conclude that $\mbox{Tr }G=0$.

We now prove the remaining property of the strain $E$ in (4.5). The strategy
of proof is the same as in the later proofs of the Euler-Lagrange equations;
we will apply the equilibrium equation (3.13) to appropriate test functions
$\phi^{h}$, such that after passing to the limit with $h\to 0$ only some
chosen terms will survive, yielding the week formulation of (4.5). One
difficulty with (3.13) is that it only allows for globaly bounded $\phi^{h}$.
For this reason, following [11], we introduce a family of truncation functions
$\theta^{h}$ which coincide with the identity on intervals
$(-\omega_{h},\omega_{h})$ with a suitable rate of convergence of
$\omega_{h}\to\infty$.

######  Lemma 5.2.

Let $\\{\omega_{h}\\}$ be a sequence of positive numbers, increasing to
$+\infty$ as $h\to 0$. There exists a sequence of nondecreasing functions
$\theta^{h}\in\mathcal{C}_{b}^{2}(\mathbb{R},\mathbb{R})$ with the following
properties:

(5.2)
$\begin{split}&\theta^{h}(t)=t\quad\forall|t|\leq\omega_{h}\qquad\mbox{and}\qquad\theta^{h}(t)=(\mathrm{sgn}~{}t)\frac{3}{2}\omega_{h}\quad\forall|t|\geq
2\omega_{h}\\\ &|\theta^{h}(t)|\leq t\quad\forall
t\qquad\mbox{and}\qquad\|\theta^{h}\|_{L^{\infty}}\leq\frac{3}{2}\omega_{h}\\\
&\|\frac{\mathrm{d}}{\mathrm{d}t}{\theta^{h}}\|_{L^{\infty}}\leq
1\qquad\mbox{and}\qquad\|\frac{\mathrm{d^{2}}}{\mathrm{d}t^{2}}{\theta^{h}}\|_{L^{\infty}}\leq\frac{C}{\omega_{h}}.\end{split}$

###### Proof.

One may take:

$\theta^{h}(t)=\left\\{\begin{array}[]{ll}t&\mbox{for }|t|\leq\omega_{h}\\\
\displaystyle{(\mbox{sgn
}t)\frac{1}{2}\left(|t|+\omega_{h}+\frac{\omega_{h}}{\pi}\sin\left(\frac{\pi|t|-\omega_{h}}{\omega_{h}}\right)\right)}&\mbox{for
}|t|\in[\omega_{h},2\omega_{h}]\\\ \displaystyle{(\mbox{sgn
}t)\frac{3}{2}\omega_{h}}&\mbox{for }|t|\geq\omega_{h}\end{array}\right.$

######  Lemma 5.3.

The limiting stress $E(x)$ satisfies: $E_{13}(x)=E_{23}(x)=0$ for almost every
$x\in\Omega^{1}$.

###### Proof.

1. Let $\eta=(\eta_{1},\eta_{2})\in\mathcal{C}_{b}^{2}(\mathbb{R}^{3},\mathbb{R}^{2})$ be a given test function, and define:

(5.3) $\eta_{3}(x^{\prime},x_{3})=-\int_{0}^{x_{3}}\mbox{div
}\eta(x^{\prime},s)~{}\mbox{d}s.$

Since $\partial_{3}\eta_{3}=-\mbox{div }\eta$, the following test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$ are
divergence-free:

$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)\vspace{2mm}}\\\
\displaystyle{h^{2}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}\end{array}\right],$

and denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:

$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}&\begin{minipage}{170.71652pt}$$\displaystyle{\left({\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\right)^{2}\partial_{3}\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$
$$\qquad\qquad+~{}\displaystyle{{\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right)\eta\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$\vspace{2mm}\end{minipage}\\\
\hline\cr\displaystyle{h^{2}\nabla_{tan}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}&\begin{minipage}{170.71652pt}$$\vspace{2mm}\displaystyle{h{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\partial_{3}\eta_{3}\left(x^{\prime},\theta^{h}\left(\frac{x_{3}}{h}\right)\right)}$$\end{minipage}\end{array}\right].$

The truncations $\theta^{h}$ are chosen as in Lemma 5.2 and such that:

(5.4) $\lim_{h\to 0}\omega_{h}=+\infty\quad\mbox{ and }\quad
h^{2}\omega_{h}\leq C.$

2. Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:

(5.5)
$\begin{split}h\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},{\theta^{h}}^{\prime\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&\qquad\qquad+h^{2}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},\nabla_{tan}\eta_{3}({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}\\\
&=h^{2}\int_{\Omega^{1}}\Big{\langle}f(x^{\prime})(\bar{R}^{h})_{31,32},\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}+h^{3}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}\eta_{3}({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)).\end{split}$

Now, we will discuss the convergence as $h\to 0$ of each term in (5.5). The
first term converges to $0$, because
$\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}$
is bounded in $L^{1}(\Omega^{1})$ in view of Lemma 3.2 (iii), while
${\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))$
is pointwise bounded by (5.2).

3. The second term in (5.5) when integrated over $\Omega^{1}\setminus B_{h}$, goes to $0$ in view of (3.11) and of the pointwise boundedness of $({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))$ by (5.2). On the other hand, the limit of this integral over $B_{h}$ is the same as the limit of:

(5.6)
$\int_{\Omega^{1}}\Big{\langle}\chi_{h}E^{h}_{13,23},({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\Big{\rangle}~{}\mbox{d}x$

because of (3.3). We now conclude that the integrals in (5.6) converge to:

$\int_{\Omega^{1}}\Big{\langle}E_{13,23},\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x.$

This follows by recalling (3.12) and observing that:

(5.7)
$({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))\rightarrow\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\quad\mbox{
in }L^{2}(\Omega^{1})$

Indeed:

$\begin{split}\int_{\Omega^{1}}&\left|({\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right))^{2}\partial_{3}\eta({y^{h}}^{\prime},\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))-\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\right|^{2}~{}\mbox{d}x\\\
&\leq
C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\right|^{4}\left(|{y^{h}}^{\prime}-x^{\prime}|^{2}+\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-(x_{3}+v(x^{\prime}))\right|^{2}\right)~{}\mbox{d}x\\\
&\qquad+C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)-1\right|^{2}~{}\mbox{d}x\\\
&\leq
C\int_{\Omega^{1}}|{y^{h}}^{\prime}-x^{\prime}|^{2}+\left|\frac{y_{3}^{h}}{h}-(x_{3}+v(x^{\prime}))\right|^{2}~{}\mbox{d}x+C\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}|}{h}\geq\omega_{h}\right\\}}1+\left|\frac{y_{3}^{h}}{h}\right|^{2}~{}\mbox{d}x\end{split}$

converges to $0$ as $h\to 0$, by (3.6), (3.7) and (5.4), proving hence (5.7).

4. The third term in (5.5) is bounded by: $\frac{C}{\omega_{h}}\int_{\Omega^{1}}|E^{h}|$ by (5.2). It therefore converges to $0$ in view of the boundedness of $E^{h}$ in $L^{1}(\Omega^{1})$ and (5.4).

The fourth term in (5.5) is bounded by:

$\begin{split}Ch^{2}\int_{\Omega^{1}}|E^{h}|\left|\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\right|~{}\mbox{d}x&\leq
Ch^{2}\omega_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|+Ch^{2}\int_{\Omega^{1}}\chi_{h}|E^{h}|\frac{|y^{h}_{3}|}{h}\\\
&\leq
Ch^{2}\omega_{h}~{}{o}(1)+Ch^{2}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\left\|\frac{y^{h}_{3}}{h}\right\|_{L^{2}(\Omega^{1})},\end{split}$

and it converges to $0$ by (3.11), (3.12), (5.4) and the boundedness of
$\frac{y^{h}_{3}}{h}$ in $L^{2}(\Omega^{1})$.

Finally, both terms in the right hand side of (5.5) are bounded by:

$\begin{split}Ch^{2}\int_{\Omega^{1}}|f(x^{\prime})|\left(\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\right|+h\left|\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\right|\right)~{}\mbox{d}x\leq
Ch^{2}\int_{\Omega^{1}}|f(x^{\prime})|(1+h\omega_{h})~{}\mbox{d}x\leq
Ch\|f\|_{L^{2}(\Omega)},\end{split}$

which clearly converges to $0$. Above, we used (5.2) and (5.4).

5. In conclusion, passing to the limit with $h\to 0$ in (5.5), results in:

(5.8)
$\int_{\Omega^{1}}\Big{\langle}E_{13,23},\partial_{3}\eta(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x=0\qquad\forall\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{3},\mathbb{R}^{2}).$

We now reproduce an argument from [11], in order to deduce that $E_{13,23}=0.$
Take an arbitrary $\phi\in\mathcal{C}^{2}_{c}(\Omega,\mathbb{R}^{2}).$ Let
$\mathcal{C}_{c}^{\infty}(\Omega,\mathbb{R})\ni{v}_{k}\rightarrow v$ in
$L^{2}(\Omega)$, and define:

$\phi_{k}(x^{\prime},x_{3})=\phi(x^{\prime},x_{3}-v_{k}(x^{\prime})),\qquad\eta(x^{\prime},x_{3})=\int_{0}^{x_{3}}\phi_{k}(x^{\prime},s)~{}\mbox{d}s$

Clearly $\phi_{k}\in\mathcal{C}^{2}_{c}(\mathbb{R}^{3},\mathbb{R}^{2})$,
$\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{3},\mathbb{R}^{2})$, and thus by (5.8)
we obtain:

$0=\int_{\Omega^{1}}\Big{\langle}E_{13,23},\phi_{k}(x^{\prime},x_{3}+v(x^{\prime}))\Big{\rangle}~{}\mbox{d}x=\int_{\Omega^{1}}\Big{\langle}E_{13,23},\phi(x^{\prime},x_{3}+v(x^{\prime})-v_{k}(x^{\prime}))\Big{\rangle}~{}\mbox{d}x$

Passing to the limit with $k\to\infty$, it follows that:

$\int_{\Omega^{1}}E_{13,23}\phi(x^{\prime},x_{3})~{}\mbox{d}x=0\qquad\forall\phi\in\mathcal{C}^{2}_{c}(\Omega,\mathbb{R}^{2})$

which concludes the proof.

##  6\. Derivation of the first Euler-Lagrange equation (1.12)

1. Let $\eta=(\eta_{1},\eta_{2})\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a given test function, and let $\eta_{3}(x^{\prime})=-\mbox{div }\eta(x^{\prime})$. Given $\theta^{h}$ as in Lemma 5.2, with:

(6.1) $\lim_{h\to 0}\omega_{h}=\lim_{h\to 0}h\omega_{h}^{2}=+\infty\quad\mbox{
and }\quad h\omega_{h}\leq C,$

consider the following divergence-free test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$:

$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})\vspace{2mm}}\\\
\displaystyle{h{\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}\end{array}\right],$

Denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:

$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta(x^{\prime})}&\begin{minipage}{85.35826pt}$$\vspace{3mm}\displaystyle{\frac{1}{h}{\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})}$$\end{minipage}\\\
\hline\cr\displaystyle{h{\theta^{h}}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{85.35826pt}$$\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}$$\end{minipage}\end{array}\right].$

2. Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:

(6.2)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\qquad+h\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},{\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta_{3}({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\qquad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},{\theta^{h}}^{\prime\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&=h\int_{\Omega^{1}}\left\langle
f(x^{\prime})(\bar{R}^{h})_{31,32},{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x+h^{2}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}{\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.\end{split}$

Now, we will check convergence as $h\to 0$ of each of the four terms in the
identity (6.2). Regarding the first term, it converges to $0$ when integrated
over $\Omega^{1}\setminus B_{h}$, by (3.11) and by the pointwise boundedness
of
${\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})$
in view of (5.2). On the other hand, the limit of this integral over $B_{h}$
is the same as the limit of:

(6.3)
$\int_{\Omega^{1}}\Big{\langle}\chi_{h}\big{(}{E^{h}}^{\prime\prime}-E^{h}_{33}\mbox{Id}_{2}\big{)}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x,$

because of the convergence in (3.3). Now, the limit of integrals in (6.3)
equals:

$\int_{\Omega^{1}}\Big{\langle}{E}^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x,$

in view of (3.12) and:

$\begin{split}\int_{\Omega^{1}}&\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})-\nabla\eta(x^{\prime})\right|^{2}~{}\mbox{d}x\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|\nabla_{tan}\eta({y^{h}}^{\prime})-\nabla\eta(x^{\prime})\right|^{2}+C\int_{\Omega^{1}}\left|{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right)-1\right|^{2}\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|{y^{h}}^{\prime}-x^{\prime}\right|^{2}~{}\mbox{d}x+C\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|\\\
&\qquad\qquad\leq
C\int_{\Omega^{1}}\left|{y^{h}}^{\prime}-x^{\prime}\right|^{2}~{}\mbox{d}x+\frac{C}{\omega_{h}^{2}},\end{split}$

where we apply (3.7), and then (3.7) and (6.1) to conclude the convergence of
both terms in the right hand side of the above displayed expression to $0$.

3. The second term in (6.2) is bounded by:

$\begin{split}Ch\int_{\Omega^{1}\setminus
B_{h}}{\theta^{h}}&\left(\frac{|y^{h}_{3}|}{h}\right)|E^{h}|~{}\mbox{d}x+Ch\int_{\Omega^{1}}|\chi_{h}E^{h}|\frac{|y^{h}_{3}|}{h}~{}\mbox{d}x\\\
&\leq Ch\omega_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x+C\|y^{h}_{3}\|_{L^{2}(\Omega^{1})}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\end{split}$

and it clearly converges to $0$ by (3.11), (3.12), (3.6) and (6.1).

The third term in (6.2) is bounded by:

$\begin{split}&\frac{C}{h\omega_{h}}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}|E^{h}|~{}\mbox{d}x\leq\frac{C}{h\omega_{h}}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}}\frac{1}{h^{2}}W(\mbox{Id}+h^{2}G^{h})+|G^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h^{3}\omega_{h}}\int_{\Omega^{1}}W\big{(}\nabla
u^{h}(x^{\prime},hx_{3})\big{)}~{}\mbox{d}x+\frac{C}{h\omega_{h}}\|G^{h}\|_{L^{2}(\Omega^{1})}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}_{3}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}\\\
&\qquad\leq
C\left(\frac{h}{\omega_{h}}+\frac{1}{h\omega_{h}^{2}}\right),\end{split}$

by (3.10), (3.7), the boundedness of $G^{h}$ in $L^{2}(\Omega^{1})$ and (1.8).
Then, the right hand side above converges to $0$ by (6.1).

Finally, the right hand side of (6.2) converges to $0$ as well, as it is
bounded by:

$Ch\int_{\Omega^{1}}|f(x^{\prime})|(1+h\omega_{h})~{}\mbox{d}x\leq
Ch\|f\|_{L^{2}(\Omega)}.$

In conclusion, passing to the limit with $h\to 0$ in (6.2) we obtain:

(6.4)
$\int_{\Omega^{1}}\Big{\langle}{E}^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x=0\quad\forall\eta\in\mathcal{C}^{2}_{b}(\mathbb{R}^{2},\mathbb{R}^{2}).$

and thus the Euler-Lagrange equation (1.12) follows directly, in view of (4.6)
and the density of test functions $\eta$ as above in
$W^{1,2}(\Omega,\mathbb{R}^{2})$.

##  7\. Derivation of the second Euler-Lagrange equation (1.13)

######  Lemma 7.1.

For every $\eta_{3}\in\mathcal{C}_{b}^{3}(\mathbb{R}^{2},\mathbb{R})$, it
follows that:

(7.1)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):\nabla
v\otimes\nabla\eta_{3}\Big{\rangle}\mathrm{d}x~{}+~{}&\lim_{h\to
0}\frac{1}{h}\int_{\Omega^{1}}\left\langle
E^{h}_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle\mathrm{d}x\\\
&\qquad=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\eta_{3}(x^{\prime})~{}\mathrm{d}x^{\prime}.\end{split}$

###### Proof.

1. Given $\eta_{3}\in\mathcal{C}_{b}^{3}(\mathbb{R}^{2},\mathbb{R})$ consider the divergence-free test functions $\phi^{h}\in\mathcal{C}_{b}^{!}(\mathbb{R}^{3},\mathbb{R}^{3})$:

$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}0\vspace{2mm}\\\
\displaystyle{\frac{1}{h}\eta_{3}(x^{\prime})}\end{array}\right],\quad\mbox{so
that}\quad\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}0&\begin{minipage}{28.45274pt}$$\vspace{3mm}0$$\end{minipage}\\\
\hline\cr\displaystyle{\frac{1}{h}\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{28.45274pt}$$\vspace{3mm}0$$\end{minipage}\end{array}\right].$

Applying the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:

(7.2)
$\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},\nabla_{tan}\eta_{3}({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}^{h}\int_{\Omega^{1}}f(x^{\prime})\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.$

Recall that the tensor field $A^{h}$ in (3.4) is defined as:
$A^{h}(x^{\prime})=\frac{1}{h}\left((\bar{R}^{h})^{T}R^{h}(x^{\prime})-\mbox{Id}\right)$.
Hence:

(7.3)
$\begin{split}\frac{1}{h}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}=A^{h}E^{h}(R^{h})^{T}\bar{R}^{h}+E^{h}(A^{h})^{T}+\frac{1}{h}E^{h},\end{split}$

and therefore the left hand side of (7.2) can be written as:

(7.4)
$\begin{split}\int_{\Omega^{1}}&\left\langle(A^{h}E^{h}(R^{h})^{T}\bar{R}^{h})_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x\\\
&+\int_{\Omega^{1}}\left\langle(E^{h}(A^{h})^{T})_{31,32},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\left\langle
E_{31,32}^{h},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mbox{d}x.\end{split}$

2. Let the sets $B_{h}$ be defined as in Lemma 3.2 (iv), for some exponent $\gamma\in(0,1)$. The first two terms in (7.4), when considered on $\Omega^{1}\setminus B_{h}$, converge to $0$ because they are bounded by:

$C\int_{\Omega^{1}\setminus
B_{h}}|A^{h}||E^{h}|~{}\mbox{d}x\leq\frac{C}{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x\leq\frac{C}{h}h^{2-\gamma},$

in view of (3.11) and $|A^{h}|\leq\frac{C}{h}$. On the other hand, the same
two terms while on $B_{h}$, converge to:

$\int_{\Omega^{1}}\left\langle(AE)_{31,32},\nabla\eta_{3}(x^{\prime})\right\rangle+\left\langle(EA^{T})_{31,32},\nabla\eta_{3}(x^{\prime})\right\rangle~{}\mbox{d}x,$

where we used the convergence (3.12) and the following strong convergences in
$L^{3}(\Omega^{1})$: of $A^{h}$ to $A$ by (3.5), of $(R^{h})^{T}\bar{R}^{h}$
to Id by (3.3), and of $\nabla\eta_{3}({y^{h}}^{\prime})$ to
$\nabla\eta_{3}(x^{\prime})$ in view of the Sobolev embedding and the strong
convergence in $W^{1,2}(\Omega^{1},\mathbb{R}^{2})$ in (3.6).

Concluding, the first two terms in (7.4) converge to:

$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}\nabla
v,\nabla\eta_{3}(x^{\prime})\Big{\rangle}-\Big{\langle}E_{33}\nabla
v,\nabla\eta_{3}(x^{\prime})\Big{\rangle}~{}\mbox{d}x$

in view of the structure of the limiting tensor $A$ in (3.5). Since the right
hand side of (7.2) converges to
$\bar{R}_{33}\int_{\Omega}f(x^{\prime})\eta_{3}(x^{\prime})$ by (3.6), passing
to the limit in all terms of (7.2) yields the desired equality (7.1) and thus
proves the lemma.

######  Lemma 7.2.

For every $\eta\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$, it
follows that:

(7.5)
$\begin{split}\int_{\Omega^{1}}&\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):(x_{3}+v(x^{\prime}))\nabla_{tan}\eta(x^{\prime})\Big{\rangle}~{}\mathrm{d}x\\\
&+\int_{\Omega^{1}}\Big{\langle}(E^{\prime\prime}-E_{33}\mathrm{Id}_{2}):\nabla
v(x^{\prime})\otimes\eta(x^{\prime})\Big{\rangle}~{}\mathrm{d}x~{}+~{}\lim_{h\to
0}\frac{1}{h}\int_{\Omega^{1}}\left\langle
E^{h}_{13,23},\nabla\eta_{3}({y^{h}}^{\prime})\right\rangle~{}\mathrm{d}x=0.\end{split}$

###### Proof.

1. Let $\eta\in\mathcal{C}_{b}^{2}(\mathbb{R}^{2},\mathbb{R}^{2})$ be a given test function, and define $\eta_{3}(x^{\prime})=-\mbox{div }\eta(x^{\prime})$. Given $\theta^{h}$ as in Lemma 5.2, with:

(7.6) $\lim_{h\to 0}\omega_{h}=\lim_{h\to 0}h\omega_{h}=+\infty\quad\mbox{ and
}\quad\lim_{h\to 0}h^{1+\frac{1-\gamma}{2}}\omega_{h}=0~{}~{}\mbox{ for some
fixed }\gamma\in(0,1),$

consider the divergence-free test functions
$\phi^{h}\in\mathcal{C}^{1}_{b}(\mathbb{R}^{3},\mathbb{R}^{3})$:

$\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta(x^{\prime})\vspace{2mm}}\\\
\displaystyle{\frac{h}{2}({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\eta_{3}(x^{\prime})}\end{array}\right].$

Denoting $\nabla_{tan}$ the gradient in the tangential directions
$e_{1},e_{2}$, we have:

$\nabla\phi^{h}(x^{\prime},x_{3})=\left[\begin{array}[]{c|c}\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\nabla_{tan}\eta(x^{\prime})}&\begin{minipage}{199.16928pt}$$\vspace{3mm}\displaystyle{\frac{1}{h}\left({\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)+({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\right)\eta(x^{\prime})}$$\end{minipage}\\\
\hline\cr\displaystyle{\frac{h}{2}({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\nabla_{tan}\eta_{3}(x^{\prime})}&\begin{minipage}{199.16928pt}\vspace{3mm}$$\displaystyle{{\theta^{h}}^{\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)\eta_{3}(x^{\prime})}$$\end{minipage}\end{array}\right].$

2. Applying now the equilibrium equation (3.13) with $\phi=\phi^{h}$, we obtain:

(7.7)
$\begin{split}\int_{\Omega^{1}}\Big{\langle}&\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}^{\prime\prime}-\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{33}\mbox{Id}_{2}:{\theta^{h}}^{\prime}\left(\frac{y^{h}_{3}}{h}\right){\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right)\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\quad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{13,23},\left({\theta^{h}}^{\prime\prime}\left(\frac{x_{3}}{h}\right){\theta^{h}}\left(\frac{x_{3}}{h}\right)+({\theta^{h}}\left(\frac{x_{3}}{h}\right))^{2}\right)\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&\qquad\quad+\frac{h}{2}\int_{\Omega^{1}}\Big{\langle}\Big{(}(\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\Big{)}_{31,32},({\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right))^{2}\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}\\\
&=h\int_{\Omega^{1}}\Big{\langle}f(x^{\prime})(\bar{R}^{h})_{31,32},{\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&\qquad\qquad+\frac{h^{2}}{2}\int_{\Omega^{1}}f(x^{\prime})(\bar{R}^{h})_{33}({\theta^{h}}\left(\frac{y^{h}_{3}}{h}\right))^{2}\eta_{3}({y^{h}}^{\prime})~{}\mbox{d}x.\end{split}$

In what follows, we will check convergence as $h\to 0$ of each of the five
terms in the identity (7.7). We first easily notice that the two terms in the
right hand side converge to $0$, as they are bounded by:

$\begin{split}C\int_{\Omega^{1}}|f(x^{\prime})|\Big{(}h\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\right|+h^{2}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\right|^{2}\Big{)}~{}\mbox{d}x&\leq
C\int_{\Omega^{1}}|f(x^{\prime})|\left(|y_{3}^{h}|+|y_{3}^{h}|^{2}\right)~{}\mbox{d}x\\\
&\leq
C\|f\|_{L^{2}(\Omega^{1})}\left(\|y_{3}^{h}\|_{L^{2}(\Omega^{1})}+\|y_{3}^{h}\|^{2}_{L^{4}(\Omega^{1})}\right).\end{split}$

Since $\frac{y_{3}^{h}}{h}$ has a strong limit in $W^{1,2}(\Omega^{1})$ by
(3.6), it results that $\|y_{3}^{h}\|_{L^{2}}$ and $\|y_{3}^{h}\|_{L^{4}}$
converge to $0$.

3. The third term in (7.7) is bounded by the following expression, in view of (5.2), (3.12), (3.6) and (3.11):

$\begin{split}Ch\int_{\Omega^{1}}\chi_{h}|E^{h}|&(\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))^{2}~{}\mbox{d}x+Ch\int_{\Omega^{1}}(1-\chi_{h})|E^{h}|(\theta^{h}\left(\frac{y_{3}^{h}}{h}\right))^{2}~{}\mbox{d}x\\\
&\leq
Ch\int_{\Omega^{1}}\chi_{h}|E^{h}|\left|\frac{y_{3}^{h}}{h}\right|^{2}~{}\mbox{d}x+Ch\omega^{2}_{h}\int_{\Omega^{1}\setminus
B_{h}}|E^{h}|~{}\mbox{d}x\\\ &\leq
Ch\|\chi_{h}E^{h}\|_{L^{2}}\left\|\frac{y^{h}_{3}}{h}\right\|^{2}_{L^{4}}+Ch\omega_{h}^{2}h^{2-\gamma}\leq
Ch+C\left(h^{1+\frac{1-\gamma}{2}}\omega_{h}\right)^{2}\end{split}$

which converges to $0$ by (7.6).

4. We will now investigate the first term in (7.7). Integrated on $\Omega^{1}\setminus B_{h}$, it is bounded by:

$C\omega_{h}\int_{\Omega^{1}}(1-\chi_{h})|E^{h}|~{}\mbox{d}x\leq
C\omega_{h}h^{2-\gamma}\leq Ch^{1+\frac{1-\gamma}{2}}\omega_{h},$

by (3.11) and hence it converges to $0$ through (7.6). The same term
integrated on $B_{h}$ equals now the following sum:

(7.8)
$\begin{split}&\int_{\Omega^{1}}\left({\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)-1\right)\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\cdot\\\
&\qquad\cdot\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)^{\prime\prime}-\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{33}\mbox{Id}_{2}:\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&+\int_{\Omega^{1}}\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)\cdot\\\
&\qquad\cdot\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)^{\prime\prime}-\left((\bar{R}^{h})^{T}R^{h}\chi_{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{33}\mbox{Id}_{2}:\nabla_{tan}\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x.\end{split}$

The first term in (7.8) goes to $0$, as it is bounded by:

$C\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\left|\frac{y_{3}^{h}}{h}\right||\chi_{h}E^{h}|~{}\mbox{d}x\leq
C\left\|\frac{y_{3}^{h}}{h}\right\|_{L^{4}(\Omega^{1})}\left|\left\\{x\in\Omega^{1};~{}\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}\right|^{1/4}\|\chi_{h}E^{h}\|_{L^{2}(\Omega^{1})}\leq\frac{C}{\omega_{h}^{1/2}},$

in view of (5.2), (3.7), (3.12) and recalling (7.6). The second term of (7.8)
converges to:

(7.9)
$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:(x_{3}+v(x^{\prime}))\nabla_{tan}\eta({x}^{\prime})\Big{\rangle}~{}\mbox{d}x$

because of (3.12) and through the following strong convergences: convergence
of $\nabla_{tan}\eta({y^{h}}^{\prime})$ to $\nabla_{tan}\eta(x^{\prime})$ in
$L^{5}(\Omega^{1})$ by (3.6), of $(\bar{R}^{h})^{T}R^{h}$ to Id in
$L^{20}(\Omega)$ by (3.3), and of $\theta^{h}\left(\frac{y^{h}_{3}}{h}\right)$
to $(x_{3}+v(x^{\prime}))$ in $L^{5}(\Omega^{1})$. The last convergence can be
seen from:

$\begin{split}\int_{\Omega^{1}}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-(x_{3}+v(x^{\prime}))\right|^{5}~{}\mbox{d}x&\leq
C\int_{\Omega^{1}}\left|\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)-\frac{y_{3}^{h}}{h}\right|^{5}~{}\mbox{d}x+C\int_{\Omega^{1}}\left|\frac{y_{3}^{h}}{h}-(x_{3}+v(x^{\prime}))\right|^{5}~{}\mbox{d}x\\\
&\leq
C\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\left|\frac{y_{3}^{h}}{h}\right|^{5}~{}\mbox{d}x+o(1)\leq\frac{C}{\omega_{h}^{1/3}}+o(1)\leq
o(1)\end{split}$

by (3.6), (3.7) and (7.6). Concluding, we obtain that the first term in (7.7)
converges to the expression in (7.9).

5. Regarding the second term in (7.7), using (3.10), (5.2), (3.1) and (3.7) we note that:

$\begin{split}\Big{|}\int_{\Omega^{1}}&\left({\theta^{h}}^{\prime\prime}\left(\frac{y_{3}^{h}}{h}\right)\theta^{h}\left(\frac{y_{3}^{h}}{h}\right)+{\theta^{h}}^{\prime}\left(\frac{y_{3}^{h}}{h}\right)^{2}-1\right)\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\Big{|}\\\
&\qquad\leq\frac{C}{h}\int_{\left\\{x\in\Omega^{1};~{}\frac{|y_{3}^{h}(x)|}{h}\geq\omega_{h}\right\\}}\left(\frac{1}{\omega_{h}}\omega_{h}+1\right)|E^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h}\int_{\left\\{\frac{|y_{3}^{h}|}{h}\geq\omega_{h}\right\\}}\frac{1}{h^{2}}W(\nabla
u^{h}(x^{\prime},hx_{3}))+|G^{h}|~{}\mbox{d}x\\\
&\qquad\leq\frac{C}{h}\left(h^{2}+\|G^{h}\|_{L^{2}(\Omega^{1}}\left|\left\\{x\in\Omega^{1};~{}\frac{|y^{h}(x)|}{h}\geq\omega_{h}\right\\}\right|^{1/2}\right)\leq\frac{C}{h}\left(h^{2}+\frac{1}{\omega_{h}}\right),\end{split}$

which converges to $0$ by (7.6). The remaining part of the second term in
(7.7) is:

(7.10)
$\begin{split}\frac{1}{h}\int_{\Omega^{1}}&\Big{\langle}\left((\bar{R}^{h})^{T}R^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&=\int_{\Omega^{1}}\Big{\langle}\left(A^{h}E^{h}(R^{h})^{T}\bar{R}^{h}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x+\int_{\Omega^{1}}\Big{\langle}\left(E^{h}(A^{h})^{T}\right)_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&\qquad+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x,\end{split}$

where we used the decomposition (7.3). Now, exactly as in the proof of Lemma
7.1 and recalling the block structure of the limiting tensor $A$ in (3.5), we
see that (7.10) converges to:

$\begin{split}&\int_{\Omega^{1}}\Big{\langle}\left(AE\right)_{13,23},\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\int_{\Omega^{1}}\Big{\langle}\left(EA^{T}\right)_{13,23},\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x\\\
&=\int_{\Omega^{1}}\Big{\langle}\left(E^{\prime\prime}-E_{33}\mbox{Id}_{2}\right)\nabla
v,\eta(x^{\prime})\Big{\rangle}~{}\mbox{d}x+\frac{1}{h}\int_{\Omega^{1}}\Big{\langle}(E^{h})_{13,23},\eta({y^{h}}^{\prime})\Big{\rangle}~{}\mbox{d}x.\end{split}$

In conclusion, passing to the limit in (7.7) clearly yields (7.5) and achieves
the lemma.

Proof of the second Euler-Lagrange equation (1.13).
Let now $\xi\in\mathcal{C}^{3}_{b}(\mathbb{R}^{2},\mathbb{R})$. Applying Lemma
7.1 with $\eta_{3}=\xi$, and Lemma 7.2 with $\eta=\nabla\xi$, it follows:

(7.11)
$-\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:(x_{3}+v(x^{\prime})\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\xi(x^{\prime})~{}\mbox{d}x^{\prime}.$

By the first Euler-Lagrange equation in (6.4) applied with $\eta=v\nabla\xi\in
W^{2,2}(\Omega,\mathbb{R}^{2})$, we see that:

$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla
v\otimes\nabla\xi+v(x^{\prime})\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=0.$

Thus, (7.11) becomes:

$\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:\nabla
v\otimes\nabla\xi\Big{\rangle}~{}\mbox{d}x-\int_{\Omega^{1}}\Big{\langle}E^{\prime\prime}-E_{33}\mbox{Id}_{2}:x_{3}\nabla^{2}\xi\Big{\rangle}~{}\mbox{d}x=\bar{R}_{33}\int_{\Omega}f(x^{\prime})\xi(x^{\prime}).$

The equality in (1.13) follows now from the above in view of (4.6), and by the
density of test functions $\xi\in\mathcal{C}^{3}_{b}$ in
$W^{2,2}(\Omega,\mathbb{R})$.

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