# CMZ_4_8_9am

elvira.zappale  Gabriel Carvalho111Instituto Superior Técnico, University of
Lisbon, Av. Rovisco Pais $1049-001$ Lisboa, e-mail:
<EMAIL_ADDRESS>, José Matias 222Department of
Mathematics, Instituto Superior Técnico, University of Lisbon, Av. Rovisco
Pais $1049-001$ Lisboa, email<EMAIL_ADDRESS>and Elvira
Zappale333Department of Basic and Applied Sciences for Engineering, Sapienza -
University of Rome, via A. Scarpa, 16 (00161) Roma (SA), e-mail:
<EMAIL_ADDRESS>
(August 2023; )

#  Asymptotic analysis of a clamped thin multidomain allowing for fractures
and discontinuities

elvira.zappale  Gabriel Carvalho111Instituto Superior Técnico, University of
Lisbon, Av. Rovisco Pais $1049-001$ Lisboa, e-mail:
<EMAIL_ADDRESS>, José Matias 222Department of
Mathematics, Instituto Superior Técnico, University of Lisbon, Av. Rovisco
Pais $1049-001$ Lisboa, email<EMAIL_ADDRESS>and Elvira
Zappale333Department of Basic and Applied Sciences for Engineering, Sapienza -
University of Rome, via A. Scarpa, 16 (00161) Roma (SA), e-mail:
<EMAIL_ADDRESS>
(August 2023; )

###### Abstract

We consider a thin multidomain of $\mathbb{R}^{3},$ consisting of a vertical
rod upon a horizontal disk. The equilibrium configurations of the thin
hyperelastic multidomain, allowing for fracture and damage, are described by
means of a bulk energy density of the kind $W(\nabla U)$, where $W$ is a Borel
function with linear growth and $\nabla U$ denotes the gradient of the
displacement, i.e. a vector valued function $U:\Omega\to\mathbb{R}^{3}$. By
assuming that the two volumes tend to zero, under suitable boundary conditions
and loads, and suitable assumptions of the rate of convergence of the two
volumes, we prove that the limit model is well posed in the union of the limit
domains, with dimensions, respectively, $1$ and $2$.

Keywords: junction in thin multidomains, beam, wire, thin film, bounded
variation, dimension reduction, ${\Gamma}$-convergence.

2020 AMS subject classifications: 49J45, 74B20, 74K10, 74K20, 74K30, 74K35,
78M30, 78M35.

##  1 Introduction

Thin structures arise in many applications, such as crystal plasticity,
modeling of magneto-elastic devices, ribbons, heterogeneous multi-structures,
etc. and their study is usually addressed through dimensional reduction
techniques, where an asymptotic analysis from high dimensions to lower ones is
conducted. In this process, one looks to find precise relations between the
initial models and the limit ones (see among a much wider literature [1], [3],
[5], [6], [7], [10], [11], [14], [15], [16],[18], [22],[24], [23], [27], [28],
[29], [44], [45, 46, 47, 48], and the references therein). For a recent survey
with an analytical perspective we refer to [4], while we refer to [17, 18] for
the basis of the theory. Following the dimension reduction problems for the
junction of two thin cylinders that have been addressed previously in [25],
[32], [33],[34], [35],[38], [39],[40], [41] and [43], we consider a thin
multidomain of ${\mathbb{R}}^{3}$ with a bulk energy density of the kind
$W(\nabla U)$, where $W$ is a Borel function with linear growth, and $\nabla
U$ denotes the gradient of a field function $U\in W^{1,1}$. By assuming that
the volumes of the two cylinders tend to zero with different assumptions on
their rates, under suitable boundary conditions on the top of the vertical
cylinder and on the lateral surface of the horizontal cylinder, and under body
loads, we derive the limit energy in the space $BV$, allowing for
discontinuities, damage and fractures, thus generalizing to the case of
multistructures, the results contained in [5], leaving for a forthcoming paper
the bending effects. We emphasize that fracture may appear in the limit model.
Moreover, in contrast with [12, 16], in this work we did not consider any
initial surface energy. More precisely, being $r_{n}$ the radius of the top
cylinder and $h_{n}$ the height of the bottom cylinder and
$\ell=\lim_{n\to\infty}\frac{h_{n}}{r_{n}^{2}}$, we analyse the cases
$\ell\in]0,+\infty[,\ell=0$ and $\ell=+\infty$ (cf. (1.21), (1.24) and
(1.27)). In this last case, due to our technical assumptions, we consider only
some specific setting, i.e. we deal with two different situations, namely the
case where $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$ and
$\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=\infty.$ In any situation described
above, we prove that the limit problem is well posed in the union of the limit
domains, with dimensions $1$ and $2$, respectively. Applications of this model
problem, in the case $\ell\in]0,+\infty[$, can be found in [36].
Weakly star converging sequences in the Sobolev space $W^{1,1}$ may develop
concentrations and, when dealing with linear growth integrands, the effect is
captured in the limit result in BV, through the recession function of the
integrand. There are different notions of recession function but, when
addressing lower semicontinuity and relaxation problems in $BV$ for integrands
with linear growth (cf. [42]), it is adequate to define its recession function
as a $limsup$ (see definition (1.30)).

Our work relies on $\Gamma$-convergence techniques ( see [21] for a
compreensive treatment) and on [8], where the problem of relaxation of
functionals in BV under trace constraints was first addressed. For the sake of
completeness, we finally address the super-linear case (i.e. $W\in
W^{1,p},\;p>1$, $W$ with growth of order p ) for the cases $\ell=0$ and
$\ell=+\infty$, which were not addressed previously (cf. [31]). The overall
plan of this work is as follows. In the following subsection, after having
introduced the problem in a thin multidomain, we reformulate it on a fixed
domain through appropriate rescalings of the kind proposed by P.G. Ciarlet and
P. Destuynder in [19] and we state our main result, Theorem 1.3. In section 2
we fix notation and state and prove some preliminary results, while section 3
is devoted to the proof of our main theorem. Finally, in section 4 we briefly
present some results in the super-linear case.

###  1.1 The original problem and the rescalings

In the sequel, $\mathbb{R}^{h{\times}k}$, denotes the set of
$h{\times}k$-matrices. In our setting, $h=3$, $k=1,2,3$ and
$x=(x_{1},x_{2},x_{3})=(x_{\alpha},x_{3})$ denotes a generic point of
$\mathbb{R}^{3}$; the gradient with respect to the first $2$ variables is
denoted by $\nabla_{\alpha}$ while the first derivative with respect to the
last variable is represented by $\nabla_{3}$.

We write a generic element in $\mathbb{R}^{3{\times}3}$ as $M=(m_{i,j})_{1\leq
i,j\leq 3}$. For our purposes, it is convenient to decompose a given vector-
valued matrix $M=(m_{i,j})_{1\leq i,j\leq 3}$ as follows: we set
$M_{\alpha}:=(m_{i,j})_{1\leq i\leq 3,1\leq j\leq
2}\in\mathbb{R}^{3{\times}2}$, $M_{3}=(m_{i,3})_{1\leq i\leq
3}\in\mathbb{R}^{3}$.

$\displaystyle M=\left(M_{\alpha}\;\;|\;\;M_{3}\right),$ (1.1)

In particular if $Id$ denotes the identity in $\mathbb{R}^{3\times 3}$, by
$\mathbb{R}^{3\times 2}\ni Id_{\alpha}$ we denote its first two columns.
Further, $|M|$ denotes the norm of $M$ given by
$\big{(}\sum_{i,j=1}^{3}|m_{i,j}|^{2}\big{)}^{1/2}.$

Let $\omega\subset{\mathbb{R}}^{2}$ be a bounded, open, and connected set,
with Lipschitz boundary, such that the origin in ${\mathbb{R}}^{2}$, denoted
by $0_{\alpha}$, belongs to $\omega$. Furthermore for the sake of exposition
we assume that $\mathcal{L}^{2}(\omega)=1$. Let
$\left\\{r_{n}\right\\}_{n\in\mathbb{N}}$,
$\left\\{h_{n}\right\\}_{n\in\mathbb{N}}\subset]0,1[$ be two sequences such
that

$\lim_{n}h_{n}=0=\lim_{n}r_{n},$ (1.2)

and

$\lim_{n}\frac{h_{n}}{r_{n}^{2}}=\ell\in[0,+\infty].$ (1.3)

For every $n\in\mathbb{N}$, consider a thin multidomain consisting of a union
between two vertical cylinders, one placed upon the other, as follows:
$\Omega_{n}=\Omega_{n}^{a}\cup\Omega_{n}^{b}$ ($a$ stands for “above” and $b$
for “below”), where $\Omega_{n}^{a}=r_{n}\omega\times[0,1[$ with small cross
section $r_{n}\omega$ and constant height,
$\Omega_{n}^{b}=\omega\times]-h_{n},0[$ with small thickness $h_{n}$ and
constant cross section (see Figure 1).

$\bullet$$x_{3}$$x_{2}$$x_{1}$$\omega\times\\{0\\}$$h_{n}$$\bullet$$\tiny
r_{n}\omega\times\\{1\\}$$\Omega^{b}_{n}$$\Omega^{a}_{n}$ Figure 1:
Multidomain $\Omega_{n}$

We consider a Borel energy density $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying the following assumptions:

$\displaystyle\frac{1}{C}|M|-C\leq W(M)\leq C(1+|M|),\quad\forall
M\in\mathbb{R}^{3{\times}3};$ (1.4)

for some $C>0$.

For every $n\in\mathbb{N}$, the energy functionals we are interested in are of
the form $I:W^{1,1}(\Omega_{n};\mathbb{R}^{3})\to\mathbb{R}$ defined by

$I[U_{n}]:=\int_{\Omega_{n}}W\left(\nabla
U_{n}(x)\right)dx=\int_{\Omega_{n}}\left(W\left(\nabla_{\alpha}U_{n}(x)\;\;|\;\;\nabla_{3}U_{n}(x)\right)+H_{n}^{{\ell}}(x)\cdot
U_{n}(x)\right)dx,$ (1.5)

where $H_{n}^{\ell}\in L^{\infty}(\Omega_{n};\mathbb{R}^{3})$ and the last
summand represent the loads, that will be specified in the sequel, according
to the considered scaling.

In addition, we require the admissible deformation $U_{n}\in
W^{1,1}(\Omega_{n};\mathbb{R}^{d})$ to satisfy the Dirichlet boundary
condition $c^{a,\ell}+d^{a,\ell}x_{\alpha}$ on the top of $\Omega_{n}^{a}$,
and $f^{b,\ell}+g^{b,\ell}x_{3}$ on the lateral surface of $\Omega_{n}^{b}$,
for some $c^{a,\ell}\in\mathbb{R}^{3}$, $d^{a,\ell}\in\mathbb{R}^{3\times 2}$
and $f^{b,\ell},g^{b,\ell}\in W^{1,1}(\omega;\mathbb{R}^{3})$.

More general loads can be imposed, but for the sake of exposition we consider
the above simplified case. We refer to Remarks 3.7, 3.17 and 3.22 below for
more detailed comments.

Our mail goal is to study the asymptotic behavior as $n\to\infty$ of the
minimization problem

$\inf_{\begin{array}[]{cc}U_{n}\in W^{1,1}(\Omega_{n};\mathbb{R}^{3})\\\
{\hbox{under the above boundary conditions}}\end{array}}I[U_{n}].$ (1.6)

As usual in dimension-reduction problems, we reformulate this minimization
problem on a fixed domain through an appropriate rescaling which maps
$\Omega_{n}$ into $\Omega:=\omega\times]-1,1[$.

Precisely, we set

$u_{n}(x)=\left\\{\begin{array}[]{ll}u^{a}_{n}(x_{\alpha},x_{3})=U_{n}(r_{n}x_{\alpha},x_{3}),\quad(x_{\alpha},x_{3})\in\Omega^{a}:=\omega\times[0,1[,\\\
\\\
u^{b}_{n}(x_{\alpha},x_{3})=U_{n}(x_{\alpha},h_{n}x_{3}),\quad(x_{\alpha},x_{3})\in\Omega^{b}:=\omega\times]-1,0[.\end{array}\right.$

Then, $u^{a}_{n}\in W^{1,1}(\Omega^{a};\mathbb{R}^{3})$ assumes the rescaled
Dirichlet boundary condition $c^{a,\ell}+r_{n}d^{a,\ell}x_{\alpha}$ on the top
of $\Omega^{a}$, while $u^{b}_{n}\in W^{1,1}(\Omega^{b};\mathbb{R}^{3})$
assumes the rescaled Dirichlet boundary condition
$f^{b,\ell}+h_{n}g^{b,\ell}x_{3}$ on the lateral boundary of $\Omega^{b}$.
Moreover, $u_{n}=(u^{a}_{n},u^{b}_{n})$ satisfies the junction condition

$u^{a}_{n}(x_{\alpha},0)=u^{b}_{n}(r_{n}x_{\alpha},0),\hbox{ for a.e.
}x_{\alpha}\in\omega,$ (1.7)

In other words, $u_{n}\in\mathcal{U}_{n}^{\ell}$, where

$\displaystyle\mathcal{U}^{\ell}_{n}=\Big{\\{}$
$\displaystyle(u^{a},u^{b})\in\left(c^{a,\ell}+r_{n}d^{a,\ell}x_{\alpha}+W_{a}^{1,1}(\Omega^{a};\mathbb{R}^{3})\right){\times}\left(f^{b,\ell}+h_{n}g^{b,\ell}x_{3}+W^{1,1}_{b}(\Omega^{b};\mathbb{R}^{3})\right):$
(1.8) $\displaystyle\qquad u^{a}\text{ and }u^{b}\text{ satisfy
\eqref{eq:bc}}\Big{\\}}$

with $W_{a}^{1,1}(\Omega^{a};\mathbb{R}^{3})$ the closure with respect to the
strong topology of $W^{1,1}(\Omega^{a};\mathbb{R}^{3})$ of

$\left\\{u^{a}\in
C^{\infty}(\overline{\Omega^{a}};\mathbb{R}^{3})\,:\,u^{a}=\mathbf{0}\hbox{ in
a neighbourhood of }\omega{\times}\\{1\\}\right\\}$

and respectively $W_{b}^{1,1}(\Omega^{b};\mathbb{R}^{3})$ is the closure with
respect to the strong topology of $W^{1,1}(\Omega^{b};\mathbb{R}^{3})$ of

$\big{\\{}u^{b}\in
C^{\infty}(\overline{\Omega^{b}};\mathbb{R}^{3})\,:\,u^{b}={\bf 0}\hbox{ in a
neighbourhood of }\partial\omega\times]-1,0[\big{\\}}.$

Without loss of generality, one can assume that

$f^{b,\ell}={\bf 0}\hbox{ a.e. in }B,$ (1.9)

for some $2$-dimensional ball $B$ such that $0_{\alpha}\in
B\subset\subset\omega$ ($\bf 0$ and $0_{\alpha}$ denote the null vectors in
$\mathbb{R}^{3}$ and $\mathbb{R}^{2}$, respectively).

We consider now three functionals, depending on the superscript $\ell$ in
(1.3), as rescalings of the energy $I$ in (1.5), and study their asymptotic
behaviour.

######  Definition 1.1.

Let $\ell\in[0,+\infty]$ be as in (1.3), i.e. the limit ratio between the
height of the cylinder $\Omega^{b}$ and the area of the cross section of the
cylinder $\Omega^{a}$.

For $\ell\in\\{q,\infty,0\\}$ with $q\in]0,+\infty[$, let
$F_{n}^{\ell}:\mathcal{U}_{n}^{\ell}\to\mathbb{R}$ be defined as follows:

$\displaystyle F_{n}^{q}:\mathcal{U}_{n}^{q}\to\mathbb{R},\quad$
$\displaystyle
F_{n}^{q}(u^{a}_{n},u^{b}_{n}):=K_{n}^{a,q}(u^{a}_{n})+\frac{h_{n}}{r_{n}^{2}}K_{n}^{b,q}(u^{b}_{n}),$
(1.10) $\displaystyle
F_{n}^{\infty}:\mathcal{U}_{n}^{\infty}\to\mathbb{R},\quad$ $\displaystyle
F_{n}^{\infty}(u^{a}_{n},u^{b}_{n}):=K_{n}^{a,\infty}(u^{a}_{n})+\frac{h_{n}}{r_{n}^{2}}K_{n}^{b,\infty}(u^{b}_{n}),$
(1.11) $\displaystyle F_{n}^{0}:\mathcal{U}_{n}^{0}\to\mathbb{R},\quad$
$\displaystyle
F_{n}^{0}(u^{a}_{n},u^{b}_{n}):=\frac{r_{n}^{2}}{h_{n}}K_{n}^{a,0}(u^{a}_{n})+K_{n}^{b,0}(u^{b}_{n}),$
(1.12)

where $\mathcal{U}_{n}^{q}$, $\mathcal{U}_{n}^{\infty}$ and
$\mathcal{U}_{n}^{0}$ are as in (1.8) with

$\displaystyle c^{a,q}\in\mathbb{R}^{3},\quad$ $\displaystyle d^{a,q}={\bf
0}\in\mathbb{R}^{3\times 2},\quad$ $\displaystyle f^{b,q}\in
W^{1,1}(\omega;\mathbb{R}^{3}),\quad$ $\displaystyle g^{b,q}={\bf
0}\in\mathbb{R}^{3},$ (1.13) $\displaystyle
c^{a,\infty}\in\mathbb{R}^{3},\quad$ $\displaystyle d^{a,\infty}={\bf
0}\in\mathbb{R}^{3\times 2},\quad$ $\displaystyle
f^{b,\infty}(x_{\alpha})=(x_{\alpha},0),\quad$ $\displaystyle
g^{b,\infty}=(0_{\alpha},1),$ (1.14) $\displaystyle
c^{a,0}=(0_{\alpha},1),\quad$ $\displaystyle d^{a,0}=Id_{\alpha},\quad$
$\displaystyle f^{b,0}\in W^{1,1}(\omega;\mathbb{R}^{3}),\quad$ $\displaystyle
g^{b,0}={\bf 0}\in\mathbb{R}^{3},$ (1.15)

and where $K_{n}^{a,\ell}:W^{1,1}(\Omega^{a};\mathbb{R}^{3})\to\mathbb{R}$ and
$K_{n}^{b,\ell}:W^{1,1}(\Omega^{b};\mathbb{R}^{3})\to\mathbb{R}$ are the
functionals defined, respectively, by

$K_{n}^{a,\ell}[u^{a}]:=\int_{{\Omega}^{a}}\left(W\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}(x)}\;\;|\;\;\displaystyle{\nabla_{3}u^{a}}(x)\right)+H_{n}^{a,\ell}(x)\cdot
u^{a}(x)\right)dx,$ (1.16)

$K_{n}^{b,\ell}[u^{b}]:=\int_{{\Omega}^{b}}\left(W\left(\displaystyle{\nabla_{\alpha}u^{b}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}(x)}\right)+H_{n}^{b,\ell}(x)\cdot
u^{b}(x)\right)dx,$ (1.17)

with $H_{n}^{a,\ell}(x):=H_{n}^{\ell}(r_{n}x_{\alpha},x_{3})$ if
$(x_{\alpha},x_{3})\in\Omega^{a}$, and
$H_{n}^{b,\ell}(x):=H_{n}^{\ell}(x_{\alpha},h_{n}x_{3})$ if
$(x_{\alpha},x_{3})\in\Omega^{b}$.

Finally, the rescaled minimization problems corresponding to (1.6) read as

$\inf_{(u^{a}_{n},u^{b}_{n})\in\mathcal{U}_{n}^{\ell}}\left\\{F_{n}^{\ell}(u_{n}^{a},u_{n}^{b})\right\\}.$
(1.18)

In other words, the aim of this paper consists of describing the limit
energies in (1.10), (1.11) and (1.12) as $n\to\infty$, when the volumes of
$\Omega_{n}^{a}$ and $\Omega_{n}^{b}$ tend to zero under different assumptions
on their rate (modelled through the superscript $\ell\in\\{q,0,\infty\\}$). In
particular, according to the values of $\ell$ in (1.3), we will make specific
assumptions on the loads.

$\displaystyle\hbox{ if }\ell=$ $\displaystyle\,q\in\,\,]0,+\infty[,\qquad$
$\displaystyle\left\\{\begin{array}[]{cc}H_{n}^{a,q}\buildrel\star\over{\rightharpoonup}H&\hbox{
in }L^{\infty}(\Omega^{a};{\mathbb{R}}^{3}),\\\
H_{n}^{b,q}\buildrel\star\over{\rightharpoonup}H&\hbox{ in
}L^{\infty}(\Omega^{b};{\mathbb{R}}^{3}),\end{array}\right.$ (1.21)
$\displaystyle\hbox{ if }\ell=$ $\displaystyle+\infty,$
$\displaystyle\left\\{\begin{array}[]{cc}H_{n}^{a,\infty}\buildrel\star\over{\rightharpoonup}H&\hbox{
in }L^{\infty}(\Omega^{a};{\mathbb{R}}^{3}),\\\
\displaystyle\frac{h_{n}}{r_{n}^{2}}H_{n}^{b,\infty}\buildrel\star\over{\rightharpoonup}{\bf
0}&\hbox{ in }L^{\infty}(\Omega^{b};{\mathbb{R}}^{3}),\end{array}\right.$
(1.24) $\displaystyle\hbox{ if }\ell=$ $\displaystyle 0,$
$\displaystyle\left\\{\begin{array}[]{cc}\displaystyle\frac{r_{n}^{2}}{h_{n}}H_{n}^{a,0}\buildrel\star\over{\rightharpoonup}{\bf
0}&\hbox{ in }L^{\infty}(\Omega^{a};{\mathbb{R}}^{3}),\\\
H_{n}^{b,0}\buildrel\star\over{\rightharpoonup}H&\hbox{ in
}L^{\infty}(\Omega^{b};{\mathbb{R}}^{3}).\end{array}\right.$ (1.27)

where $H\in L^{\infty}(\Omega;\mathbb{R}^{3})$.

Our analysis will be conducted by means of $\Gamma$-convergence (referring to
[21] for a comprehensive treatment), with respect to the weak * convergence in
$BV(\Omega;\mathbb{R}^{3})$, which in view of Propositions 3.2, 3.8, and 3.18,
is the natural one. We also stress that, due to the growth condition (1.4) on
$W$, there is no loss of generality in replacing this convergence by the
$L^{1}(\Omega;\mathbb{R}^{3})$ \- strong convergence. In particular we observe
that

######  Remark 1.2.

The parcels in $F_{n}^{\ell}(u_{n}^{a},u_{n}^{b})$ regarding the loads are
continuous with respect to $\Gamma$-convergence, i.e. they converge to the
same value independently of the approximant sequences
$(u_{n}^{a},u_{n}^{b})\in{\mathcal{U}}_{n}^{\ell}$ strongly converging in
$L^{1}$ to $(u^{a},u^{b})$, if we consider the convergences (1.21), (1.24),
(1.27).

For instance if $\ell=q$, then

$\lim_{n}\int_{\Omega^{a}}H_{n}^{a,q}\cdot u_{n}^{a}dx=\int_{\Omega^{a}}H\cdot
u^{a}dx,\quad\lim_{n}\int_{\Omega^{b}}H_{n}^{b,q}\cdot
u_{n}^{b}dx=\int_{\Omega^{b}}H\cdot u^{b}dx,$

along any sequences $u_{n}^{a}\to u^{a}$ in
$L^{1}(\Omega^{a};{\mathbb{R}}^{3})$ and $u_{n}^{b}\to u^{b}$ in
$L^{1}(\Omega^{b};{\mathbb{R}}^{3})$.

For the remaining cases, $\ell=+\infty$ and $\ell=0$ a similar reasoning can
be made.

In view of our latter observation, for which the loads represent continuous
terms with respect to $\Gamma$-convergence, we can neglect them in our
subsequent analysis.

We underline that, for the sake of exposition, in the proof of Propositions
3.2, 3.8, and 3.18, we will consider null loads, referring to the arguments in
[32, Lemma 2.2] and [37, Proposition 5.1] for the case of not null loads. We
stress that in this case either we can choose the coercivity constant in (1.4)
according to the $L^{\infty}$ bounds on the loads, or we can start our
asymptotic analysis considering $L^{1}$ strong converging sequences
$(u_{n}^{a})_{n}$ and $(u_{n}^{b})_{n}$.

The following result summarizes the content of this paper. We refer to Section
2 for the notation adopted in the theorem to represent the derivatives of
fields with Bounded Variation.

######  Theorem 1.3.

Let $W\;:\;{\mathbb{R}}^{3{\times}3}\to{\mathbb{R}}$ be a Borel function
satisfying (1.4), and $F_{n}^{\ell}$ be as in Definition 1.1 depending on the
superscript $\ell$ defined in (1.3). Denote by
$\hat{W}:{\mathbb{R}}^{3}\to\mathbb{R}$ the function given by

$\displaystyle\hat{W}(\xi_{3}):=\inf_{\xi_{\alpha}\in\mathbb{R}^{3\times
2}}W(\xi_{\alpha}|\xi_{3}),$ (1.28)

and let $W_{0}:\mathbb{R}^{3\times 2}\to\mathbb{R}$ be the function defined

$\displaystyle
W_{0}(\xi_{\alpha}):=\inf_{\xi_{3}\in\mathbb{R}^{3}}W(\xi_{\alpha}|\xi_{3}).$
(1.29)

Assume that the recession functions of $\hat{W}$ and $W_{0}$,
$\hat{W}^{\infty}:\mathbb{R}^{3}\to\mathbb{R}$, and
$W_{0}^{\infty}:\mathbb{R}^{3\times 2}\to\mathbb{R}$, defined as

$\hat{W}^{\infty}(\xi):=\limsup_{t\to+\infty}\frac{W(t\xi)}{t},\hbox{ and
}W_{0}^{\infty}(\xi):=\limsup_{t\to+\infty}\frac{W_{0}(t\xi)}{t},$ (1.30)

respectively, satisfy the following: there exist $C>0,0<m<1,L>0$ such that

$\displaystyle\left|(\hat{W})^{\infty}(\xi_{3})-\frac{\hat{W}(t\xi_{3})}{t}\right|\leq\frac{C}{t^{m}}$
(1.31)

for $\xi_{3}\in\mathbb{R}^{3},\|\xi_{3}\|=1$, $t>L$, and

$\displaystyle\left|W_{0}^{\infty}(\xi_{\alpha})-\frac{W_{0}(t\xi_{\alpha})}{t}\right|\leq\frac{C}{t^{m}}$
(1.32)

for $\xi_{\alpha}\in\mathbb{R}^{3\times 2},\|\xi_{\alpha}\|=1$, $t>L$. Let
$\mathcal{U}_{n}^{\ell}$ be the space in (1.8).

Define further $K^{a,{\ell}}:BV(]0,1[;\mathbb{R}^{3})\to{\mathbb{R}}$ and
$K^{b,{\ell}}:BV(\omega;\mathbb{R}^{3})\to{\mathbb{R}}$ as

$\displaystyle
K^{a,\ell}(u^{a}):=\displaystyle{\int_{]0,1[}{\hat{W}}^{**}\left(\nabla_{3}u^{a}(x_{3})\right)}dx$
$\displaystyle+\displaystyle{\int_{]0,1[}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}u^{a}}{d|D^{s}_{3}u^{a}|},\right)}d|D^{s}_{3}u^{a}|$
$\displaystyle+(\hat{W}^{\ast\ast})^{\infty}(c^{a,\ell}-u^{a}(1^{-}))+\int_{\Omega^{a}}H(x)\cdot
u^{a}(x_{3})dx,$ (1.33) $\displaystyle K^{b,\ell}(u^{b}):=$
$\displaystyle\displaystyle{\int_{\omega}QW_{0}\left(\nabla_{\alpha}u^{b}(x_{\alpha})\right)dx_{\alpha}}+\int_{\omega}(QW_{0})^{\infty}\left(\frac{D^{s}_{\alpha}u^{b}}{|D^{s}_{\alpha}u^{b}|}\right)d|D^{s}_{\alpha}u^{b}|+$
$\displaystyle+\int_{\partial\omega\times]-1,0[}(QW_{0})^{\infty}((f^{b,\ell}-(u^{b}))^{-}\otimes\nu_{u^{b}})d\mathcal{H}^{2}+\int_{\Omega^{b}}H(x)\cdot
u^{b}(x_{\alpha})dx,$ (1.34)

where $u^{a}(1^{-})$ represent the internal trace of $u^{a}$, $Q(W_{0})$ and
$\hat{W}^{\ast\ast}$ represent the quasiconvex and the convex envelopes (see
subsection 2.1) of $W_{0}$ and $\hat{W}$ in (1.29) and (1.28), respectively,
and the superscript $\infty$ stands for their recession functions in the sense
of (1.30).

Then the following representations hold:

Case $\ell=q$: Assume that $q\in]0,+\infty[$, that the Dirichlet conditions in
${\mathcal{U}}_{n}^{q}$ are as (1.13) and the loads satisfy (1.21). Then for
every $(u^{a},u^{b})\in
BV(]0,1[;\mathbb{R}^{3}){\times}BV(\omega;\mathbb{R}^{3})$,

$\displaystyle\inf\Big{\\{}$
$\displaystyle\liminf_{n}F_{n}^{q}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q},$
$\displaystyle u_{n}^{a}\buildrel\star\over{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\buildrel\star\over{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\Big{\\}}=K^{a,q}(u^{a})+qK^{b,q}(u^{b}).$
(1.35)

Case $\ell=+\infty$: Assume that the Dirichlet conditions in
${\mathcal{U}}_{n}^{\infty}$ are as in (1.14) and the loads satisfy (1.24).
Assume further that there exists $C>0$ such that

$\frac{1}{C}|M-Id|\leq W(M),\quad W(Id)=0,$ (1.36)

for every $M\in\mathbb{R}^{3\times 3}$, with $Id$ being the identity matrix in
$\mathbb{R}^{3\times 3}$.

We consider two different subcases.

  * –

If $\displaystyle\frac{h_{n}}{r_{n}}\to 0$ then for every $u^{a}\in
BV(]0,1[;\mathbb{R}^{3})$,

$\displaystyle\inf\Big{\\{}$
$\displaystyle\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},$
$\displaystyle u_{n}^{a}\buildrel\star\over{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\Big{\\}}=K^{a,\infty}(u^{a})$ (1.37)

  * –

If $\displaystyle\frac{h_{n}}{r_{n}}\to\infty$ then for every $u^{a}\in
BV(]0,1[;\mathbb{R}^{3})$,

$\displaystyle\inf\Big{\\{}$
$\displaystyle\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},u_{n}^{a}\buildrel\star\over{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),$ $\displaystyle
u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\Big{\\}}=K^{a,\infty}(u^{a})+(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+}))$
(1.38)

where $u^{a}(0^{+})$ is the internal trace of the function $u^{a}$ at $0$.

Case $\ell=0$: Assume further that the Dirichlet conditions in
${\mathcal{U}}_{n}^{0}$ are as in (1.15) and the loads satisfy (1.27). Assume
further (1.36). Then for every $u^{b}\in BV(\omega;\mathbb{R}^{3})$,

$\displaystyle\inf\Big{\\{}$
$\displaystyle\liminf_{n}F_{n}^{0}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{o},$
$\displaystyle u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{ in
}W^{1,1}(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\buildrel\star\over{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\Big{\\}}=K^{b,0}(u^{b}).$ (1.39)

######  Remark 1.4.

It is worth to emphasize that in view of Propositions 3.2, 3.8 and 3.18, the
strong convergences appearing in the definitions of problems (1.37), (1.38)
and (1.3) are redundant. We could still consider only the $BV$ weak*
convergence as in the problem (1.3).

We also underline that in most of the cases $\ell\in[0,+\infty]$, the limiting
energies are decoupled. In particular, due to the choices of the boundary
conditions and to the well on $W$, in the cases $\ell=\infty$ and $\ell=0$ the
sample is fixed below and above respectively, but if $\ell=+\infty$ and
$\lim_{n}\frac{h_{n}}{r_{n}}=\infty$, then an energetic penalization appears
at the junction point $\bf 0$. Namely, while in the case $\ell=+\infty$ and
$\lim_{n}\frac{h_{n}}{r_{n}}=0$, the infimizing sequences in the subdomain
below can ”stretch” near the joining area $r_{n}\omega{\times}\\{0\\}$ and
still keep their energy infinitesimal, in the case where $h_{n}$ tends ”much
slower” to $0$ than $r_{n}^{2}$ ( i.e. $h_{n}/r_{n}\to\infty$) than this
stretching does not allow any value because $u_{n}^{b}$ is approaching the
rigid limit function in such a manner that its values
$r_{n}\omega{\times}\\{0\\}$ are in fact tending to ${\bf 0}$ thus leading to
the aforementioned penalization.

We also observe that in the some parts of the proof we consider $W$ continuous
(Lipschitz), since, in view of Lemma 2.8, it can be assumed to be quasiconvex
(see [20]).

##  2 Preliminary results

In this section we fix notation, we collect the main properties of the space
of BV functions that we need to prove our results, recall some convexity
notions and prove some preliminary results.

We will use the following notations

  * -

$O\subset\mathbb{R}^{h}$ is an open set;

  * -

${\mathcal{A}}(O)$ is the family of all open subsets of $O$;

  * -

$\bf 0$ and $0_{\alpha}$ denote the null vectors in $\mathbb{R}^{3}$ and
$\mathbb{R}^{2}$, respectively;

  * -

$C^{\infty}_{c}(O;\mathbb{R}^{k})\coloneqq\\{u\colon
U\to\mathbb{R}^{k}:\text{$u$ is smooth and has compact support in $O$}\\}$; if
$k=1$, we just denote this set by $C^{\infty}_{c}(O)$; $C^{\infty}_{0}(O)$ and
$C^{\infty}_{0}(O;\mathbb{R}^{k})$ denote the closures of $C^{\infty}_{c}(O)$
and $C^{\infty}_{c}(O;\mathbb{R}^{k})$, respectively, in the $\sup$ norm;

  * -

$\mathcal{M}(O)$ and $\mathcal{M}(O;\mathbb{R}^{k})$ are the sets of (signed)
finite real-valued or vector-valued Radon measures on $O$, respectively;
$\mathcal{M}^{+}(O)$ is the set of non-negative finite Radon measures on $O$;

  * -

given $\mu\in\mathcal{M}(O)$ or $\mu\in\mathcal{M}(O;\mathbb{R}^{k})$, the
measure $|\mu|\in\mathcal{M}^{+}(O)$ denotes the total variation of $\mu$;

  * -

$\mathcal{L}^{h}$ and $\mathcal{H}^{h-1}$ denote the $h$-dimensional Lebesgue
measure and the $\left(h-1\right)$-dimensional Hausdorff measure in
$\mathbb{R}^{h}$, respectively; the symbol $dx$ will also be used to denote
integration with respect to $\mathcal{L}^{h}$, while $d{\mathcal{H}}^{h-1}$
will be used to denote surface integration with respect to
$\mathcal{H}^{h-1}$;

  * -

$\mathcal{S}^{h-1}$ denotes the unit sphere in $\mathbb{R}^{h}$;

  * -

$Q$ will denote the open unit cube in $\mathbb{R}^{h}$ i.e. $Q:=]0,1[^{h}$
and, given $\nu\in\mathcal{S}^{h-1}$, $Q_{\nu}$ denotes some open unit cube
with two faces perpendicular to $\nu$.

  * -

for $x\in\mathbb{R}^{h},Q(x,\varepsilon):=x+\varepsilon Q$ and similarly,
$Q_{\nu}(x;\varepsilon):=x+\varepsilon Q_{\nu}$;

  * -

$B$ will denote the open unit ball in $\mathbb{R}^{h}$ centered at the origin
and $B(x,\varepsilon):=x+\varepsilon B$;

  * -

$\overline{B}$ will denote the close unit ball centered at the origin and
$\overline{B}(x,\varepsilon):=x+\varepsilon\overline{B}$;

  * -

$\langle\cdot,\cdot\rangle$ denotes the scalar product;

  * -

where not otherwise specified, $C$ denotes a positive constant which may vary
from line to line;

######  Definition 2.1.

A function $w\in L^{1}(O;{\mathbb{R}}^{k})$ is said to be of _bounded
variation_ , and we write $w\in BV(O;{\mathbb{R}}^{k})$, if all its first
distributional derivatives $D_{j}w_{i}$ belong to $\mathcal{M}(\Omega)$ for
$1\leq i\leq h$ and $1\leq j\leq k$.

The matrix-valued measure whose entries are $D_{j}w_{i}$ is denoted by $Dw$
and $|Dw|$ stands for its total variation. We observe that if $w\in
BV(O;\mathbb{R}^{k})$ then $w\mapsto|Dw|(O)$ is lower semicontinuous in
$BV(O;\mathbb{R}^{k})$ with respect to the
$L_{\mathrm{loc}}^{1}(O;\mathbb{R}^{k})$ topology.

We briefly recall some facts about functions of bounded variation. For more
details we refer the reader to [2, 49].

######  Definition 2.2.

Given $w\in BV\left(O;\mathbb{R}^{k}\right)$ the _approximate upper_  _limit_
and the _approximate lower limit_ of each component $w^{i}$, $i=1,\dots,k$,
are defined by

$\left(w^{i}\right)^{+}\left(x\right):=\inf\left\\{t\in\mathbb{R}:\,\lim_{\varepsilon\rightarrow
0^{+}}\frac{\mathcal{L}^{h}\left(\left\\{y\in O\cap
Q\left(x,\varepsilon\right):\,w^{i}\left(y\right)>t\right\\}\right)}{\varepsilon^{h}}=0\right\\}$

and

$\left(w^{i}\right)^{-}\left(x\right):=\sup\left\\{t\in\mathbb{R}:\,\lim_{\varepsilon\rightarrow
0^{+}}\frac{\mathcal{L}^{h}\left(\left\\{y\in O\cap
Q\left(x,\varepsilon\right):\,w^{i}\left(y\right)<t\right\\}\right)}{\varepsilon^{h}}=0\right\\},$

respectively. The _jump set_ of $w$ is given by

$J_{w}:=\bigcup_{i=1}^{k}\left\\{x\in
O:\,\left(w^{i}\right)^{-}\left(x\right)<\left(w^{i}\right)^{+}\left(x\right)\right\\}.$

It can be shown that $J_{w}$ and the complement of the set of Lebesgue points
of $w$ differ, at most, by a set of $\mathcal{H}^{h-1}$ measure zero.
Moreover, $J_{w}$ is $\left(h-1\right)$-rectifiable, i.e., there are $C^{1}$
hypersurfaces $\Gamma_{i}$ such that
$\mathcal{H}^{h-1}\left(J_{w}\setminus\cup_{i=1}^{\infty}\Gamma_{i}\right)=0.$

######  Proposition 2.3.

If $w\in BV\left(O;\mathbb{R}^{k}\right)$ then

  1. i)

for $\mathcal{L}^{h}-$a.e. $x\in O$

$\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon}\left\\{\frac{1}{\mathcal{\varepsilon}^{h}}\int_{Q\left(x,\varepsilon\right)}\left|w(y)-w(x)-\nabla
w\left(x\right)\cdot(y-x)\right|^{\frac{h}{h-1}}dy\right\\}^{\frac{h-1}{h}}=0;$

  2. ii)

for $\mathcal{H}^{h-1}$-a.e. $x\in J_{w}$ there exist $w^{+}\left(x\right),$
$w^{-}\left(x\right)\in\mathbb{R}^{k}$ and $\nu\left(x\right)\in S^{h-1}$
normal to $J_{w}$ at $x,$ such that

$\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon^{h}}\int_{Q_{\nu}^{+}\left(x,\varepsilon\right)}\left|w\left(y\right)-w^{+}\left(x\right)\right|dy=0,\qquad\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{\varepsilon^{h}}\int_{Q_{\nu}^{-}\left(x,\varepsilon\right)}\left|w\left(y\right)-w^{-}\left(x\right)\right|dy=0,$

where $Q_{\nu}^{+}\left(x,\varepsilon\right):=\left\\{y\in
Q_{\nu}\left(x,\varepsilon\right):\,\left\langle
y-x,\nu\right\rangle>0\right\\}$ and
$Q_{\nu}^{-}\left(x,\varepsilon\right):=\left\\{y\in
Q_{\nu}\left(x,\varepsilon\right):\,\left\langle
y-x,\nu\right\rangle<0\right\\}$;

  3. iii)

for $\mathcal{H}^{h-1}$-a.e. $x\in O\backslash J_{w}$

$\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{\mathcal{\varepsilon}^{h}}\int_{Q\left(x,\varepsilon\right)}\left|w(y)-w\left(x\right)\right|dy=0.$

We observe that in the vector-valued case in general
$\left(w^{i}\right)^{\pm}\neq\left(w^{\pm}\right)^{i}.$ In the sequel $w^{+}$
and $w^{-}$ denote the vectors introduced in $ii)$ above.

Choosing a normal $\nu_{w}\left(x\right)$ to $J_{w}$ at $x,$ we denote the
_jump_ of $w$ across $J_{w}$ by $\left[w\right]:=w^{+}-w^{-}.$ The
distributional derivative of $w\in BV\left(\Omega;\mathbb{R}^{d}\right)$
admits the decomposition

$Dw=\nabla w\mathcal{L}^{h}\lfloor
O+\left(\left[w\right]\otimes\nu_{w}\right)\mathcal{H}^{h-1}\lfloor
J_{w}+D^{c}w,$

where $\nabla w$ represents the density of the absolutely continuous part of
the Radon measure $Dw$ with respect to the Lebesgue measure. The _Hausdorff_ ,
or _jump_ , _part_ of $Dw$ is represented by
$\left(\left[w\right]\otimes\nu_{w}\right)\mathcal{H}^{h-1}\lfloor J_{w}$ and
$D^{c}w$ is the _Cantor part_ of $Dw$. The measure $D^{c}w$ is singular with
respect to the Lebesgue measure and it is diffuse, i.e., every Borel set
$B\subset\Omega$ with $\mathcal{H}^{N-1}\left(B\right)<\infty$ has Cantor
measure zero.

The following result, that will be exploited in the sequel, can be found in
[26, Lemma 2.6].

######  Lemma 2.4.

Let $w\in BV(O;\mathbb{R}^{k})$, for ${\cal H}^{h-1}$ a.e. $x$ in $J_{w}$,

$\displaystyle{\lim_{{\varepsilon}\to
0^{+}}\frac{1}{{\varepsilon}^{h-1}}\int_{J_{w}\cap
Q_{\nu(x)}(x,{\varepsilon})}|w^{+}(y)-w^{-}(y)|d{\cal
H}^{h-1}=|w^{+}(x)-w^{-}(x)|.}$

We state here the compactness result in $BV$ (Proposition $3.23$ in [2]).

######  Proposition 2.5.

Every sequence $(u_{n})_{n}\subset BV_{\text{loc}}(O;\mathbb{R}^{k})$
satisfying

$\sup\left\\{\int_{A}|u_{n}|\;dx+|Du_{n}|(A),\;n\in\mathbb{N}\right\\}<+\infty\qquad\forall
A\in\mathcal{A}(O),\;A\subset\subset O\\}$

admits a subsequence $(u_{n(k)})_{k}$ converging in
$L^{1}_{\text{loc}}(O;\mathbb{R}^{k})$ to $u\in
BV_{\text{loc}}(O;\mathbb{R}^{k}).$ If $O$ is a bounded extension domain and
the sequence is bounded in $BV(O;\mathbb{R}^{k})$ we can say that $u\in
BV(O;\mathbb{R}^{k})$ and the subsequence weakly* converges to $u$.

Finally, we recall that any open set $O$ with compact Lipschitz boundary is an
extension domain (Proposition $3.21$ in [2]).

###  2.1 Convexity notions

In order to obtain the representation result we need to recall some well known
notions of convex analysis, as well as some more general properties,
essentially concerning convexity and quasiconvexity. Let
$g:\mathbb{R}^{h}\to\mathbb{R}$ be a function, the convex envelope (or
convexification) of $g$ is the function

$\displaystyle g^{\ast\ast}:=\sup\\{h\leq g:h\hbox{ convex}\\}.$

######  Definition 2.6.

A Borel function $f:\mathbb{R}^{k\times h}\to\mathbb{R}$ is said to be
quasiconvex if

$f(v)\leq\int_{Q}f(v+\nabla w(x))dx$

for every $w\in W^{1,\infty}_{0}(\mathbb{R}^{h};\mathbb{R}^{k})$

######  Definition 2.7.

Given a Borel function $f:\mathbb{R}^{k\times h}\to\mathbb{R}$, the
quasiconvexification of $f$ at $v\in\mathbb{R}^{k\times h}$ is given by

$Qf(v):=\inf\left\\{\int_{Q}f(v+\nabla w(x))dx:w\in
C^{\infty}_{0}(Q;\mathbb{R}^{k})\right\\}.$

The above Definition allows to recover the usual notion of quasiconvex
envelope, i.e. the greatest quasiconvex function below $f$, see [20].

The following result allows us to prove that there is no loss of generality in
assuming $W$ quasiconvex, hence continuous.

######  Lemma 2.8.

Let $W\;:\;{\mathbb{R}}^{3{\times}3}\to{\mathbb{R}}$ be a Borel function
satisfying (1.4), and $F_{n}^{\ell}$ be as in Definition 1.1 depending on the
superscript $\ell$ defined in (1.3). Let also $\overline{F}_{n}^{\ell}$ be
defined as $F_{n}^{\ell}$ but by replacing $K_{n}^{a,\ell}$ and
$K_{n}^{b,\ell}$ respectively with:

$\overline{K_{n}^{a,\ell}}(u^{a}_{n})=\int_{{\Omega}^{a}}\left(QW\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}(x)}\;\;|\;\;\displaystyle{\nabla_{3}u^{a}_{n}}(x)\right)+H_{n}^{a,\ell}(x)\cdot
u^{a}_{n}(x)\right)dx,$
$\overline{K_{n}^{b,\ell}}(u^{b}_{n}):=\int_{{\Omega}^{b}}\left(QW\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)+H_{n}^{b,\ell}(x)\cdot
u_{n}^{b}(x)\right)dx.$

Here $QW$ stands for the quasiconvexification of the integrand $W$.

Let

$\displaystyle\mathcal{R}^{\ell}:=\Big{\\{}$ $\displaystyle(u^{a},u^{b})\in
BV(\Omega^{a};\mathbb{R}^{3})\times
BV(\Omega^{b};\mathbb{R}^{3}):\exists(u^{a}_{n},u^{b}_{n})\in\mathcal{U}_{n}^{\ell};$
$\displaystyle(u^{a}_{n},u^{b}_{n})\overset{\ast}{\rightharpoonup}(u^{a},u^{b})\in
BV(\Omega^{a};\mathbb{R}^{3})\times BV(\Omega^{b};\mathbb{R}^{3})$ (2.1)
$\displaystyle\hbox{ and
}\limsup_{n}F_{n}^{\ell}(u^{a}_{n},u^{b}_{n})<+\infty\Big{\\}}.$

Then, the following functionals $G^{\ell}$,
$\overline{G^{\ell}}:\mathcal{R}^{\ell}\to\mathbb{R}$ are equal:

$\displaystyle G^{\ell}(u^{a},u^{b})$
$\displaystyle:=\inf\left\\{\liminf_{n\to\infty}F_{n}^{\ell}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\ell},\right.$
$\displaystyle\left.u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\},$ (2.2)
$\displaystyle\overline{G^{\ell}}(u^{a},u^{b})$
$\displaystyle:=\inf\left\\{\liminf_{n\to\infty}\overline{F}_{n}^{\ell}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\ell},\right.$
$\displaystyle\left.u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\buildrel\star\over{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\}.$ (2.3)

###### Proof.

From the inequality $QW\leq W$ it easily follows that $\overline{G}^{\ell}\leq
G^{\ell}$. Now we prove the other inequality. Let $(u^{a},u^{b})\in{\cal
R}^{\ell}$. There is no loss of generality in assuming
$\overline{G}^{\ell}(u^{a},u^{b})<+\infty$.

As already observed above, the contribution of the loads can be neglected
since it is continuous with respect to $\Gamma$-convergence.

Let $(u_{n}^{a},u_{n}^{b})\subset\mathcal{U}^{\ell}_{n}$ be an infimizing
sequence of (2.8), namely

$\lim_{n}\overline{F}^{\ell}_{n}(u_{n}^{a},u_{n}^{n})=\overline{G}^{\ell}(u^{a},u^{b})<+\infty.$

For any $n\in{\mathbb{N}}$, from Theorem $9.1$ in [20], we have that

$\displaystyle\int_{\Omega^{a}}QW_{n}^{a}(\nabla
u_{n}^{a}(x))dx=\inf\left\\{\liminf_{k\to\infty}\int_{\Omega^{a}}W_{n}^{a}(\nabla
u_{k}(x))dx:\;u_{k}\in
u_{n}^{a}+W_{0}^{1,1}(\Omega^{a};{\mathbb{R}}^{3})\right.$
$\displaystyle\left.u_{k}\to u_{n}^{a}\hbox{ in
}L^{1}(\Omega^{a};{\mathbb{R}}^{3})\right\\},$ (2.4)
$\displaystyle\int_{\Omega^{b}}QW_{n}^{b}(\nabla
u_{n}^{b}(x))dx=\inf\left\\{\liminf_{k\to\infty}\int_{\Omega^{b}}W_{n}^{b}(\nabla
u_{k}(x))dx:\;u_{k}\in
u_{n}^{b}+W_{0}^{1,1}(\Omega^{b};{\mathbb{R}}^{3})\right.$
$\displaystyle\left.u_{k}\to u_{n}^{b}\hbox{ in
}L^{1}(\Omega^{b};{\mathbb{R}}^{3})\right\\},$ (2.5)

where we define, for every $\xi\in\mathbb{R}^{3\times 3}$,

$W_{n}^{a}(\xi):=W\left(\frac{1}{r_{n}}\xi_{\alpha}\;|\;\xi_{3}\right),\quad
W_{n}^{b}(\xi):=W\left(\xi_{\alpha}\;|\;\frac{1}{h_{n}}\xi_{3}\right).$

Notice that

$QW^{a}_{n}(\xi)=QW\left(\frac{1}{r_{n}}\xi_{\alpha}\;|\;\xi_{3}\right),\quad
QW_{n}^{b}(\xi)=QW\left(\xi_{\alpha}\;|\;\frac{1}{h_{n}}\xi_{3}\right),$

for every $\xi\in\mathbb{R}^{3\times 3}$, (see [9] for the details of similar
equalities). Based on (2.4) and (2.5) we can therefore choose for fixed
$n\in{\mathbb{N}}$, $u^{a}_{n,k}\in
u_{n}^{a}+W_{0}^{1,1}(\Omega^{a};{\mathbb{R}}^{3})$ and $u^{b}_{n,k}\in
u_{n}^{b}+W_{0}^{1,1}(\Omega^{a};{\mathbb{R}}^{3})$ such that

$\displaystyle u^{a}_{n,k}\to u_{n}^{a}\hbox{ in
}L^{1}(\Omega^{a};{\mathbb{R}}^{3}),$ $\displaystyle\quad u^{b}_{n,k}\to
u_{n}^{b}\hbox{ in }L^{1}(\Omega^{a};{\mathbb{R}}^{3})\hbox{ as }k\to\infty,$
$\displaystyle\lim_{k}\int_{\Omega^{a}}W\left(\tfrac{1}{r_{n}}\nabla_{\alpha}u_{n,k}^{a}(x)\;|\;\nabla_{3}u_{n,k}^{a}(x)\right)dx$
$\displaystyle=\int_{\Omega^{a}}QW\left(\tfrac{1}{r_{n}}\nabla_{\alpha}u_{n}^{a}(x)\;|\;\nabla_{3}u_{n}^{a}(x)\right)dx,$
$\displaystyle\lim_{k}\int_{\Omega^{a}}W\left(\nabla_{\alpha}u_{n,k}^{b}(x)\;|\;\tfrac{1}{h_{n}}\nabla_{3}u_{n,k}^{b}(x)\right)dx$
$\displaystyle=\int_{\Omega^{a}}QW\left(\nabla_{\alpha}u_{n}^{b}(x)\;|\;\tfrac{1}{h_{n}}\nabla_{3}u_{n}^{b}(x)\right)dx.$

Since the traces of $u_{n,k}^{a}$ and $u_{n,k}^{b}$ are respectively the
traces of $u_{n}^{a}$ and $u_{n}^{b}$ it results that
$(u_{n,k}^{a}$,$u_{n,k}^{b})\in\mathcal{U}^{\ell}_{n}$. By a diagonal argument
we can construct sequences
$(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\in\mathcal{U}^{\ell}_{n}$ such that:

$\displaystyle u^{a}_{n,k(n)}\to u^{a}\hbox{ in
}L^{1}(\Omega^{a};{\mathbb{R}}^{3})$ $\displaystyle\quad u^{b}_{n,k(n)}\to
u^{b}\hbox{ in }L^{1}(\Omega^{a};{\mathbb{R}}^{3})\hbox{ as }n\to\infty,$
(2.6)
$\displaystyle\left|F_{n}^{\ell}(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\;\right.$
$\displaystyle-\left.\;\overline{F_{n}^{\ell}}(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\right|\leq
1/n.$

Therefore

$\lim_{n\to\infty}F_{n}^{\ell}(u_{n,k(n)}^{a},u_{n,k(n)}^{b})=\lim_{n\to\infty}\overline{F^{\ell}_{n}}(u_{n}^{a},u_{n}^{b})=\overline{G^{\ell}}(u^{a},u^{b})<+\infty.$
(2.7)

Recall that $\ell$ is defined as in (1.3). From the above inequality and (1.4)
we can easily see that for any case $\ell\in[0,+\infty]$, we have that

$\sup_{n\in\mathbb{N}}\|\nabla
u_{n,k(n)}^{a}\|_{L^{1}(\Omega^{a};{\mathbb{R}}^{3})}<+\infty,\quad\sup_{n\in\mathbb{N}}\|\nabla
u_{n,k(n)}^{b}\|_{L^{1}(\Omega^{b};{\mathbb{R}}^{3})}<+\infty.$

From (2.6) it follows

$u_{n,k(n)}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3})\quad
u_{n,k(n)}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3}),\hbox{ as }n\to\infty.$

We stress that our arguments cover all the cases for $\ell$. Indeed, as
already done in Remark 1.4, the strong convergences appearing for the scaled
gradients in the definitions of the $\Gamma$-limits in (1.37), (1.38), and
(1.3) are redundant (see Proposition 3.2 and 3.8) in the sense that it will be
enough to assume that there is weak* convergence in $BV$, then the strong will
follow by (1.4) and the imposed boundary conditions. Since the sequences
$(u_{n,k(n)}^{a},u_{n,k(n)}^{b})$ are admissible to problem (2.8), from (2.7)
we conclude $G^{\ell}(u^{a},u^{b})\leq\overline{G}^{\ell}(u^{a},u^{b})$. ∎

The following result is a consequence of standard diagonalization arguments,
the coercivity assumption in (1.4), the metrizability of weak* topology in
$BV$ on compact sets, and the attainment of the functional $G^{\ell}$ in
(2.8), by a subsequence of $(F_{n}^{\ell}(u^{a}_{n},u^{b}_{n}))$ with
$(u^{a}_{n},u^{b}_{n})\in\mathcal{U}_{n}^{\ell}$. Once again, we emphasize
that the arguments below cover all the energies appearing in Theorem 1.3.

######  Lemma 2.9.

Let $W\;:\;{\mathbb{R}}^{3{\times}3}\to{\mathbb{R}}$ be a Borel function
satisfying (1.4), and $F_{n}^{\ell}$ be as in Definition 1.1 depending on the
superscript $\ell$ defined in (1.3). Let $G^{\ell}$ be the functional defined
in (2.8). Then it is lower semicontinuous on the space $\mathcal{R}^{\ell}$,
in (2.1), with respect to the $BV(\Omega^{a};\mathbb{R}^{3})\times
BV(\Omega^{b};\mathbb{R}^{3})$ weak * convergence.

###### Proof.

As observed above, the lower semicontinuity result can be proven with respect
to weak* convergence in $BV$ in all the cases $\ell\in[0,+\infty]$, since the
compactness results will provide autonomously stronger convergences in the
cases $\ell\in\\{0,+\infty\\}$. Let $(u^{a},u^{b})\in{\cal R}^{\ell}$ and
$(u^{a}_{n},u^{b}_{n})\in{\cal R}^{\ell}$ such that
$u^{a}_{n}\buildrel\star\over{\rightharpoonup}u^{a}$ in
$BV(\Omega^{a};{\mathbb{R}}^{3})$,
$u^{b}_{n}\buildrel\star\over{\rightharpoonup}u^{b}$ in
$BV(\Omega^{b};{\mathbb{R}}^{3})$. Let
$(u_{n,k}^{a},u_{n,k}^{b})\in{\mathcal{U}}_{k}^{\ell}$ be sequences such that

$\displaystyle u^{a}_{n,k}\buildrel\star\over{\rightharpoonup}u_{n}^{a}\hbox{
in }BV(\Omega^{a};{\mathbb{R}}^{3}),$ $\displaystyle\quad
u^{b}_{n,k}\buildrel\star\over{\rightharpoonup}u_{n}^{b}\hbox{ in
}BV(\Omega^{b};{\mathbb{R}}^{3})\quad\hbox{as }k\to\infty,$
$\displaystyle\lim_{k}F_{k}^{\ell}(u_{n,k}^{a},u_{n,k}^{b})$
$\displaystyle=G^{\ell}(u_{n}^{a},u_{n}^{b}).$

By a diagonal argument we can construct a sequence
$(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\in{\mathcal{U}}_{k(n)}^{\ell}$ for
$n\in{\mathbb{N}}$ such that

$\displaystyle u^{a}_{n,k(n)}\buildrel\star\over{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};{\mathbb{R}}^{3}),$ $\displaystyle\quad
u^{b}_{n,k(n)}\buildrel\star\over{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};{\mathbb{R}}^{3}),$
$\displaystyle\left|F_{k(n)}^{\ell}(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\;\right.$
$\displaystyle-\left.\;G^{\ell}(u_{n}^{a},u_{n}^{b})\right|\leq\frac{1}{n}.$

The sequences $(u_{n,k(n)}^{a},u_{n,k(n)}^{b})$ are admissible for the
infimizing problem $G^{\ell}(u^{a},u^{b})$ in (2.8) and from the above
inequality

$\liminf_{n\to\infty}G^{\ell}(u_{n}^{a},u_{n}^{b})=\liminf_{n\to\infty}F_{k(n)}^{\ell}(u_{n,k(n)}^{a},u_{n,k(n)}^{b})\geq
G^{\ell}(u^{a},u^{b}).$

∎

##  3 Main results

We are dividing the section in three subsections according to the cases
(1.21), (1.24) and (1.27).

In each subsection we detail the compactness result, and prove the lower bound
and the upper bound inequalities that lead to the statement in Theorem 1.3.

###  3.1 Finite rate

We start with the case (1.21). We emphasize that our analysis is performed in
the realm of $\Gamma$-convergence, hence the term
$\int_{\Omega^{a}}H^{a,q}_{n}(x)\cdot
u^{a}_{n}(x)dx+\frac{h_{n}}{r_{n}^{2}}\int_{\Omega^{b}}H^{b,q}_{n}(x)\cdot
u^{b}_{n}(x)dx$ with $H^{a,q}_{n}$ and $H^{b,q}_{n}$ satisfying (1.21) can be
neglected, i.e. $H$ can be assumed to be $0$ since it is a continuous
perturbation, which won’t affect our conclusion (see Remark 1.2). In order to
provide our asymptotic analysis, we start by providing a compactness result.
To this end, we begin with the introduction of the following limiting
functions space.

######  Definition 3.1 (The limit space $V$).

Let $V$ be the space defined as

$V=BV(]0,1[;\mathbb{R}^{3}){\times}BV(\omega;\mathbb{R}^{3}).$ (3.1)

######  Proposition 3.2.

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be such that
(1.2), (1.3) and (1.21) hold. Let the spaces $\mathcal{U}_{n}^{q}$ and $V$ be
introduced in (1.8) and (3.1), respectively where the Dirichlet conditions in
${\mathcal{U}}_{n}^{q}$ are as in (1.13). Let $F_{n}^{q}$ be the energy
functional defined in (1.10), where the loads $H_{n}^{a,q},H_{n}^{b,q}$
satisfy condition (1.21).

Then, for every $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{q}(u_{n}^{a},u_{n}^{b})|<+\infty$, there exist
an increasing sequence of positive integer numbers $(n_{i})_{i\in\mathbb{N}}$,
depending possibly on the selected subsequence $(n_{i})_{i\in\mathbb{N}}$, and
$({u}^{a},{u}^{b})\in V$, such that

$\left\\{\begin{array}[]{l}u^{a}_{n_{i}}\overset{\ast}{\rightharpoonup}{u}^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\\\ \\\
u^{b}_{n_{i}}\overset{\ast}{\rightharpoonup}{u}^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3}).\end{array}\right.$ (3.2)

###### Proof.

We prove (3.2) in the simplified case in which the loads $H^{a,q}_{n}$ and
$H^{b,q}_{n}$ are 0, referring to Remark 1.2 for more precise setting in the
general case. Furthermore we prove only the first convergence, the proof of
the second one being similar, and relying on (1.21).

Assumption (1.4), and the fact that, the subset of $\mathcal{U}^{q}_{n}$ of
fields with finite energy is not empty, ensures that

$\displaystyle{\sup_{n\in\mathbb{N}}}\Bigg{\\{}\displaystyle{\int_{\Omega^{a}}\left(\frac{1}{{r_{n}}}|\nabla_{\alpha}u^{a}_{n}(x)|+|\nabla_{3}u^{a}_{n}(x)|\right)dx}+\displaystyle{\frac{h_{n}}{r^{2}_{n}}\int_{\Omega^{b}}\left(|\nabla_{\alpha}u^{b}_{n}(x)|+\frac{1}{{h_{n}}}|\nabla_{3}u^{b}_{n}(x)|\right)dx\quad\Bigg{\\}}<+\infty.}$
(3.3)

This estimate, conditions (1.2), (1.21), and the boundary conditions in (1.13)
in the definition of $\mathcal{U}^{q}_{n}$ combined with Poincaré’s inequality
(see [49]), lead us to conclude that $(u_{n}^{a})_{n}$ is uniformly bounded in
$W^{1,1}(\Omega^{a};\mathbb{R}^{3})$ and $(u_{n}^{b})_{n}$ is uniformly
bounded in $W^{1,1}(\Omega^{b};\mathbb{R}^{3})$, with
$\|\nabla_{x_{\alpha}}u^{a}_{n}\|_{L^{1}}\leq Cr_{n}^{2}$,
$\|\nabla_{3}u^{a}_{n}\|_{L^{1}}\leq C,$
$\|\nabla_{\alpha}u^{b}_{n}\|_{L^{1}}\leq\tilde{C}$, and
$\|\nabla_{3}u^{b}_{n}\|_{L^{1}}\leq\tilde{C}h_{n}$ for some positive
constants $C$ and $\tilde{C}$ that are independent of $n$.

Thus, using the compactness properties in $BV$ of bounded sequences in
$W^{1,1}$ (see Proposition 2.5), the compact embedding of $W^{1,1}$ in
$L^{1^{*}}$ (see [2]) we can find an increasing sequence of positive integer
numbers $(n_{i})_{i\in\mathbb{N}}$ and functions ${u}^{a}\in
BV(\Omega^{a};\mathbb{R}^{3})$ and ${u}^{b}\in BV(\Omega^{b};\mathbb{R}^{3})$
for which (3.2) holds and, in addition, $u^{a}_{n_{i}}$ converges to $u^{a}$
in $L^{1^{*}}(\Omega^{a};\mathbb{R}^{3})$ and weakly* in
$BV(\Omega^{a};\mathbb{R}^{3})$, $u^{b}_{n_{i}}$ converges to $u^{b}$ in
$L^{1^{*}}(\Omega^{b};\mathbb{R}^{3})$, and weakly* in
$BV(\Omega^{b};\mathbb{R}^{3})$ as $i\to\infty$.

$\displaystyle u^{a}=c^{a,q}\text{ on
}\omega{\times}\\{1\\},\enspace\nabla_{\alpha}{u}^{a}=0\text{ a.e.~{}in
}\Omega^{a},$ $\displaystyle u^{b}=f^{b,q}\text{ on
}\partial\omega\times]-1,0[,\enspace\nabla_{3}u^{b}=0\text{ a.e.~{}in
}\Omega^{b}.$

In particular, $u^{a}$ is independent of $x_{\alpha}$ and ${u}^{b}$ is
independent of $x_{3}$, which allows us to identify $(u^{a},u^{b})$ with an
element in $V$. ∎

######  Proposition 3.3 (Lower bound $q$).

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be such that
(1.2), (1.3) and (1.21) hold. Let $\mathcal{U}_{n}^{q}$ and $V$ be the spaces
introduced in (1.8) and (3.1), respectively where the Dirichlet conditions in
${\mathcal{U}}_{n}^{q}$ are as in (1.13). Let $F_{n}^{q}$ be the energy
functional defined in (1.10), with the loads $H_{n}^{a,q},H_{n}^{b,q}$ which
satisfy (1.21).

Assume also that the recession functions of $\hat{W}$ and of $W_{0}$ in (1.28)
and (1.29), respectively, defined as in (1.30), satisfy (1.31) and (1.32).

Then for every $(u^{a},u^{b})\in V$,

$\displaystyle\inf\Big{\\{}$
$\displaystyle\liminf_{n}F_{n}^{q}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q},$
$\displaystyle u_{n}^{a}\buildrel\star\over{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\buildrel\star\over{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\Big{\\}}\geq
K^{a,q}({u}^{a})+qK^{b,q}({u}^{b}),$

where $K^{a,q}$ and $K^{b,q}$ are defined in (1.33) and (1.34).

###### Proof..

As in the proof of Proposition 3.2, we assume for the sake of exposition, that
the loads are null.

Considering the convergences in (3.2), take a sequence
$(u^{a}_{n},u^{b}_{n})\in\mathcal{U}^{q}_{n}$ for which

$\displaystyle\liminf_{n\to\infty}\left(\int_{{\Omega}^{a}}W\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}(x)}\;\;|\;\;\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}W\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)dx\right)<+\infty.$

We want to provide a lower bound. Then it is easily seen that

$\displaystyle\liminf_{n\to\infty}\left(\int_{{\Omega}^{a}}W\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}(x)}\;\;|\;\;\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}W\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)dx\right)\geq$
$\displaystyle\liminf_{n\to\infty}\left(\int_{{\Omega}^{a}}\hat{W}\left(\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}W_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\right)dx\right)\geq$
$\displaystyle\liminf_{n\to\infty}\left(\int_{{\Omega}^{a}}\hat{W}^{\ast\ast}\left(\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\right)dx\right)\geq$
$\displaystyle\liminf_{n\to\infty}\int_{{\Omega}^{a}}\hat{W}^{**}\left(\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+q\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\right)dx,$
(3.4)

with $\hat{W}$ and $W_{0}$ as in (1.28) and (1.29). We observe also that in
the last but one inequality, we have exploited the following estimates
$\hat{W}\geq\hat{W}^{**}$ and $W_{0}\geq QW_{0}$ and in the latter, the
superadditivity of $\liminf$.

Moreover, following [44, page 556], the function $QW_{0}$ can be extended into
a function $Z:\mathbb{R}^{3}\times\mathbb{R}^{3}$ which is quasiconvex. We
recall that, by [8, Proposition 2.6], $(QW_{0})^{\infty}=Q(W_{0}^{\infty})$
and $((\hat{W})^{\ast\ast})^{\infty}=((\hat{W})^{\infty})^{\ast\ast}$, and the
analogous properties of (1.31) and (1.32) hold. Notice that
$u_{n}^{a}|_{\omega{\times}\\{1\\}}=c^{a,q}$ and
$u_{n}^{b}|_{\partial\omega{\times}]0,1[}=f^{b,q}$. We apply now [8, Example
4.1] in both terms to obtain

$\displaystyle\liminf_{n\to\infty}\int_{{\Omega}^{a}}\hat{W}^{**}\left(\displaystyle{\nabla_{3}u^{a}_{n}(x)}\right)dx+q\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\right)dx\geq$
$\displaystyle\geq$
$\displaystyle\;\int_{{\Omega}^{a}}\hat{W}^{**}\left(\displaystyle{\nabla_{3}u^{a}(x_{3})}\right)dx+\int_{\Omega^{a}}(\hat{W}^{**})^{\infty}\left(\frac{dD^{s}_{\alpha}u^{a}}{d|D^{s}_{\alpha}u^{a}|}\right)d|D^{s}_{\alpha}u^{a}|+(\hat{W}^{\ast\ast})^{\infty}(c^{a,q}-u^{a}(1^{-}))+$
$\displaystyle+\;$ $\displaystyle
q\left(\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}(x_{\alpha})}\right)dx+\int_{{\Omega}^{b}}(QW_{0})^{\infty}\left(\displaystyle{\frac{dD^{s}_{\alpha}u^{b}}{d|D^{s}_{\alpha}u^{b}|}}\right)d|D^{s}_{\alpha}u^{b}|+\right.$
$\displaystyle+\;$
$\displaystyle\left.\int_{\partial\omega\times]-1,0[}(QW_{0})^{\infty}((f^{b,q}-(u^{b})^{-})\otimes\nu_{\partial\omega\times]-1,0[})d\mathcal{H}^{2}\right)=K^{a,q}(u^{a})+qK^{b,q}(u^{b}).$

This concludes the proof. ∎

Let

$U:=\left\\{(u^{a},u^{b})\in(c^{a,q}+C^{\infty}_{a}([0,1];\mathbb{R}^{3}))\times(f^{b,q}+C^{\infty}_{0}(\omega;\mathbb{R}^{3})),u^{a}(0)=u^{b}(0_{\alpha})\right\\}$
(3.5)

with $C^{\infty}_{a}([0,1];\mathbb{R}^{3})=\\{u^{a}\in
C^{\infty}([0,1];\mathbb{R}^{3}):u^{a}(1)=0\\}.$

######  Proposition 3.4.

Let $U$ be as in (3.5). Then it is dense in
$\left(c^{a,q}+W^{1,1}_{a}(]0,1[;\mathbb{R}^{3})\right){\times}$
$\left(f^{b,q}+W^{1,1}_{0}(\omega;\mathbb{R}^{3})\right)$ with respect to
strong topology of $W^{1,1}(]0,1[;\mathbb{R}^{3})\times
W^{1,1}(\omega;\mathbb{R}^{3})$.

###### Proof.

It follows as the case $p<N-1$ in Proposition 3.1 of [32], with $v^{(1)}$ and
$v^{(2)}$ therein, coinciding with $u^{a}$ and $u^{b}$, respectively. Indeed,
we rely on equations (3.2), (3.3) and (3.4) in Proposition 3.1 of [32],
considering a sequence of cut-off functions $\varphi_{n}$ as in (3.3) of class
$C^{\infty}$ with the same radii as in (3.2), so that (3.4) holds. Hence the
sequences

$u^{a}_{n}(x_{3}):=u^{a}(x_{3})\hbox{ for }x_{3}\in]0,1[,$

and

$u^{b}_{n}(x_{\alpha}):=\left\\{\begin{array}[]{ll}u^{a}(0)&\hbox{ in
}\overline{B}(0_{\alpha},\varepsilon_{n}),\\\
\varphi_{n}u^{a}(0)+(1-\varphi_{n}(x_{\alpha}))u^{b}(x_{\alpha})&\hbox{ in
}\overline{B}(0_{\alpha},\eta_{n})\setminus\overline{B}(0_{\alpha},\varepsilon_{n}),\\\
u^{b}(x_{\alpha})&\hbox{ in
}\omega\setminus\overline{B}(0_{\alpha},\eta_{n})\end{array}\right.$

for $x_{\alpha}\in\omega$, converge, as in the assertion, to $u^{a}$ and
$u^{b}$, respectively.

∎

######  Proposition 3.5.

Under the same assumptions of Proposition 3.3, for every $(u^{a},u^{b})\in U$
in (3.5),

$\displaystyle\inf\Big{\\{}$
$\displaystyle\limsup_{n}F_{n}^{q}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q},\;u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),$ $\displaystyle
u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3})\Big{\\}}\leq\int_{0}^{1}\hat{W}(\nabla_{3}{u}^{a}(x_{3}))dx_{3}+q\int_{\omega}W_{0}(\nabla_{\alpha}{u}^{b}(x_{\alpha}))dx_{\alpha}+$
$\displaystyle\int_{\Omega^{a}}H(x)\cdot
u^{a}(x_{3})dx+\int_{\Omega^{b}}H(x)\cdot u^{b}(x_{\alpha})dx.$ (3.6)

###### Proof.

As in Proposition 3.3 we assume that the loads are null, so we can avoid to
consider them in the analysis below.

Following the strategy in [31, Section 5], for every $(u^{a},u^{b})\in U$ and
$z^{a}\in W^{1,1}_{0}(]0,1[;\mathbb{R}^{3})$ and $z^{b}\in
W^{1,1}_{0}(\omega;\mathbb{R}^{3})$ we define

$\displaystyle
u_{n}^{a}(x_{\alpha},x_{3}):=\left\\{\begin{array}[]{ll}(z^{a}(\varepsilon_{n})r_{n}x_{\alpha}+u^{a}(\varepsilon_{n}))\frac{x_{3}}{\varepsilon_{n}}+u^{b}(r_{n}x_{\alpha})\frac{\varepsilon_{n}-x_{3}}{\varepsilon_{n}}&\hbox{
if }(x_{\alpha},x_{3})\in\omega\times]0,\varepsilon_{n}[,\\\
z^{a}(x_{3})r_{n}x_{\alpha}+u^{a}(x_{3})&\hbox{ if
}(x_{\alpha},x_{3})\in\omega\times[\varepsilon_{n},1[,\end{array}\right.$
$\displaystyle
u^{b}_{n}(x_{\alpha},x_{3}):=h_{n}x_{3}z^{b}(x_{\alpha})+u^{b}(x_{\alpha})\hbox{
if }(x_{\alpha},x_{3})\in\Omega^{b}.$

Arguing as in [32, (4.11), (4.12), (4.5) in Proposition 4.1], for $p=1$, it
results that

$\displaystyle u^{a}_{n}\to u^{a}$ $\displaystyle\hbox{ strongly in
}L^{1}(\Omega^{a};\mathbb{R}^{3})$
$\displaystyle\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}\to z^{a}$
$\displaystyle\hbox{ strongly in }L^{1}(\Omega^{a};\mathbb{R}^{3{\times}2}),$
(3.7) $\displaystyle\nabla_{3}u^{a}_{n}\to\nabla_{3}u^{a}$
$\displaystyle\hbox{strongly in }L^{1}(\Omega^{a};\mathbb{R}^{3}).$

Moreover, as in [44],

$\displaystyle u^{b}_{n}\to u^{b}$ $\displaystyle\hbox{ strongly in
}L^{1}(\Omega^{b};\mathbb{R}^{3}),$
$\displaystyle\nabla_{\alpha}u^{b}_{n}\to\nabla_{\alpha}u^{b}$
$\displaystyle\hbox{ strongly in }L^{1}(\Omega^{b};\mathbb{R}^{3{\times}2}),$
(3.8) $\displaystyle\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}=z^{b}$
$\displaystyle\hbox{ in }\Omega^{b},$
$\displaystyle\lim_{n\to\infty}\left(K^{a,q}_{n}(u^{a}_{n})+\frac{h_{n}}{r_{n}^{2}}K^{b,q}_{n}(u^{b}_{n})\right)=\int_{\Omega^{a}}W(z^{a}(x_{3}),\nabla_{3}u^{a}(x_{3}))dx+q\int_{\Omega^{b}}W(\nabla_{\alpha}u^{b}(x_{\alpha}),z^{b}(x_{\alpha}))dx.$
(3.9)

Recalling that $z^{a}\in W^{1,1}_{0}(]0,1[;\mathbb{R}^{3})$ and $z^{b}\in
W^{1,1}_{0}(\omega;\mathbb{R}^{3})$ can be arbitrarily chosen, then arguing as
in the proof of [31, Theorem 5.1], replacing the spaces
$W^{1,p}_{0}(]0,1[;\mathbb{R}^{3})$ and $W^{1,p}_{0}(\omega;\mathbb{R}^{3})$
by $W^{1,1}_{0}(]0,1[;\mathbb{R}^{3})$ and
$W^{1,1}_{0}(\omega;\mathbb{R}^{3})$ and the spaces
$L^{p}(]0,1[;\mathbb{R}^{3})$ and $L^{p}(\omega;\mathbb{R}^{3})$ by
$L^{1}(]0,1[;\mathbb{R}^{3})$ and $L^{1}(\omega;\mathbb{R}^{3})$,
respectively, using Aumann’selection lemma, and definitions $W_{0}$ and
$\hat{W}$ in (1.28) and (1.29) respectively, we obtain

$\displaystyle\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{q}(u_{n}^{a},u^{b}_{n}):(u_{n}^{a},u^{b}_{n})\in\mathcal{U}_{n}^{q},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3})\right\\}\leq\int_{0}^{1}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx_{3}+q\int_{\omega}W_{0}(\nabla_{\alpha}{u}^{b}(x_{\alpha}))dx_{\alpha}.$

The proof is concluded by a standard diagonalization result applying
Proposition 3.4.

∎

######  Proposition 3.6 (Upper bound $q$).

Under the same assumptions of Proposition 3.3, for every $(u^{a},u^{b})\in V$

$\displaystyle\inf\left\\{\limsup_{n}F_{n}^{q}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3})\right\\}\leq K^{a,q}(u^{a})+qK^{b,q}(u^{b}),$
(3.10)

where $K^{a,q}$ and $K^{b,q}$ are defined in (1.33) and (1.34), respectively.

###### Proof.

In view of Propositions 3.4 and 3.5, for every $u^{a}\in
c^{a,q}+W^{1,1}_{a}(]0,1[;\mathbb{R}^{3})$ and $u^{b}\in
f^{b,q}+W^{1,1}_{0}(\omega;\mathbb{R}^{3})$, inequality (3.6) holds.

Recall that $\mathcal{R}^{q}$ in (2.1) coincides with $V$ in Definition 3.1
and let the functional $G^{q}:V\to\mathbb{R}$ defined in (2.8), be given by

$\displaystyle G^{q}(u^{a},u^{b}):=\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{q}(u_{n}^{a},u^{b}_{n}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{q},\;u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\}.$

Its lower semicontinuity with respect to the weak * topology of
$BV(]0,1[;\mathbb{R}^{3})\times BV(\omega;\mathbb{R}^{3})$ (see Lemma 2.9) and
the fact that the functional in the right hand side of (3.6), i.e. the
functional
$\int_{0}^{1}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx_{3}+q\int_{\omega}W_{0}(\nabla_{\alpha}{u}^{b}(x_{\alpha}))dx_{\alpha}$
defined in $c^{a,q}+W^{1,1}_{a}(]0,1[;\mathbb{R}^{3})\times
f^{b,q}+W^{1,1}_{0}(\omega;\mathbb{R}^{3})$, admits, in view of [8, Example
4.1] as relaxation in $V$, (with respect to the $BV$ weak * convergence in
$BV(]0,1[;\mathbb{R}^{3})\times BV(\omega;\mathbb{R}^{3})$) the same
functional, with $K^{a,q}$ and $K^{b,q}$ given by (1.33) and (1.34),
respectively, proves our result. ∎

######  Remark 3.7.

Propositions (3.3), (3.5) and (3.6) also hold if in (1.13) we consider more
general boundary conditions, i.e. any $d^{a,q}\in{\mathbb{R}}^{3{\times}2}$
and $g^{b,q}\in W^{1,1}(\omega;{\mathbb{R}}^{3})$.

In fact, suppose that in the proof of Proposition 3.3, in (3.4) we have
$u_{n}^{a}|_{\omega{\times}\\{1\\}}=c^{a,q}+r_{n}d^{a,q}x_{\alpha}$ for
$d^{a,q}\in{\mathbb{R}}^{3{\times}2}$. Then for
$\tilde{u}_{n}^{a}:=u_{n}^{a}-r_{n}d^{a,q}x_{\alpha}$, since $d^{a,q}$ is
constant, we have that $\nabla_{3}\tilde{u}_{n}^{a}=\nabla_{3}u_{n}^{a}$ and
$\tilde{u}_{n}^{a}|_{\omega{\times}\\{1\\}}=c^{a,q}$ therefore

$\liminf_{n\to\infty}\int_{{\Omega}^{a}}\hat{W}^{**}\left(\displaystyle{\nabla_{3}u^{a}_{n}}(x)\right)dx=\liminf_{n\to\infty}\int_{{\Omega}^{a}}\hat{W}^{**}\left(\displaystyle{\nabla_{3}\tilde{u}^{a}_{n}}(x)\right)dx.$

For $u_{n}^{b}\in W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$ with
$u_{n}^{b}|_{\partial\omega{\times}]0,1[}=f^{b,q}+h_{n}g^{b,q}x_{3}$ consider
$\tilde{u}_{n}^{b}:=u_{n}^{b}-h_{n}g^{b,q}x_{3}$.

$\displaystyle\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}\tilde{u}^{b}_{n}(x)}\right)dx=\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)-h_{n}x_{3}\nabla_{\alpha}g^{b,q}(x_{\alpha})}\right)dx.$

Since $QW_{0}$ is quasiconvex and has linear growth it is Lipschitz (see
[20]), hence

$\displaystyle\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}\tilde{u}^{b}_{n}(x)}\right)dx=\liminf_{n\to\infty}\int_{{\Omega}^{b}}QW_{0}\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\right)dx.$

Therefore, we can reproduce the rest of the proof in Proposition 3.3 with
$\tilde{u}_{n}^{a}$ and $\tilde{u}_{n}^{b}$.

Finally, in Proposition 3.5 by replacing $z^{a}$ and $z^{b}$ respectively with
$z^{a}+d^{a,q}$ and $z^{b}+g^{b,q}$ we still reach the same conclusion.

###  3.2 Infinite rate: $\ell=+\infty$.

We now consider the case (1.24). For the same reasons as in the preceding
subsection, we can neglect the term $\int_{\Omega^{a}}H_{n}^{a,\infty}(x)\cdot
u^{a}_{n}(x)dx+\frac{h_{n}}{r_{n}^{2}}\int_{\Omega^{b}}H_{n}^{b,\infty}(x)\cdot
u^{b}_{n}(x)dx$ since (1.24) provides continuity of these latter integrals
with respect to $\Gamma$-convergence.

######  Proposition 3.8.

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4) and (1.36). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be as
in (1.2), with $\ell$ in (1.3) coinciding with $+\infty.$ Let the spaces
$\mathcal{U}_{n}^{\infty}$ be as in (1.8) with Dirichlet conditions as in
(1.14), and the loads $H_{n}^{a},H_{n}^{b}$ satisfy (1.24). Let
$F_{n}^{\infty}$ be the energy functional defined in (1.11).

Then, for every $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{\infty}[(u_{n}^{a},u_{n}^{b})]|<+\infty$,
there exist an increasing sequence of positive integer numbers
$(n_{i})_{i\in\mathbb{N}}$, ${u}^{a}\in BV(]0,1[;{\mathbb{R}}^{3})$ depending
possibly on the selected subsequence $(n_{i})_{i\in\mathbb{N}}$, such that

$\left\\{\begin{array}[]{l}u^{a}_{n_{i}}\overset{\ast}{\rightharpoonup}{u}^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\\\ \\\ u^{b}_{n_{i}}\to(x_{\alpha},0)\hbox{
in }W^{1,1}(\Omega^{b};\mathbb{R}^{3}).\end{array}\right.$ (3.11)

Furthermore, it results

$\displaystyle{\frac{1}{h_{n_{i}}}}\nabla_{x_{3}}u^{b}_{n_{i}}\to(0_{\alpha},1)\hbox{
in }L^{1}(\Omega^{b};\mathbb{R}^{3}).$

###### Proof.

Arguing as in Proposition 3.2, we first observe that we neglect to loads, then
we can say that there exist a not relabelled subsequence
$(u^{a}_{n},u^{b}_{n})\in\mathcal{U}_{n}^{\infty}$, and $(u^{a},u^{b})\in V$
(where V is defined in (3.1)) such that (3.2) holds. From (1.36) and the
assumption
$\sup_{n\in{\mathbb{N}}}|F_{n}^{\infty}(u_{n}^{a},u_{n}^{b})|<+\infty$, there
exists a $C>0$, such that, for all $n\in{\mathbb{N}}$

$\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}\left|\;\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)\;\;-\;Id\right|dx\leq\frac{h_{n}}{r_{n}^{2}}K_{n}^{b}(u_{n}^{b})\leq
C$

and so,

$\displaystyle\int_{{\Omega}^{b}}\left|\nabla_{\alpha}u^{b}_{n}(x)\;-\;Id_{\alpha}\right|dx\leq\frac{r_{n}^{2}}{h_{n}}C$
(3.12)
$\displaystyle\int_{{\Omega}^{b}}\left|\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)\;-\;(0_{\alpha},1)\right|dx\leq\frac{r_{n}^{2}}{h_{n}}C,$
(3.13)

wity $Id_{\alpha}$ as in (1.1), which from (1.24) implies

$\displaystyle\nabla u_{n}^{b}\to\left(Id_{\alpha}\;\;|\;\;{\bf
0}\right)\;\hbox{in}\;L^{1}(\Omega^{b};{\mathbb{R}}^{3\times 3})$ (3.14)
$\displaystyle\displaystyle{\frac{1}{h_{n}}}\nabla_{3}u^{b}_{n}\to(0_{\alpha},1)\hbox{
in }L^{1}(\Omega^{b};\mathbb{R}^{3}),$

where $0$ in the (3.14) stands for the null column in $\mathbb{R}^{3\times
1}.$

From (3.2) and (3.14) we get $u^{b}_{n_{i}}\to(x_{\alpha},0)+y_{0}\;\hbox{
in}\;W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$ for some $y_{0}\in{\mathbb{R}}$.
Since $u_{n}^{b}|_{\partial\omega{\times}]-1,0[}=(x_{\alpha},r_{n}x_{3})$,
from the strong $W^{1,1}$ convergence we conclude that the traces also
converge strongly in $L^{1}$ to $(x_{\alpha},0)+y_{0}$ and therefore
$y_{0}=0$.

∎

In conclusion, in view of Proposition 3.8, the functional $G^{\infty}$ in
(2.8) becomes the left hand side of (1.38), i.e.

$\displaystyle
G^{\infty}(u^{a},(x_{\alpha},0)):=\inf\left\\{\liminf_{n\to\infty}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},\right.$
$\displaystyle\left.u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\},$ (3.15)

where $\mathcal{U}_{n}^{\infty}$ is given by (1.8) with the specific boundary
datum as in (1.14) and $u^{a}\in BV(]0,1[;\mathbb{R}^{3})$.

The functional $G^{\infty}$ is defined in $\mathcal{R}^{\infty}$, namely in
$BV(]0,1[;\mathbb{R}^{3})\times\\{(x_{\alpha},0)\\}$.

We will study problem (3.2) in two separate subcases: first
$\frac{h_{n}}{r_{n}}\to 0$, then $\frac{h_{n}}{r_{n}}\to\infty$. In what
follows, this subcases will be denoted respectively by $(\infty,0)$ and
$(\infty,\infty)$.

####  3.2.1 Case $\displaystyle\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$

As in the previous cases, the integral representation will be achieved in two
steps.

######  Proposition 3.9 (Lower bound ($\infty,0$)).

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4) and (1.36). Assume also that the recession function of
$\hat{W}$ (1.28) defined as in (1.30), satisfies (1.31). For $n\in\mathbb{N}$
let $r_{n}$, $h_{n}$ be as in (1.2), with $\ell$ in (1.3) coinciding with
$+\infty,$ with $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$. Let the spaces
$\mathcal{U}_{n}^{\infty}$ be as in (1.8) with Dirichlet conditions as in
(1.14), and the loads $H_{n}^{a,\infty},H_{n}^{b,\infty}$ satisfy (1.24). Let
$F_{n}^{\infty}$ be the energy functional defined in (1.11), and let
$G^{\infty}(u^{a},(x_{\alpha},0))$ be as in (1.37).

Then for every $u^{a}\in BV(]0,1[;\mathbb{R}^{3})$

$\displaystyle G^{\infty}(u^{a},(x_{\alpha},0))\geq K^{a,\infty}(u^{a}).$

###### Proof..

Recall that we consider null loads. Let $K^{a,\infty}_{n}$ be the functional
in (1.16). The energy

$\displaystyle\inf\left\\{\lim_{n\to\infty}\left(K^{a,\infty}_{n}(u^{a}_{n})\right):u_{n}^{a}\in\left(c^{a,\infty}+W_{a}^{1,1}(\Omega^{a};\mathbb{R}^{3})\right),u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3})\right\\}$ (3.16)

is clearly a lower bound to (3.2). Using the same arguments as in Proposition
3.3 we can get $K^{a,\infty}(u^{a})$ as a lower bound to (3.16) and
consequently to (3.2), which concludes the proof. ∎

In order to prove the upper bound we assume first
$u^{a}\in(c^{a,\infty}+W_{a}^{1,1}(\;]0,1[;{\mathbb{R}}^{3}))$, where
$c^{a,\infty}$ is as in (1.14), and we get the following result.

######  Lemma 3.10.

Under the same assumptions of Proposition 3.9, it results that

$\displaystyle\inf\left\\{\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u^{b}_{n}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\;\leq\;\int_{0}^{1}\hat{W}(\nabla_{3}{u}^{a}(x_{3}))dx_{3},$

for every $u^{a}\in(c^{a,\infty}+W_{a}^{1,1}(\;]0,1[;{\mathbb{R}}^{3}))$.

###### Proof.

For $z^{a}\in W^{1,1}_{0}(]0,1[;{\mathbb{R}}^{3{\times}2})$ consider the
following sequences:

$\displaystyle
u_{n}^{a}(x_{\alpha},x_{3})=z^{a}(x_{3})r_{n}x_{\alpha}+u^{a}(x_{3})$
$\displaystyle(x_{\alpha},x_{3})\in\Omega^{a},$ (3.17) $\displaystyle
u_{n}^{b}(x_{\alpha},x_{3})=(x_{\alpha},h_{n}x_{3})(1-\phi_{n}(x_{\alpha}))\;+\;u_{n}^{a}\left(\frac{1}{r_{n}}x_{\alpha},-h_{n}x_{3}\right)\phi_{n}(x_{\alpha})\quad$
$\displaystyle(x_{\alpha},x_{3})\in\Omega^{b},$ (3.18)

where $\phi_{n}$ is a smooth cut-off function defined in ${\mathbb{R}}^{2}$
satisfying:

$\displaystyle\phi_{n}(x_{\alpha}):=\left\\{\begin{array}[]{ll}1&\hbox{ if
}x_{\alpha}\in r_{n}\omega,\\\ 0&\hbox{ if
}x_{\alpha}\notin(1+\beta)r_{n}\omega,\end{array}\right.$ (3.21)
$\displaystyle|\nabla\phi_{n}(x_{\alpha})|\leq\frac{C}{\beta
r_{n}}\quad\hbox{and}\quad 0\leq\phi_{n}(x_{\alpha})\leq 1.$ (3.22)

Here $\beta>0$ is a fixed arbitrary value.

Notice that $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty}$ because for
$x_{\alpha}\in r_{n}\omega$ we get
$u_{n}^{b}(x_{\alpha},0)=u_{n}^{a}(x_{\alpha}/r_{n},0)$.

$\displaystyle\left(\frac{1}{r_{n}}\nabla_{\alpha}u_{n}^{a}\;|\;\nabla_{3}u_{n}^{a}\right)=$
$\displaystyle\;(z^{a}\;|\;\nabla_{3}u^{a}+\nabla_{3}z^{a}r_{n}x_{\alpha})$
$\displaystyle\left(\nabla_{\alpha}u_{n}^{b}\;|\;\frac{1}{h_{n}}\nabla_{3}u_{n}^{b}\right)(x_{\alpha},x_{3})=$
$\displaystyle
Id(1-\phi_{n}(x_{\alpha}))+\left(\frac{1}{r_{n}}\nabla_{\alpha}u_{n}^{a}\;|\;-\nabla_{3}u_{n}^{a}\right)\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)\cdot\phi_{n}(x_{\alpha})$
$\displaystyle+\left(\nabla_{\alpha}\phi_{n}\otimes\left[u_{n}^{a}\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)-(x_{\alpha},h_{n}x_{3})\right]\;|\;0\right).$

Arguing as in Proposition 3.5 it results that

$\displaystyle u^{a}_{n}\to u^{a}$ $\displaystyle\hbox{ strongly in
}L^{1}(\Omega^{a};\mathbb{R}^{3})$
$\displaystyle\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}\to z^{a}$
$\displaystyle\hbox{ strongly in }L^{1}(\Omega^{a};\mathbb{R}^{3{\times}2}),$
(3.23) $\displaystyle\nabla_{3}u^{a}_{n}\to\nabla_{3}u^{a}$
$\displaystyle\hbox{ strongly in }L^{1}(\Omega^{a};\mathbb{R}^{3}),$ (3.24)

which implies, since $W$ is continuous,

$K^{a,\infty}_{n}(u_{n}^{a})=\int_{\Omega^{a}}W\left(\frac{1}{r_{n}}\nabla_{\alpha}u_{n}^{a}(x)\;|\;\nabla_{3}u_{n}^{a}(x)\right)dx\to\int_{\Omega^{a}}W\left(z^{a}(x_{3})+d^{a,\infty}\;|\;\nabla_{3}u^{a}(x_{3})\right)dx.$

For the sequence $(u_{n}^{b})_{n}$ in (3.18), it is easily seen that
$u_{n}^{b}|_{\partial\omega{\times}]-1,0[}=(x_{\alpha},h_{n}x_{3})$. To prove
that $(u_{n}^{b})_{n}$ is admissible, we observe that, by the Proposition 3.8,
if
$\sup_{n\in{\mathbb{N}}}\frac{h_{n}}{r_{n}^{2}}K_{n}^{b,\infty}(u_{n}^{b})<+\infty$
then $u_{n}^{b}\to(x_{\alpha},0)$ in $W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$ as
it is shown below.

From (1.4) and (3.22) we have the following estimation for the energy, where

$B_{n}:=(1+\beta)r_{n}\omega{\times}]-1,0[,\quad
A_{n}:=r_{n}\omega{\times}]-1,0[,\quad\psi_{n}^{b}(x_{\alpha},x_{3}):=(x_{\alpha},h_{n}x_{3})$

$\displaystyle\frac{h_{n}}{r_{n}^{2}}K_{n}^{b,\infty}$
$\displaystyle(u_{n}^{b}(x))=\;\frac{h_{n}}{r_{n}^{2}}\int_{{\Omega}^{b}}W\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;\;|\;\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)dx$
$\displaystyle\leq$
$\displaystyle\frac{h_{n}}{r_{n}^{2}}\left[\int_{{\Omega}^{b}\setminus
B_{n}}W\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;|\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)dx+C\int_{B_{n}}\left[1+\left|\left(\displaystyle{\nabla_{\alpha}u^{b}_{n}(x)}\;|\;\displaystyle{\frac{1}{h_{n}}\nabla_{3}u^{b}_{n}(x)}\right)\right|\right]dx\right]$
$\displaystyle\leq$
$\displaystyle\frac{h_{n}}{r_{n}^{2}}C|B_{n}|+\frac{h_{n}}{r_{n}^{2}}C\int_{B_{n}\setminus
A_{n}}(1-\phi_{n}(x_{\alpha}))\left|Id\right|dx$ $\displaystyle+$
$\displaystyle\frac{h_{n}}{r_{n}^{2}}C\int_{B_{n}\setminus
A_{n}}\left|\left(u^{a}_{n}\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)\;-\;\psi_{n}^{b}(x)\right)\otimes\left(\displaystyle{\nabla_{\alpha}\phi_{n}(x_{\alpha})}\;\;|\;\;0\right)\right|dx$
$\displaystyle+$
$\displaystyle\frac{h_{n}}{r_{n}^{2}}C\int_{B_{n}}\phi_{n}(x_{\alpha})\cdot\left|\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}}\;\;|\;\;\displaystyle{-\nabla_{3}u^{a}_{n}}\right)\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)\right|dx$
$\displaystyle\leq$ $\displaystyle
O(h_{n})+\frac{h_{n}}{r_{n}^{2}}C\int_{B_{n}}\frac{1}{\beta
r_{n}}\left|\left(u^{a}_{n}\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)\;-\;\psi_{n}^{b}(x)\right)\right|dx\
+$ (3.25)
$\displaystyle\frac{h_{n}}{r_{n}^{2}}C\int_{B_{n}}\left|\left(\displaystyle{\frac{1}{r_{n}}\nabla_{\alpha}u^{a}_{n}}\;\;|\;\;\displaystyle{-\nabla_{3}u^{a}_{n}}\right)\left(\frac{x_{\alpha}}{r_{n}},-h_{n}x_{3}\right)\right|dx.$
(3.26)

Since $u^{a}\in W^{1,1}(]0,1[\;{\mathbb{R}}^{3})$, by the continuous embedding
in $L^{\infty}$ (see [13]), it follows that $u^{a}$ is bounded in
$L^{\infty}(]0,1[;\mathbb{R}^{3}])$. Hence, from (3.17) $(u^{a}_{n})_{n}$ is
also uniformly bounded in $L^{\infty}(]0,1[;\mathbb{R}^{3})$. Noticing that
$(\psi_{n}^{b})_{n}$ is also bounded in
$L^{\infty}(\Omega^{b};\mathbb{R}^{3})$ it follows that the integral in (3.25)
is $O(h_{n}/r_{n})$ and therefore it goes to 0 as $n\to\infty$.

On the other hand, (3.26), after a change of variables, equals to

$C\int_{\omega{\times}]0,h_{n}[}\left|\left(\frac{1}{r_{n}}\nabla_{\alpha}u_{n}^{a}\;|\;\nabla_{3}u_{n}^{a}\right)\right|dx$

From (3.23), (3.24) and uniform integrability it results that the integral
tends to 0.

Consequently for all $z^{a}\in W_{0}^{1,1}(]0,1[;{\mathbb{R}}^{3{\times}2})$
and with $u^{a}\in(c^{a,\infty}+W_{a}^{1,1}(]0,1[;{\mathbb{R}}^{3}))$, we have

$\displaystyle\inf\left\\{\liminf_{n}\left(K^{a,\infty}_{n}(u^{a}_{n})+\frac{h_{n}}{r_{n}^{2}}K^{b,\infty}_{n}(u^{b}_{n})\right):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\;\leq\;\int_{\Omega^{a}}W\left(z^{a}(x_{3})\;|\;\nabla_{3}u^{a}(x_{3})\right)dx,$

where we recall that $z^{a}$ has been arbitrarily chosen.

Following along the lines the arguments in [44, Proposition 7], relying on the
density of $W^{1,1}_{0}(]0,1[;\mathbb{R}^{3\times 2})$ in
$L^{1}(]0,1[;\mathbb{R}^{3\times 2})$ with respect to the $L^{1}$-strong
convergence, and exploiting the dominated convergence theorem, and the
definition of $\hat{W}$ in (1.28), we have

$\inf\left\\{\int_{\Omega^{a}}W\left(z^{a}(x_{3})\;|\;\nabla_{3}u^{a}(x_{3})\right)dx\;:\>z^{a}\in
W_{0}^{1,1}(\Omega^{a};{\mathbb{R}}^{3{\times}2})\right\\}\leq\int_{\Omega^{a}}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx,$

thus the proof is completed. ∎

The following result, together with Proposition 3.9, concludes the proof of
Theorem 1.3 (case $\ell=+\infty$, $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$),
since it provides an upper bound for the relaxed energy in (3.2).

######  Proposition 3.11 (Upper bound $(\infty,0)$).

Under the same assumptions of Proposition 3.9, it results

$\displaystyle\inf\left\\{\liminf_{n}F_{n}^{\infty}(u^{a}_{n},u^{b}_{n}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\leq K^{a,\infty}(u^{a}),$

for every $u^{a}\in BV(\;]0,1[;{\mathbb{R}}^{3}))$, where $K^{a,\infty}$ is
defined in (1.33).

###### Proof.

The proof relies on the same arguments exploited in the proof of Proposition
3.6, namely on the lower semicontinuity of the functional in (3.2) with
respect to the weak * convergence in $BV(\Omega^{a};\mathbb{R}^{3})$ (see
Lemma 2.9, and the relaxation with respect to the same topology of the
functional
$F_{a}:c^{a,\infty}+W^{1,1}_{a}(]0,1[;\mathbb{R}^{3})\to\mathbb{R}^{+}$
defined as $\int_{\Omega^{a}}\hat{W}(\nabla_{3}u^{a}(x_{3})dx$). Indeed the
latter relaxed functional coincides with $K^{a,\infty}$ in (1.33), in view of
[8, Example 4.1]. ∎

####  3.2.2 Case $\displaystyle\frac{h_{n}}{r_{n}}\to\infty$

In the next lemma we prove that the sequence of deformations
$u^{b}_{n}(x_{\alpha},x_{3})-(x_{\alpha},h_{n}x_{3})$ on the thin film below
admits the origin as a Lebesgue point. This result will be instrumental to
deduce a penalization term in the final relaxed energy regarding the uncoupled
behaviour of $u^{a}$ at the origin.

######  Proposition 3.12.

Let $(g_{n})_{n\in{\mathbb{N}}}\subset W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$
satisfy $g_{n}\to\bf 0$ in $W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$. Assume
that,

$\displaystyle|\nabla_{\alpha}g_{n}|_{L^{1}(\Omega^{b};{\mathbb{R}}^{3})}\leq
C\frac{r_{n}^{2}}{h_{n}},$ $\displaystyle\forall n\in{\mathbb{N}},$ (3.27)
$\displaystyle|\nabla_{3}g_{n}|_{L^{1}(\Omega^{b};{\mathbb{R}}^{3})}\to 0,$
$\displaystyle\hbox{as }n\to\infty,$ (3.28) $\displaystyle
g_{n}|_{\partial\omega{\times}]-1,0[}=\bf 0,$ $\displaystyle\forall
n\in{\mathbb{N}},$ (3.29) $\displaystyle\frac{h_{n}}{r_{n}}\to\infty,\quad
r_{n},h_{n}\to 0$ $\displaystyle\hbox{as }n\to\infty.$ (3.30)

Then,

$\lim_{n\to\infty}\int_{\omega}|g_{n}|(r_{n}x_{\alpha},0)dx_{\alpha}=0$

###### Proof.

Let $\delta>0$ and consider $n\in{\mathbb{N}}$ large enough satisfying

$\displaystyle]-\delta r_{n},\delta r_{n}[^{2}\;\subset\omega\;$
$\displaystyle\subset\;]-\delta,\delta[^{2}.$ (3.31)

Recalling (3.29) we can extend $g_{n}\in
W^{1,1}_{0}(\Omega^{b};\mathbb{R}^{3})$ to
$W^{1,1}_{0}({\mathbb{R}}^{3};\mathbb{R}^{3})$ as follows:

$\tilde{g}_{n}=\left\\{\begin{array}[]{ll}g_{n}&\hbox{ in }\Omega^{b},\\\ \bf
0&\hbox{ in }{\mathbb{R}}^{3}\setminus\Omega^{b}.\end{array}\right.$ (3.32)

For easier notation we simply denote $g_{n}$ as this extension.

From the absolute continuity on lines of a function in
$W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$, we can apply the Fundamental Theorem
of Calculus on the lines between $(\overline{x}_{1},\delta,\overline{x}_{3})$
and $(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})$ for a.e.
$(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})\in]-\delta r_{n},\delta
r_{n}[^{2}{\times}]-1,0[$.

$g_{n}(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})=g_{n}(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})-g_{n}(\overline{x}_{1},\delta,\overline{x}_{3})=\int_{\overline{x}_{2}}^{\delta}\frac{\partial}{\partial
x_{2}}g_{n}(\overline{x}_{1},y,\overline{x}_{3})dy$

where in the first inequality we notice that from (3.31) we have
$(\overline{x}_{1},\delta,\overline{x}_{3})\in{\mathbb{R}}^{3}\setminus\Omega^{b}$
therefore from (3.32) $g_{n}(\overline{x}_{1},\delta,\overline{x}_{3})=0$.

We can now estimate $|g_{n}(\cdot,x_{3})|$ in $]-\delta r_{n},\delta
r_{n}[^{2}$. For $(\overline{x}_{1},\overline{x}_{2})\in]-\delta r_{n},\delta
r_{n}[^{2}$.

$\displaystyle|g_{n}(\overline{x}_{1},\overline{x}_{2},\overline{x}_{3})|\leq\int_{\overline{x}_{2}}^{\delta}\left|\frac{\partial}{\partial
x_{2}}g_{n}(\overline{x}_{1},y,\overline{x}_{3})\right|dy$
$\displaystyle\leq\int_{\overline{x}_{2}}^{\delta}\left|\nabla_{\alpha}g_{n}(\overline{x}_{1},y,\overline{x}_{3})\right|dy\leq\int_{-\delta
r_{n}}^{\delta}\left|\nabla_{\alpha}g_{n}(\overline{x}_{1},y,\overline{x}_{3})\right|dy.$

By noticing that this last term doesn’t depend on $\overline{x}_{2}$ we get
that by integrating in $]-\delta r_{n},\delta r_{n}[^{2}$ and using (3.31) in
the first inequality

$\displaystyle\|g_{n}(\cdot,\overline{x}_{3})\|_{L^{1}(r_{n}\omega)}\leq\|g_{n}(\cdot,\overline{x}_{3})\|_{L^{1}(]-\delta
r_{n},\delta r_{n}[^{2})}$ $\displaystyle\leq 2\delta r_{n}\int_{-\delta
r_{n}}^{\delta r_{n}}\int^{\delta}_{-\delta
r_{n}}\left|\nabla_{\alpha}g_{n}(x_{1},x_{2},\overline{x}_{3})\right|dx_{2}dx_{3}.$
(3.33)

Notice that these arguments are valid for $\overline{x}_{3}$ a.e.. Since from
(3.32) $\nabla_{\alpha}g_{n}(x)=0$ in
$]-\delta,\delta[^{2}\setminus\omega\;{\times}]-1,0[$, then integrating (3.33)
on $x_{3}$ we get

$\displaystyle\|g_{n}\|_{L^{1}(r_{n}\omega{\times}]-1,0[)}\;\leq\;2\delta
r_{n}\|\nabla_{\alpha}g_{n}\|_{L^{1}(]-\delta r_{n},\delta[{\times}]-\delta
r_{n},\delta r_{n}[{\times}]-1,0[)}$ $\displaystyle=2\delta
r_{n}\|\nabla_{\alpha}g_{n}\|_{L^{1}(r_{n}\omega{\times}]-1,0[)}$
$\displaystyle\leq 2\delta r_{n}\|\nabla_{\alpha}g_{n}\|_{L^{1}(\Omega^{b})}.$
(3.34)

Let $v_{n}(x_{\alpha},x_{3}):=g_{n}(r_{n}x_{\alpha},x_{3})$ be defined on
$\Omega^{b}$. From (3.27),(3.28) and (3.30) it is straightforward that $\nabla
v_{n}\to 0$ in $L^{1}(\Omega^{b};{\mathbb{R}}^{3})$. On the other hand,
(3.34), (3.27) and (3.30) imply that

$\int_{\Omega^{b}}|v_{n}|dx\;=\;\frac{1}{r_{n}^{2}}\int_{r_{n}\omega{\times}]-1,0[}|g_{n}|dx\;\leq\;C\frac{r_{n}}{h_{n}}\;\to\;0.$

Therefore, we have $v_{n}\to 0$ in $W^{1,1}(\Omega^{b};{\mathbb{R}}^{3})$ and
by the continuity of the trace operator, we get

$\lim_{n\to\infty}\int_{\omega\times\\{0\\}}|g_{n}|(r_{n}x_{\alpha},0)dx_{\alpha}=\lim_{n\to\infty}\int_{\omega\times\\{0\\}}|v_{n}|dx_{\alpha}=0.$

∎

Based on the result just stated we can extend Proposition 3.8, specifically
for $\frac{h_{n}}{r_{n}}\to\infty$.

######  Lemma 3.13 (Compactness $(\infty,\infty)$).

Under the same assumptions of Proposition 3.8 and assuming (3.30) then, for
every sequence $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{\infty}[(u_{n}^{a},u_{n}^{b})]|<+\infty$, it
follows that (3.11) holds and

$u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in }L^{1}(\omega;{\mathbb{R}}^{3}).$

###### Proof.

As in the other analogous results above, we assume null loads, referring to
Remark 1.2 for the general case. From Proposition 3.8, the assumptions on the
boundary (1.14), i.e.

$\displaystyle
u_{n}^{b}|_{\partial\omega{\times}]-1,0[}=(x_{\alpha},h_{n}x_{3}),$

and the second condition in (3.11), namely

$\displaystyle u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};{\mathbb{R}}^{3}),$

we have that (3.13) and (3.12) hold, which, in turn, imply

$\displaystyle\int_{{\Omega}^{b}}\left|\nabla_{\alpha}u^{b}_{n}\;-\;Id_{\alpha}\right|dx\leq\frac{r_{n}^{2}}{h_{n}}C,$
$\displaystyle\left(\nabla_{3}u^{b}_{n}\;-\;(0_{\alpha},h_{n})\right)\to{\bf
0}\hbox{ in }L^{1}(\Omega^{b};{\mathbb{R}}^{3}),\hbox{ as }n\to\infty,$

with $Id_{\alpha}$ as in (1.1).

Applying Lemma 3.12 to the function $g_{n}:=u_{n}^{b}-(x_{\alpha},h_{n}x_{3})$
and recalling that $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n}$ and in
particular (1.7) holds, we get

$\displaystyle\lim_{n\to\infty}\int_{\omega}|u_{n}^{b}(r_{n}x_{\alpha},0)-(r_{n}x_{\alpha},0)|dx_{\alpha}=$
$\displaystyle\lim_{n\to\infty}\int_{\omega}|u_{n}^{a}(x_{\alpha},0)-(r_{n}x_{\alpha},0)|dx_{\alpha}=0,$

which implies

$\displaystyle u_{n}^{a}(\cdot,0)\to\mathbf{0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}),$

that concludes the proof. ∎

In view of Lemma 3.13, we can rewrite problem (3.2) as

$\displaystyle
G^{\infty}(u^{a},(x_{\alpha},0))=\inf\left\\{\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}),u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}.$ (3.35)

The representation of Theorem 1.3, given by (1.38), will be achieved by
proving a double inequality.

######  Proposition 3.14 (Lower bound $(\infty,\infty)$).

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4) and (1.36). For $n\in\mathbb{N}$ let the spaces
$\mathcal{U}_{n}^{\infty}$ be introduced in (1.8) where the Dirichlet
conditions in ${\mathcal{U}}_{n}^{\infty}$ are as in (1.14). Let
$F_{n}^{\infty}$ be the energy functional defined in (1.11), where $r_{n}$,
$h_{n}$ and the loads $H_{n}^{a,\infty},H_{n}^{b,\infty}$ satisfy conditions
(1.2), (1.24) and $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=\infty$. Assume also
that the recession function of $\hat{W}$ (1.28) defined as in (1.30), satisfy
(1.31).

Then for every $u^{a}\in BV(]0,1[;{\mathbb{R}}^{3})$,

$\displaystyle\inf\left\\{\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\geq
K^{a,\infty}(u^{a})+\hat{W}^{**}(u^{a}(0^{+})),$

where $K^{a,\infty}$ is defined in (1.33).

###### Proof.

The following functional $J^{a}:BV(]0,1[;\mathbb{R}^{3})\to\mathbb{R}$ is
clearly a lower bound to the left hand side of (3.2.2).

$\displaystyle
J(u^{a}):=\inf\left\\{\liminf_{n}\int_{\Omega^{a}}\hat{W}^{**}(\nabla_{3}u_{n}^{a}(x))dx:u_{n}^{a}\in(W_{a}^{1,1}(\Omega^{a};{\mathbb{R}}^{3})+c^{a,\infty}),\right.$
$\displaystyle\left.u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3})\right\\}.$ (3.36)

Now we prove $J(u^{a})\geq K^{a}(u^{a})+\hat{W}^{**}(u^{a}(0^{+}))$. Let
$u^{a}\in BV(]0,1[;\mathbb{R}^{3})$, and let $(u_{n}^{a})_{n}$ be a recovery
sequence for (3.36), i.e.

$\displaystyle
u_{n}^{a}\in(W_{a}^{1,1}(\Omega^{a};{\mathbb{R}}^{3})+c^{a,\infty}),\quad\lim_{n\to\infty}\int_{\Omega^{a}}\hat{W}^{**}(\nabla_{3}u_{n}^{a}(x))dx=J(u^{a}),$
$\displaystyle u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),\quad u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}).$

Let $\Omega^{a}_{\varepsilon}:=\omega{\times}]-\varepsilon,1[$ and
$\Omega^{a-}_{\varepsilon}:=\omega{\times}]-\varepsilon,0[$. Let also
$\gamma_{n}\in L^{1}(\partial\Omega^{a-}_{\varepsilon};{\mathbb{R}}^{3})$ be
defined as

$\gamma_{n}=\left\\{\begin{array}[]{ll}u_{n}^{a}(\cdot,0)&\hbox{ in
}\;\;\;\partial\Omega^{a-}_{\varepsilon}\cap\\{x_{3}=0\\},\\\ {\bf 0}&\hbox{
in
}\;\;\;\partial\Omega^{a-}_{\varepsilon}\setminus\\{x_{3}=0\\}.\end{array}\right.$
(3.37)

By Gagliardo’s trace theorem (see [30]), we can find a function $g_{n}\in
W^{1,1}(\Omega^{a-}_{\varepsilon};{\mathbb{R}}^{3})$ such that

$\hbox{tr}(g_{n})=\gamma_{n},\quad|g_{n}|_{W^{1,1}(\Omega^{a-}_{\varepsilon})}\leq
C|\gamma_{n}|_{L^{1}(\partial\Omega^{a-}_{\varepsilon})},$ (3.38)

for some $C=C(\Omega^{a-}_{\varepsilon})>0$.

Consider now the following extensions:

$\displaystyle\tilde{u}_{n}^{a}(x_{\alpha},x_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}u_{n}^{a}(x_{\alpha},x_{3})&\hbox{
if }\;\;\;0<x_{3}<1,\\\ g_{n}(x_{\alpha},x_{3})&\hbox{ if
}-\varepsilon<x_{3}<0,\end{array}\right.$ $\displaystyle\tilde{u}^{a}(x_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}u^{a}(x_{3})&\hbox{ if
}\;\;\;0<x_{3}<1,\\\ {\bf 0}&\hbox{ if
}-\varepsilon<x_{3}<0.\end{array}\right.$

It is easily checked using (1.4), (3.37) and (3.38) that :

$\displaystyle(\tilde{u}_{n}^{a})_{n}\subset
W^{1,1}(\Omega^{a}_{\varepsilon};\mathbb{R}^{3}),\;\tilde{u}_{n}^{a}\to\tilde{u}^{a}\hbox{
in
}L^{1}(\Omega^{a}_{\varepsilon};\mathbb{R}^{3}),\;\sup_{n\in{\mathbb{N}}}|\nabla\tilde{u}_{n}^{a}|_{L^{1}(\Omega^{a}_{\varepsilon})}<+\infty.$

We have that $(\tilde{u}_{n}^{a})_{n}$ is admissible to the following extended
problem defined on $\Omega^{a}_{\varepsilon}$,

$\displaystyle\inf\left\\{\liminf_{n}\int_{\Omega^{a}_{\varepsilon}}\hat{W}^{**}(\nabla_{3}u_{n}(x))dx:u_{n}\in(W_{a}^{1,1}(\Omega^{a}_{\varepsilon};{\mathbb{R}}^{3})+c^{a,\infty}),u_{n}\overset{\ast}{\rightharpoonup}\tilde{u}^{a}\hbox{
in }BV(\Omega^{a}_{\varepsilon};\mathbb{R}^{3})\right\\}.$

And so, arguing as in Proposition 3.3, it results that

$\displaystyle\liminf_{n\to\infty}\int_{\Omega^{a}_{\varepsilon}}\hat{W}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx\geq\int_{\Omega^{a}_{\varepsilon}}{\hat{W}}^{**}(\nabla_{3}u^{a}(x_{3}))dx\;$
$\displaystyle+\;\int_{\Omega^{a}_{\varepsilon}}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}u^{a}}{d|D^{s}_{3}u^{a}|},\right)d|D^{s}_{3}u^{a}|\;+$
$\displaystyle+\;(\hat{W}^{\ast\ast})^{\infty}(c^{a,\infty}-u^{a}(1^{-}))$
(3.39)

Now we estimate from below (3.36) using the inequality just stated.

$\displaystyle\liminf_{n}\int_{\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}u_{n}^{a}(x))dx=$
$\displaystyle\liminf_{n}\left(\int_{\Omega^{a}_{\varepsilon}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx-\int_{\Omega^{a}_{\varepsilon}\setminus\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx\right)$
$\displaystyle\geq$
$\displaystyle\liminf_{n}\int_{\Omega^{a}_{\varepsilon}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx-\limsup_{n}\int_{\Omega^{a}_{\varepsilon}\setminus\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx$
$\displaystyle\geq$ $\displaystyle\geq$
$\displaystyle\int_{\Omega^{a}_{\varepsilon}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}^{a}(x_{3}))dx+\int_{\Omega^{a}_{\varepsilon}}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}\tilde{u}^{a}}{d|D^{s}_{3}\tilde{u}^{a}|},\right)d|D^{s}_{3}\tilde{u}^{a}|$
$\displaystyle+$
$\displaystyle(\hat{W}^{\ast\ast})^{\infty}(c^{a,\infty}-u^{a}(1^{-}))-\limsup_{n}\int_{\Omega^{a}_{\varepsilon}\setminus\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}_{n}^{a}(x))dx$
$\displaystyle=$ $\displaystyle=$
$\displaystyle\int_{\Omega^{a}_{\varepsilon}}{\hat{W}}^{**}(\nabla_{3}\tilde{u}^{a}(x_{3}))dx+\int_{\Omega^{a}_{\varepsilon}}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}\tilde{u}^{a}}{d|D^{s}_{3}\tilde{u}^{a}|},\right)d|D^{s}_{3}\tilde{u}^{a}|$
$\displaystyle+$
$\displaystyle(\hat{W}^{\ast\ast})^{\infty}(c^{a,\infty}-u^{a}(1^{-}))-\int_{\Omega^{a}_{\varepsilon}\setminus\Omega^{a}}{\hat{W}}^{**}({\bf
0})dx$ $\displaystyle=$ (3.40) $\displaystyle=$
$\displaystyle\int_{\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}u^{a}(x_{3}))dx+\int_{\Omega^{a}}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}u^{a}}{d|D^{s}_{3}u^{a}|},\right)d|D^{s}_{3}u^{a}|$
$\displaystyle+$
$\displaystyle(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+}))+(\hat{W}^{\ast\ast})^{\infty}(c^{a,\infty}-u^{a}(1^{-}))$
(3.41)

In (3.40) we exploited Dominated Convergence theorem, the continuity of
$\hat{W}^{\ast\ast}$ and $\nabla_{3}\tilde{u}_{n}^{a}\to 0$ in
$L^{1}(\Omega^{a}_{\varepsilon}\setminus\Omega^{a};{\mathbb{R}}^{3})$, due to
(3.36), the definition of $\tilde{u}^{a}_{n}$ and $\tilde{u}^{a}$, and (3.37)
together with (3.38).

In (3.41) we exploited that $\mathcal{L}^{2}(\omega)=1$ and

$\displaystyle\int_{\omega{\times}]-\varepsilon,0]}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}u^{a}}{d|D^{s}_{3}u^{a}|},\right)d|D^{s}_{3}u^{a}|$
$\displaystyle=\int_{\omega{\times}\\{0\\}}({\hat{W}}^{**})^{\infty}\left(\frac{dD^{s}_{3}u^{a}}{d|D^{s}_{3}u^{a}|},\right)d|D^{s}_{3}u^{a}|=$
$\displaystyle=(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+}))$

Consequently, by letting $\varepsilon\to 0$, from (3.41) it results that

$J(u^{a})=\lim_{n\to\infty}\int_{\Omega^{a}}{\hat{W}}^{**}(\nabla_{3}u_{n}^{a}(x))dx\geq
K^{a,\infty}(u^{a})+(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+})).$

∎

Now we are in position to prove the upper bound.

A construction similar to Proposition 3.5 gives rise to the proof of the
following result, which is key for the attainment of the upper bound

######  Proposition 3.15.

Under the same assumptions of Proposition 3.14

$\displaystyle\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{\infty}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{\infty},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}),u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\leq\int_{0}^{1}\hat{W}(\nabla
u^{a}(x_{3}))dx,$

for every $u^{a}\in c^{a,\infty}x_{3}+W^{1,1}_{0}(]0,1[;{\mathbb{R}}^{3})$.

###### Proof.

Considering the following sequences and with the arguments used in Proposition
3.5 we get the desired result.

For $z^{a}\in W^{1,1}(]0,1[;{\mathbb{R}}^{3{\times}2})$, recalling that, by
Lemma 3.13, the limit function of every converging sequence $(u^{b}_{n})_{n}$
above, is $u^{b}=(x_{\alpha},0)$,

$\displaystyle
u_{n}^{a}(x_{\alpha},x_{3}):=\left\\{\begin{array}[]{ll}[z^{a}(\varepsilon_{n})r_{n}x_{\alpha}+u^{a}(\varepsilon_{n})]\frac{x_{3}}{\varepsilon_{n}}+u^{b}(r_{n}x_{\alpha})\frac{\varepsilon_{n}-x_{3}}{\varepsilon_{n}}&\hbox{
if }(x_{\alpha},x_{3})\in\omega\times]0,\varepsilon_{n}[,\\\
z^{a}(x_{3})r_{n}x_{\alpha}+u^{a}(x_{3})&\hbox{ if
}(x_{\alpha},x_{3})\in\omega\times[\varepsilon_{n},1[,\end{array}\right.$
$\displaystyle u^{b}_{n}(x_{\alpha},x_{3}):=(x_{\alpha},h_{n}x_{3})\hbox{ if
}(x_{\alpha},x_{3})\in\Omega^{b}.$

Indeed it is easily seen that
$(u_{n}^{a},u^{b}_{n})\in\mathcal{U}^{\infty}_{n}$. Furthermore (3.7), (3.8)
and (3.9) hold, with $K^{a,q}_{n}$ and $K^{b,q}_{n}$ replaced by
$K^{a,\infty}_{n}$ and $K^{b,\infty}_{n}$, respectively, in the last equality.
∎

Notice that $c^{a,\infty}x_{3}+W^{1,1}_{0}(]0,1[;{\mathbb{R}}^{3})=\\{u\in
W^{1,1}(]0,1[;{\mathbb{R}}^{3}):\;u(0)=0\hbox{ and }u(1)=c^{a,\infty}\\}$.
When relaxing to BV and applying [8, Example 4.1] we get the subsequent
result.

######  Proposition 3.16 (Upper bound $(\infty,\infty)$).

Under the same assumptions as Proposition 3.14,

$\displaystyle\inf\left\\{\liminf_{n}F_{n}^{\infty}(u^{a}_{n},u^{b}_{n}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty}_{n},u_{n}^{a}\overset{\ast}{\rightharpoonup}u^{a}\hbox{
in }BV(\Omega^{a};\mathbb{R}^{3}),u_{n}^{a}(\cdot,0)\to{\bf 0}\hbox{ in
}L^{1}(\omega;{\mathbb{R}}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3})\right\\}\leq
K^{a,\infty}(u^{a})+(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+})).$

###### Proof.

The proof relies on the same arguments exploited in the proof of Proposition
3.11, namely on the lower semicontinuity of the functional in (3.2.2) with
respect to the weak * convergence in $BV(\Omega^{a};\mathbb{R}^{3})$ (see
Lemma 2.9, and the relaxation with respect to the same topology of the
functional
$F_{a}:c^{a,\infty}x_{3}+W^{1,1}_{a}(]0,1[;\mathbb{R}^{3})\to\mathbb{R}^{+}$
defined as $\int_{\Omega^{a}}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx$). Indeed the
latter relaxed functional coincides with
$K^{a,\infty}(u^{a})+(\hat{W}^{\ast\ast})^{\infty}(u^{a}(0^{+}))$, in view of
[8, Example 4.1].

∎

######  Remark 3.17.

Notice that in this section, either when $h_{n}/r_{n}\to\infty$ or
$h_{n}/r_{n}\to 0$, the arguments regarding the ”above” functions
$(u_{n}^{a})_{n}$ in the upper bounds and lower bounds are similar to the
first case, $\ell=q\in]0,+\infty[$. Notice also that Remark 3.7 applies to the
lower bound Propositions 3.9 and 3.14. For the upper bounds in Propositions
3.11 and 3.15 we note that if in the construction of the recovery sequences we
replace $z^{a}$ by $z^{a}+d^{a,\infty}$ with $z^{a}\in
W^{1,1}_{0}(]0,1[;{\mathbb{R}}^{3{\times}2})$ and a general
$d^{a,\infty}\in{\mathbb{R}}^{3{\times}2}$, we achieve the same results.

###  3.3 Case $l=0$

We now consider the case (1.27). For the same reasons as in the preceding
subsection, we can neglect the term
$\frac{r_{n}^{2}}{h_{n}}\int_{\Omega^{a}}H_{n}^{a,0}(x)\cdot
u^{a}_{n}(x)dx+\int_{\Omega^{b}}H_{n}^{b,0}(x)\cdot u^{b}_{n}(x)dx$ based on
(1.27). We can therefore also assume $H_{n}^{a,0}=0$ and $H_{n}^{b,0}=0$.

######  Proposition 3.18.

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4) and (1.36). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be as
in (1.2), with $\ell$ in (1.3) coinciding with $0$. For $n\in\mathbb{N}$ let
the spaces $\mathcal{U}_{n}^{0}$ be introduced in (1.8) where the Dirichlet
conditions in ${\mathcal{U}}_{n}^{0}$ are as in (1.15). Let $F_{n}^{0}$ be the
energy functional defined in (1.12), and the loads $H_{n}^{a,0},H_{n}^{b,0}$
satisfy (1.27).

Then, for every $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{0}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{0}[(u_{n}^{a},u_{n}^{b})]|<+\infty$, there
exist an increasing sequence of positive integer numbers
$(n_{i})_{i\in\mathbb{N}}$, ${u}^{b}\in BV(\omega;{\mathbb{R}}^{3})$,
depending possibly on the selected subsequence $(n_{i})_{i\in\mathbb{N}}$,
such that

$\left\\{\begin{array}[]{l}u^{a}_{n_{i}}\to(0_{\alpha},x_{3})\hbox{ in
}W^{1,1}(\Omega^{b};\mathbb{R}^{3}),\\\ \\\
u^{b}_{n_{i}}\overset{\ast}{\rightharpoonup}{u}^{b}\hbox{ in
}BV(\Omega^{a};\mathbb{R}^{3}),\end{array}\right.$

$\displaystyle{\frac{1}{r_{n_{i}}}}\nabla_{\alpha}u^{a}_{n_{i}}\to
Id_{\alpha}\hbox{ in }L^{1}(\Omega^{b};\mathbb{R}^{3{\times}2}).$

with $Id_{\alpha}\in{\mathbb{R}}^{3{\times}2}$ as in (1.1)

###### Proof.

The proof is omitted since it is identical to the one of Proposition 3.8. ∎

In view of Proposition 3.18, for $u^{b}\in BV(\omega;\mathbb{R}^{3})$, the
left hand side of (1.3) can be written as

$\displaystyle G^{0}((0_{\alpha},x_{3}),u^{b})=\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{0}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{0},\right.$
$\displaystyle\left.u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{ in
}W^{1,1}(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\}.$

where $G^{0}$ is defined in (2.8).

######  Proposition 3.19 (Lower bound, $0$).

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (1.4) and (1.36). Let $r_{n}$, $h_{n}$ and the loads
$H_{n}^{a,0},H_{n}^{b,0}$ satisfy conditions (1.2), (1.3), with $\ell=0$, and
(1.27) respectively.

For every $n\in\mathbb{N}$, let the spaces $\mathcal{U}_{n}^{0}$ be introduced
in (1.8) where the Dirichlet conditions in ${\mathcal{U}}_{n}^{0}$ are as in
(1.15). Let $F_{n}^{0}$ be the energy functional defined in (1.12). Assume
also that the recession function of $W_{0}$ (1.29), defined as in (1.30),
satisfies (1.32).

Then,

$\displaystyle\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{0}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{0},u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{
in
}W^{1,1}(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\}\geq K^{b,0}(u^{b}),$ (3.42)

for every $u^{b}\in BV(\Omega^{b};{\mathbb{R}}^{3})$, where $K^{b,0}$ is
defined as in (1.34).

###### Proof.

The relaxed energy (3.42) has a clear lower bound

$\displaystyle\inf\left\\{\liminf_{n}K^{b,0}_{n}(u^{b}_{n}):u_{n}^{b}\in(W_{b}^{1,1}(\Omega^{b};{\mathbb{R}}^{3})+f^{b,0}),\;u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }BV(\Omega^{b};\mathbb{R}^{3})\right\\}.$ (3.43)

where $f^{b,0}$ is as in (1.15), and extended as a function of $\Omega^{b}$,
constantly in the $x_{3}$ direction. By the same token as in Proposition 3.3,
$K^{b,0}(u^{b})$ is a lower bound for (3.43) which implies that it is a lower
bound also for (1.3).

∎

######  Proposition 3.20 (Upper bound $0$).

Under the same assumptions of Proposition 3.19,

$\displaystyle\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{0}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n},u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{
in }W^{1,1}(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3})\right\\}\leq\int_{\Omega^{b}}W_{0}(\nabla_{\alpha}u^{b}(x_{\alpha}))dx,$

for every $u^{b}\in f^{b,0}+C^{\infty}_{0}(\omega;{\mathbb{R}}^{3})$.

###### Proof.

The construction of the recovery sequence is similar to the construction in
Proposition 3.10.

For $z^{b}\in W^{1,1}_{0}(\omega;{\mathbb{R}}^{3})$ consider the following
sequences:

$\displaystyle u_{n}^{a}(x_{\alpha},x_{3})$
$\displaystyle=(r_{n}x_{\alpha},x_{3}),$
$\displaystyle(x_{\alpha},x_{3})\in\Omega^{a},$ $\displaystyle
u_{n}^{b}(x_{\alpha},x_{3})$
$\displaystyle=(u^{b}+z^{b}h_{n}x_{3})\cdot(1-\phi_{n}(x_{\alpha}))\;+\;u_{n}^{a}\left(\frac{1}{r_{n}}x_{\alpha},-h_{n}x_{3}\right)\cdot\phi_{n}(x_{\alpha}),$
$\displaystyle=(u^{b}+z^{b}h_{n}x_{3})\cdot(1-\phi_{n}(x_{\alpha}))\;+\;(x_{\alpha},-h_{n}x_{3})\phi_{n}(x_{\alpha})$
$\displaystyle(x_{\alpha},x_{3})\in\Omega^{b},$

where, for every $n$, $\phi_{n}$ is the cut-off function defined in (3.22)
satisfying:

$\displaystyle\phi_{n}(x_{\alpha}):=\left\\{\begin{array}[]{ll}1&\hbox{ if
}x_{\alpha}\in r_{n}\omega,\\\ 0&\hbox{ if
}x_{\alpha}\notin(1+\beta)r_{n}\omega,\end{array}\right.$
$\displaystyle|\nabla\phi_{n}(x_{\alpha})|\leq\frac{C}{\beta
r_{n}}\quad\hbox{and}\quad 0\leq\phi_{n}(x_{\alpha})\leq 1.$

Here $\beta>0$ is a fixed arbitrary value. It results that

$\displaystyle\nabla_{\alpha}u_{n}^{b}(x_{\alpha},x_{3})=$
$\displaystyle\;[\nabla_{\alpha}u^{b}(x_{\alpha})+\nabla_{\alpha}z^{b}(x_{\alpha})h_{n}x_{3}](1-\phi_{n}(x_{\alpha})),$
$\displaystyle+\nabla_{\alpha}\phi_{n}(x_{\alpha})\otimes\left[(x_{\alpha},-h_{n}x_{3})-(u^{b}(x_{\alpha})+z^{b}(x_{\alpha})h_{n}x_{3}\right]+\phi_{n}(x_{\alpha})Id_{\alpha},$
$\displaystyle\nabla_{3}u_{n}^{b}(x_{\alpha},x_{3})=$ $\displaystyle
h_{n}\left(z^{b}(x_{\alpha})-\left(Id_{3}+z^{b}(x_{\alpha})\right)\phi_{n}(x_{\alpha})\right).$

with $Id_{\alpha}\in{\mathbb{R}}^{3{\times}2}$ as in (1.1).

Hence,

$\frac{1}{h_{n}}\nabla_{3}u_{n}^{b}\to z^{b}\hbox{ in
}L^{1}(\Omega^{b};{\mathbb{R}}^{3}).$

Defining, as above, $B_{n}:=(1+\beta)r_{n}\omega\times]-1,0[$ and
$A_{n}:=r_{n}\omega\times]-1,0[$,

$\displaystyle\int_{\Omega^{b}}|\nabla_{\alpha}u_{n}^{b}-\nabla_{\alpha}u^{b}|dx\leq$
$\displaystyle\int_{\Omega^{b}\setminus
A_{n}}|\nabla_{\alpha}z^{b}h_{n}x_{3}|dx+\frac{C}{r_{n}}\int_{B_{n}\setminus
A_{n}}\left[|(x_{\alpha},-h_{n}x_{3})|+|u^{b}|+|z^{b}h_{n}x_{3}|\right]dx\;+$
$\displaystyle+$
$\displaystyle\int_{B_{n}}\left|\phi_{n}(x_{\alpha})Id_{\alpha}\right|+|\nabla_{\alpha}u^{b}|dx$
(3.44)

Because $|B_{n}|=O(r_{n}^{2})$ and for $n\in{\mathbb{N}}$ large enough $u_{b}$
is bounded in $L^{\infty}(B_{n};\mathbb{R}^{3})$ (since $f^{b,0}$ satisfies
(1.9) and $u^{b}\in f^{b,0}+C^{\infty}_{0}(\omega;{\mathbb{R}}^{3})$). then we
have that (3.44) tends to $0$ as $n\to\infty$, i.e.

$\nabla_{\alpha}u_{n}^{b}\to\nabla_{\alpha}u^{b}\hbox{ in
}L^{1}(\Omega^{b};{\mathbb{R}}^{3}).$

By the continuity of $W$, the dominated convergence theorem and $W(Id)=0$ we
get:

$\displaystyle\lim_{n\to\infty}\left(\frac{r_{n}^{2}}{h_{n}}K^{a,0}_{n}(u^{a}_{n})+K^{b,0}_{n}(u^{b}_{n})\right)=$
$\displaystyle\lim_{n\to\infty}\int_{\Omega^{b}}W\left(\nabla_{\alpha}u_{n}^{b}(x)\;\;|\;\;\frac{1}{h_{n}}\nabla_{3}u_{n}^{b}(x)\right)dx=$
$\displaystyle=$
$\displaystyle\int_{\Omega^{b}}W(\nabla_{\alpha}u^{b}(x_{\alpha})\;\;|\;\;z^{b}(x_{\alpha}))dx$

Once again by the arguments in [44, Proposition 7] we get the desired result
by infimizing in $z^{b}\in W^{1,1}_{0}(\omega;{\mathbb{R}}^{3})$.

∎

######  Proposition 3.21 (Upper bound $0$).

Under the same assumptions as Proposition 3.19 we have

$\displaystyle\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{0}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}_{n}^{0},u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{
in }W^{1,1}(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{ in
}BV(\Omega^{b};\mathbb{R}^{3})\right\\}\leq K^{b,0}(u^{b}),$

for every $u^{b}\in BV(\omega;{\mathbb{R}}^{3})$, where $K^{b,0}$ is defined
in (1.34).

###### Proof.

From Proposition 3.20, by applying a similar reasoning as in Proposition 3.11
or Proposition 3.6 we get the desired result. ∎

######  Remark 3.22.

We note that, similarly to Remark 3.17, in this section the arguments
regarding the ”below” functions $(u_{n}^{b})_{n}$ in the upper bound and lower
bound results are similar to the first case, $\ell=q\in]0,+\infty[$. Remark
3.7 also applies to Proposition 3.19. For Proposition 3.20 we note that if in
the construction of the recovery sequence we replace $z^{b}$ by
$z^{b}+g^{b,0}$ with $z^{b}\in W^{1,1}_{0}(\omega;{\mathbb{R}}^{3})$ and a
general $g^{b,0}\in W^{1,1}(\omega;{\mathbb{R}}^{3})$ then we achieve the same
results.

##  4 The super-linear case

For the sake of completeness we comment here on the case of super-linear
growth for $\ell=+\infty$ and $\ell=0$, cases that were not addressed in [31].
We underline that our subsequent analysis is quite complete in the case
$\ell=\infty$, in the sense that all the $p>1$ are taken into account, while
in the case $\ell=0$, due to the adopted techniques we are able to consider
only $1<p<2$.

For $p>1$, assume that $W$ is a Borel function satisfying the following
hypotheses:

$W(M)\leq C(1+|M|^{p}),\quad\forall M\in\mathbb{R}^{3{\times}3}$ (4.1)

for some $C>0$ and

$\frac{1}{C}|M-Id|^{p}\leq W(M)\quad\forall
M\in\mathbb{R}^{{3{\times}3}},\quad\;\;W(Id)=0.$ (4.2)

###  4.1 Case $\ell=+\infty$

We start defining

$\displaystyle\mathcal{U}^{\infty,p}_{n}=\Big{\\{}$
$\displaystyle(u^{a},u^{b})\in\left(c^{a,\infty}+W_{a}^{1,p}(\Omega^{a};\mathbb{R}^{3})\right){\times}\left(f^{b,\infty}+h_{n}g^{b,\infty}x_{3}+W^{1,p}_{b}(\Omega^{b};\mathbb{R}^{3})\right):$
(4.3) $\displaystyle\qquad u^{a}\text{ and }u^{b}\text{ satisfy
\eqref{eq:bc}}\Big{\\}}$

with the boundary data given by (1.14) and
$W_{a}^{1,p}(\Omega^{a};\mathbb{R}^{3})$ the closure, with respect to
$W^{1,p}$-norm, of

$\left\\{u^{a}\in
C^{\infty}(\overline{\Omega^{a}};\mathbb{R}^{3})\,:\,u^{a}={\bf 0}\hbox{ in a
neighbourhood of }\omega{\times}\\{1\\}\right\\},$

and $W_{b}^{1,p}(\Omega^{b};\mathbb{R}^{3})$ the closure, with respect to
$W^{1,p}$-norm, of

$\big{\\{}u^{b}\in
C^{\infty}(\overline{\Omega^{b}};\mathbb{R}^{3})\,:\,u^{b}={\bf 0}\hbox{ in a
neighbourhood of }\partial\omega\times]-1,0[\big{\\}}.$

Moreover, we define

$\displaystyle
F_{n}^{\infty,p}(u_{n}^{a},u_{n}^{b})=K_{n}^{a,\infty}(u_{n}^{a})+\frac{h_{n}}{r_{n}^{2}}K_{n}^{b,\infty}(u_{n}^{b}),\quad(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty,p}_{n},$
(4.4)

where $K_{n}^{a,\infty},K_{n}^{b,\infty}$ are defined by (1.16) and (1.17),
taking into account that the loads are

$\displaystyle H_{n}^{a,\infty}\rightharpoonup H\hbox{ in
}L^{p^{\prime}}(\Omega^{a};\mathbb{R}^{3}),\hbox{ and
}\frac{h_{n}}{r^{2}_{n}}H_{n}^{b,\infty}\rightharpoonup 0\hbox{ in
}L^{p^{\prime}}(\Omega^{b};\mathbb{R}^{3})$ (4.5)

where $H\in L^{p^{\prime}}(\Omega;\mathbb{R}^{3})$, with
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

In this case, the compactness result becomes

######  Proposition 4.1.

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (4.1) and (4.2). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be as in
(1.2), with $\ell$ in (1.3) coinciding with $+\infty.$ Let the spaces
$\mathcal{U}_{n}^{p,\infty}$ be as in (4.3) with Dirichlet conditions as in
(1.14), and the loads $H_{n}^{a,\infty},H_{n}^{b,\infty}$ satisfy (4.5). Let
$F_{n}^{\infty,p}$ be the energy functional defined in (4.4). Then, for every
$(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty,p}_{n}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{\infty,p}[(u_{n}^{a},u_{n}^{b})]|<+\infty$,
there exist an increasing sequence of positive integer numbers
$(n_{i})_{i\in\mathbb{N}}$, ${u}^{a}\in W^{1,p}(]0,1[;{\mathbb{R}}^{3})$
depending possibly on the selected subsequence $(n_{i})_{i\in\mathbb{N}}$,
such that

$\left\\{\begin{array}[]{l}u^{a}_{n_{i}}\rightharpoonup{u}^{a}\hbox{ in
}W^{1,p}(\Omega^{a};\mathbb{R}^{3}),\\\ \\\
u^{b}_{n_{i}}\to(x_{\alpha},0)\hbox{ in
}W^{1,p}(\Omega^{b};\mathbb{R}^{3}),\end{array}\right.$

and, it results

$\displaystyle{\frac{1}{h_{n_{i}}}}\nabla_{3}u^{b}_{n_{i}}\to(0_{\alpha},1)\hbox{
in }L^{p}(\Omega^{b};\mathbb{R}^{3}).$

Furthermore, if $p>2$, $u^{a}(0)={\bf 0}$. Moreover, for every $p>1$, if
$\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=\infty$, then
$\lim_{n\to\infty}u^{a}_{n}(\cdot,0)={\bf 0}$ in
$L^{p}(\omega;\mathbb{R}^{3})$.

The proof is omitted, being very similar to the other compactness results
proven above, exploiting also the same arguments as in [32, Proposition 2.1]
to obtain the null condition at the origin for $u^{a}$. The last convergence,
which holds even if $p\leq 2$, is based on arguments entirely identical to
those of Proposition 3.12 and Lemma 3.13.

The following result holds

######  Proposition 4.2 ($\Gamma$ \- limit $(p,\infty)$).

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (4.1) and (4.2). For $n\in\mathbb{N}$ let the spaces
$\mathcal{U}_{n}^{\infty}$ be introduced in (4.3) where the Dirichlet
conditions in ${\mathcal{U}}_{n}^{\infty,p}$ are as in (1.14). Let
$F_{n}^{\infty,p}$ be the energy functional defined in (4.4), where $r_{n}$,
$h_{n}$ satisfy conditions (1.2) with $\ell=\infty$ and the loads
$H_{n}^{a,\infty},H_{n}^{b,\infty}$ satisfy conditions (4.5).

Then, if $1<p<2$ and $\sup_{n\in\mathbb{N}}\frac{h_{n}}{r_{n}}<+\infty$ or if
$p=2$ and $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$ it results that for every
$u^{a}\in c^{a,\infty}+W^{1,p}_{a}(]0,1[;{\mathbb{R}}^{3})$

$\displaystyle G^{\infty,p}(u^{a},(x_{\alpha},0)):=\inf$
$\displaystyle\left\\{\liminf_{n}F_{n}^{\infty,p}(u_{n}^{a},u_{n}^{b}):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{\infty,p}_{n},u_{n}^{a}\rightharpoonup
u^{a}\hbox{ in }W^{1,p}(\Omega^{a};\mathbb{R}^{3}),\right.$
$\displaystyle\left.u_{n}^{b}\to(x_{\alpha},0)\hbox{ in
}W^{1,p}(\Omega^{b};\mathbb{R}^{3})\right\\}=\displaystyle{\int_{]0,1[}{\hat{W}}^{**}\left(\nabla_{3}u^{a}(x_{3})\right)}dx+\int_{\Omega^{a}}H(x)\cdot
u^{a}(x_{3})dx,$ (4.6)

On the other hand, if either $p>2$ (with no assumption on the ratio
$h_{n}/r_{n}$) or $1<p\leq 2$ and
$\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=\infty$, then, the same representation
holds for every $u^{a}\in
c^{a,\infty}x_{3}+W^{1,p}_{0}(]0,1[;{\mathbb{R}}^{3})$.

###### Proof.

As in the case $p=1$ we assume that $H={\bf 0}$. Then, it is easily observed
that the functional
$\int_{]0,1[}{\hat{W}}^{**}\left(\nabla_{3}u^{a}(x_{3})\right)dx$ is trivially
a lower bound for the left hand side energy in (4.6), due to standard lower
semicontinuity results in the Sobolev setting, see [20].

For what concerns the upper bound in the first case, i.e. if $1<p<2$ and
$\sup_{n\in\mathbb{N}}\frac{h_{n}}{r_{n}}<\infty$ or if $p=2$ and
$\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=0$, one can argue as in the case $p=1$
and $(\infty,0)$, Proposition 3.10, changing slightly the cut-off function in
the argument for the upper bound therein. That is, the cut-off functions in
the recovery sequence are functions $\phi_{n}$ such that

$\phi_{n}(x_{\alpha}):=\left\\{\begin{array}[]{ll}1&\hbox{ if }x_{\alpha}\in
r_{n}^{1/p}\omega,\\\ 0&\hbox{ if
}x_{\alpha}\notin(1+\beta)r_{n}^{1/p}\omega\end{array}\right.$

Noticing that $\|\nabla\phi_{n}\|_{L^{\infty}}\leq\frac{1}{\beta
r_{n}^{1/p}}$, estimates analogous to (3.25) and (3.26) hold. The same
conclusion to get an upper bound of the type
$\int_{0}^{1}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx_{3}$ can be easily achieved in
$c^{a,\infty}+W^{1,p}_{a}(]0,1[;\mathbb{R}^{3})$. To get the desired upper
bound we apply a standard relaxation.

As regards the upper bound in the cases $\ell=+\infty$ and $p>2$ or
$\ell=+\infty$, $\lim_{n\to\infty}\frac{h_{n}}{r_{n}}=\infty$ for every $p>1$,
we can argue as in the proof of Proposition 3.15 to get

$\displaystyle
G^{\infty,p}(u^{a},(x_{\alpha},0))\leq\int_{0}^{1}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx_{3}$

for every $u^{a}\in c^{a,\infty}x_{3}+W^{1,p}_{0}(]0,1[;\mathbb{R}^{3})$. To
conclude it is enough to apply once again standard relaxation results for
$\int_{0}^{1}\hat{W}(\nabla_{3}u^{a}(x_{3}))dx_{3}$ in
$c^{a,\infty}x_{3}+W^{1,p}_{0}(]0,1[;\mathbb{R}^{3})$ (see [20]) and exploit
the weak lower semicontinuity in $W^{1,p}(]0,1[;\mathbb{R}^{3})$ of
$G^{\infty,p}$. ∎

###  4.2 Case $\ell=0$

In this case, we are lead to study the asymptotic behaviour, with respect the
$W^{1,p}$-weak convergence, as $n\to\infty$ of the rescaled energy
$F_{n}^{0,p}:{\mathcal{U}}_{n}^{0,p}\to\mathbb{R}$ defined in analogy with
(1.12),

$\displaystyle
F_{n}^{0,p}(u_{n}^{a},u_{n}^{b})=\frac{r_{n}^{2}}{h_{n}}K_{n}^{a,0}(u_{n}^{a})+K_{n}^{b,0}(u_{n}^{b}),\quad(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{0,p}_{n},$
(4.7)

where $K_{n}^{a,0},K_{n}^{b,0}$ are defined by (1.16) and (1.17), taking into
account that the loads satisfy

$\displaystyle H_{n}^{b,0}\rightharpoonup H,\hbox{ and
}\frac{r^{2}_{n}}{h_{n}}H_{n}^{a,0}\rightharpoonup 0\;\;\hbox{ in
$L^{p^{\prime}}(\Omega^{a},{\mathbb{R}}^{3})$},$ (4.8)

where $H$ and $p^{\prime}$ are as in (4.5), and

$\displaystyle\mathcal{U}^{0,p}_{n}=\Big{\\{}$
$\displaystyle(u^{a},u^{b})\in\left(c^{a,0}+r_{n}d^{a,0}x_{\alpha}+W_{a}^{1,p}(\Omega^{a};\mathbb{R}^{3})\right){\times}\left(f^{b,0}+h_{n}g^{b,0}x_{3}+W^{1,p}_{b}(\Omega^{b};\mathbb{R}^{3})\right):$
(4.9) $\displaystyle\qquad u^{a}\text{ and }u^{b}\text{ satisfy
\eqref{eq:bc}}\Big{\\}},$

with the boundary data given by (1.15) (with $f^{b,0}\in
W^{1,p}(\omega;\mathbb{R}^{3})$).

The $W^{1,p}$ counterpart of Proposition 3.18 asserts the following

######  Proposition 4.3.

Consider the Borel function $W:\mathbb{R}^{3{\times}3}\to\mathbb{R}$
satisfying (4.1) and (4.2). For $n\in\mathbb{N}$ let $r_{n}$, $h_{n}$ be as in
(1.2), with $\ell$ in (1.3) coinciding with $0$. Let the spaces
$\mathcal{U}_{n}^{0,p}$ be as in (4.9) with Dirichlet conditions as in (1.15),
and the loads $H_{n}^{a,0},H_{n}^{b,0}$ satisfy (4.8). Let $F_{n}^{0,p}$ be
the energy functional defined in (4.7).

Then, for every $(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{0,p}_{n}$ such that
$\sup_{n\in{\mathbb{N}}}|F_{n}^{0,p}[(u_{n}^{a},u_{n}^{b})]|<+\infty$, there
exist an increasing sequence of positive integer numbers
$(n_{i})_{i\in\mathbb{N}}$, ${u}^{b}\in
f^{b,0}+W^{1,p}_{b}(\omega;{\mathbb{R}}^{3})$, depending possibly on the
selected subsequence $(n_{i})_{i\in\mathbb{N}}$, with
$u^{b}(0_{\alpha})=u^{a}(0)=0$ if $p>2$ such that

$\left\\{\begin{array}[]{l}u^{a}_{n_{i}}\to(0_{\alpha},x_{3})\hbox{ in
}W^{1,p}(\Omega^{b};\mathbb{R}^{3}),\\\ \\\
u^{b}_{n_{i}}\overset{\ast}{\rightharpoonup}{u}^{b}\hbox{ in
}W^{1,p}(\Omega^{a};\mathbb{R}^{3}),\end{array}\right.$
$\displaystyle{\frac{1}{r_{n_{i}}}}\nabla_{\alpha}u^{a}_{n_{i}}\to
Id_{\alpha}\hbox{ in }L^{p}(\Omega^{b};\mathbb{R}^{3{\times}2}).$

where $Id_{\alpha}\in\mathbb{R}^{3\times 2}$ as in (1.1).

The proof is omitted, being very similar to the one of Proposition 3.18,
relying in turn on [32, Proposition 2.1] and the results contained in [31,
Section 5].

The relaxation now has the form, for $u^{b}\in
f^{b,0}+W^{1,p}_{a}(\omega,{\mathbb{R}}^{3})$

$\displaystyle
G^{0,p}(u^{b},(0_{\alpha},x_{3})):=\inf\left\\{\liminf_{n}\left(\frac{r_{n}^{2}}{h_{n}}K^{a}_{n}(u^{a}_{n})+K^{b}_{n}(u^{b}_{n})\right):(u_{n}^{a},u_{n}^{b})\in\mathcal{U}^{0,p}_{n},\right.$
$\displaystyle\left.u_{n}^{a}\to(0_{\alpha},x_{3})\hbox{ in
}W^{1,p}(\Omega^{a};\mathbb{R}^{3}),u_{n}^{b}\overset{\ast}{\rightharpoonup}u^{b}\hbox{
in }W^{1,p}(\Omega^{b};\mathbb{R}^{3})\right\\}.$

The following representation result holds

######  Proposition 4.4 ($\Gamma$-limit $(p,0)$).

Under the same assumptions as in Proposition 4.3, and with $1<p<2$, it results

$\displaystyle
G^{0,p}((0_{\alpha},x_{3}),u^{b})=\int_{\Omega^{b}}QW_{0}(\nabla_{\alpha}u^{b}(x_{\alpha}))dx+\int_{\Omega^{b}}H(x)\cdot
u^{b}(x_{\alpha})dx_{\alpha},$

for every $u^{b}\in f^{0,b}+W^{1,p}_{a}(\omega;\mathbb{R}^{3})$.

###### Proof.

The proof is sketched, (assuming once again for simplicity that the loads are
null) relying on arguments entirely similar to the lower bound in Proposition
3.19 and the upper bound in Proposition 3.20, taking into account that, with
the latter argument we get

$G^{0,p}((0_{\alpha},x_{3}),u^{b})\leq\int_{\Omega^{b}}W_{0}(\nabla_{\alpha}u^{b}(x_{\alpha}))dx,$

for every $u^{b}\in f^{0,b}+C^{\infty}_{0}(\omega;\mathbb{R}^{3})$, with
$u^{b}(0_{\alpha})=0$. The inequality

$G^{0,p}((0_{\alpha},x_{3}),u^{b})\leq\int_{\Omega^{b}}QW_{0}(\nabla_{\alpha}u^{b}(x_{\alpha}))dx,$

for every $u^{b}\in f^{0,b}+W^{1,p}_{a}(\omega;\mathbb{R}^{3}),$ is obtained
via standard density results as those in [31, Section 4]. ∎

## Acknowledgements

Part of this paper has been written during the Erasmus traineeship of GC in
the second semester of 2023 at Sapienza-University of Rome and while JM was
visiting professor there at Dipartimento di Scienze di Base ed Applicate per
l’Ingegneria, whose hospitality both gratefully acknowledge. JM also
acknowledges finantial support from GNAMPA project, professori visitatori 2022
and FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and
UIDP/04459/2020. EZ is a member of INdAM-GNAMPA and thanks for its support,
through GNAMPA Project 2023 ‘Prospettive nelle Scienze dei Materiali: modelli
variazionali, analisi asintotica e omogeneizzazione’.

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