# Asymptotic analysis of thin structures with
point dependent energy growth

Michela Eleuteri
Università degli Studi di Modena e Reggio Emilia
Dipartimento di Scienze Fisiche Informatiche e Matematiche
via Campi 213/b, 41125 Modena (Italy)
<EMAIL_ADDRESS>
Francesca Prinari
Università degli Studi di Pisa
Dipartimento di Scienze Agrarie, Alimentari e Agro-ambientali
via del Borghetto, Pisa (Italy)
<EMAIL_ADDRESS>
Elvira Zappale
Sapienza - Università di Roma
Dipartimento di Scienze di Base e Applicate per l’Ingegneria
via Antonio Scarpa, 16, 00161, Roma (Italy)
<EMAIL_ADDRESS>
###### Abstract

$3d-2d$ dimensional reduction for hyperelastic thin films modeled through
energies with point dependent growth, assuming that the sample is clamped on
the lateral boundary, is performed in the framework of $\Gamma$-convergence.
Integral representation results, with a more regular lagrangian related to the
original energy density, are provided for the lower dimensional limiting
energy, in different contexts.

Keywords: $\Gamma$-convergence, dimension reduction, variable exponents

MSC2020: 49J45, 74K35, 74E05, 74E10.

###### Contents

  1. 1 Introduction
  2. 2  Notation and preliminary results
    1. 2.1 Variable exponents Lebesgue spaces
    2. 2.2 Trace operator in Sobolev spaces
    3. 2.3 Variable exponents Sobolev spaces
    4. 2.4 $\Gamma$-convergence
  3. 3 Some lower semicontinuity results
    1. 3.1 The convex case
    2. 3.2 The quasiconvex case
  4. 4 Dimensional reduction
    1. 4.1 Compactness of bounded energy sequences
    2. 4.2 The case of convex densities
    3. 4.3 The nonconvex case

##  1 Introduction

In solid mechanics, the equilibrium state of a body can be described by an
energy minimization problem, over a suitable class of fields, of an integral
functional of the type

$\int_{{\Omega}}f(x,\nabla v(x))dx,$

where $\Omega\subset\mathbb{R}^{N}$ is a bounded open set, $v$ is the
deformation or the displacement, defined in a suitable function space
$X(\Omega;\mathbb{R}^{d})$, e.g. $W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{d})$,
and $f:\Omega\times\mathbb{R}^{d\times N}\rightarrow\mathbb{R}$ is the
hyperelastic energy density, assumed as a Carathéodory function satisfying
suitable growth conditions.

In many applications, for instance to deal with multiphase materials, finer
and finer heterogeneities, shape optimization, thin structures, phase
transitions, etc. other parameters may come into play and the equilibrium
configurations arise as solutions of the following $\varepsilon$\-
parameterized problems

$\displaystyle\min_{v\in
X(\varepsilon)(\Omega_{\varepsilon};\mathbb{R}^{d})}{\mathcal{F}}_{\varepsilon}(v):=\min_{v\in
X(\varepsilon)(\Omega_{\varepsilon};\mathbb{R}^{d})}\frac{1}{\varepsilon}\int_{\Omega_{\varepsilon}}f(\varepsilon)(x,\nabla
v(x))dx$ (1.1)

where $\\{\Omega_{\varepsilon}\\}$ may represent a $\varepsilon$\- dependent
family of bounded open subsets in $\mathbb{R}^{N}$ and
$X(\varepsilon)(\Omega_{\varepsilon};\mathbb{R}^{d})$ is a suitable functional
space, possible encoding the anisotropy of the model, expressed through a
point-dependent integrability condition on the deformation gradients. Hence
one is interested in detecting, as $\varepsilon\rightarrow 0^{+}$, the
asymptotic behaviour of such configurations.

Two main types of questions are addressed in this paper, one dealing with the
detection of sufficient conditions on $X(\varepsilon)$, that, under special
structure conditions on $f(\varepsilon),$ ensure existence of solutions of the
problem (1.1), when the domain $\Omega_{\varepsilon}$ is a thin domain of the
form

$\displaystyle\Omega_{\varepsilon}:=\omega\times(-\varepsilon/2,\varepsilon/2),$
(1.2)

with $\omega\subset\mathbb{R}^{2}$ a bounded, connected, open set with
Lipschitz boundary, the other one concerning the asymptotic behaviour of
problems (1.1), under more general assumptions on $f(\varepsilon)$. In the
latter case, by means of a suitable variational convergence, we derive a
limiting problem defined on $\omega$ with a structure analogous to the
original one, which admits solutions. This approach rigorously justifies the
fact that, in the applications, when dealing with very thin structures, it is
convenient to work with a lower dimensional model. We refer to the pioneering
papers [1] and [44] where this approach has been formally obtained, in the
framework of $\Gamma$-convergence, in the hyperelastic setting for strings and
membranes, in the classical functional setting of Sobolev spaces.

The main novelty of this manuscript is that the energy density
$f(\varepsilon)$ depends on the parameter $\varepsilon$ also through its
growth which involves a so called variable growth exponent
$p_{\varepsilon}=p_{\varepsilon}(x)$ or better, a family of variable growth
exponents converging to a fixed one in some suitable topology. This leads to
non-trivial mathematical questions, because it appears to be challenging the
idea to control the dependence of the parameter $\varepsilon$ both in the set
and in the energy density.

In this context it is convenient to rephrase the problem (1.1) on a fixed
domain

$\Omega:=\Omega_{1}=\omega\times(-1/2,1/2)$ (1.3)

through a re-scaling in the transverse direction $x_{3}$, thus obtaining the
equivalent minimization problem

$\displaystyle\min_{u\in
X_{\varepsilon}(\Omega;\mathbb{R}^{3})}\mathcal{F}_{\varepsilon}(u):=\min_{u\in
X_{\varepsilon}(\Omega;\mathbb{R}^{3})}\int_{\Omega}f_{\varepsilon}\left(x,\nabla
u(x)\right)dx,$ (1.4)

where $f_{\varepsilon},$ $X_{\varepsilon}(\Omega;\mathbb{R}^{3})$, and $u$
have been obtained by $f(\varepsilon)$,
$X(\varepsilon)(\Omega_{\varepsilon};\mathbb{R}^{d})$ with $d=3$ and $v$,
respectively, via the change of variables which maps $x_{3}$ from
$(-\varepsilon/2,\varepsilon/2)$ into $(-1/2,1/2)$. In the first case, we
consider $p_{\varepsilon}:\Omega\rightarrow(1,+\infty)$,
$X_{\varepsilon}(\omega;{R}^{3})=W^{1,p_{\varepsilon}(\cdot)}(\Omega;\mathbb{R}^{3})$
and $f_{\varepsilon}:\Omega\times\mathbb{R}^{3\times 3}\rightarrow\mathbb{R}$
of the following type

$f_{\varepsilon}(x,\xi):=W^{p_{\varepsilon}(x)}(x,\xi),\quad\hbox{or }\quad
f_{\varepsilon}(x,\xi):=a_{\varepsilon}(x)W^{p_{\varepsilon}(x)}(\xi),$

with suitable functions $W$ and $a_{\varepsilon}$ (which are made precise in
the sequel). In particular we provide lower semicontinuity with respect to the
(strong $L^{\infty}\times L^{1}$) joint convergence of the sequences
$\\{p_{\varepsilon}\\}\subseteq L^{\infty}(\Omega;(1,+\infty))$ and
$\\{u_{\varepsilon}\\}\subseteq W^{1,1}(\Omega;\mathbb{R}^{d})$ (see Theorems
1.1 and 1.2).

In the second case, we deal with the very dimensional reduction problem where
the function $f_{\varepsilon}$ in (1.4), representing the hyperelastic energy
density of a $3d$ thin structure, is of the form

$\displaystyle f_{\varepsilon}(x,\xi):=f(x_{\alpha},\varepsilon
x_{3},p_{\varepsilon}(x),\xi_{\alpha},\tfrac{1}{\varepsilon}\xi_{3}),$ (1.5)

for every

$x\equiv(x_{\alpha},x_{3})\in\Omega,\,\xi\equiv(\xi_{\alpha},\xi_{3})\in\mathbb{R}^{3\times
3},\alpha=1,2.$

Here $p_{\varepsilon}$ plays the role of a variable exponent defined in
$\Omega$, i.e.

$p_{\varepsilon}(x)\equiv
p_{\varepsilon}(x_{\alpha},x_{3}):=p(x_{\alpha},\varepsilon x_{3})$ (1.6)

for ${\mathcal{L}^{3}}$\- a.e. $x\in\Omega$, and the re-scaled variable
$(\xi_{\alpha},\frac{1}{\varepsilon}\xi_{3})\in\mathbb{R}^{3\times 3}$ appears
to take into account the scaling of the gradient $\nabla
v=(\nabla_{\alpha}v,\nabla_{3}v)$ after the change of variable which maps it
in the scaled deformation gradient of $u$ in (1.4), cf. Section 4 for precise
definition.

Dimensional reduction problems of the type (1.1) have attracted much attention
in the past decades due to the many applications in engineering, materials
science, conductors, micromagnetics, chemistry and biology. Indeed a wide
literature has been developed to rigorously deduce a simplified formulation,
in a lower dimensional setting via a variational approach. Among a wide
literature, we refer to [2, 34] for micromagnetic and ferromgnetic materials,
to [5], [13] for brittle materials, to [15] in the case of materials which
allow for fracture and plastic behaviour, to [42, 20] in the case of
magnetoelasticity, to [30, 32, 33, 35, 37, 39] for the case of hyperelastic
multistructures, to [31], to model delamination, to [29, 49] to detect bending
effect, in particular in shells as in [38], or to [7, 12, 11] to describe thin
structures made by heterogeneous materials with fine distribution, also in the
discrete setting. Besides the above far from exhaustive bibliography, it is
worth to point out that particular minimum problems like the one in (1.4)
appear also in the modeling of conductors to detect the dielectric breakdown,
see e.g. [8], or in the context of damaging and optimal design (cf. in [12],
[16], and [41]). It is worth to point out that, besides it has been analyzed
the case of multicomponent materials which exhibit a very different behaviour
from one point to another, and imposing a perimeter penalization among the
interfaces of the components, see [9, 10], at the best of our knowledge, it
has been always considered the case where the energetic pointwise dependent
behaviour has a rough change from one point to another of the sample. The case
where the position dependent energetic behaviour is suitably continuous has
not yet been considered. This is indeed one of the topic of this paper, in the
pure hyperelastic setting, neglecting damage or shape optimization. Indeed,
despite the last mentioned results, the regularity of the energetic point
dependent behaviour allows us to obtain an explicit representation of the
limiting problems in any dimensional reduction setting.

We aim now to present the main results of the paper. To this purpose,
referring to [23] for a comprehensive treatment of the topic, we recall that a
(Lebesgue) measurable function $p:\Omega\rightarrow[1,+\infty]$, playing the
role of an exponent, is called a variable exponent and we denote by
$\mathcal{P}(\Omega)$ the class of variable exponents on $\Omega$. Moreover,
$\mathcal{P}_{b}(\Omega)$ and $\mathcal{P}^{log}(\Omega)$ are the subsets of
$\mathcal{P}(\Omega)$ whose elements satisfy the boundedness condition

$\displaystyle 1<{p^{-}}:=\mathop{\rm ess\>inf}_{\Omega}p(x)\leq\mathop{\rm
ess\>sup}_{\Omega}p(x)=:{p^{+}}<+\infty,$ (1.7)

and the so called $log$-Hölder continuity property:

$\displaystyle\exists\gamma>0:|p(x)-p(y)|\leq\frac{\gamma}{|\log|x-y||}\hbox{
for every }x,y\in\Omega,0<|x-y|<\frac{1}{2},$ (1.8)

respectively. Finally we set
$\mathcal{P}^{log}_{b}(\Omega):=\mathcal{P}^{log}(\Omega)\cap\mathcal{P}_{b}(\Omega)$
(according to Definitions 2.1 and 2.5). Condition (1.8) has been first
introduced by V.V. Zhikov in [50]; the failure of this condition is a possible
cause of discontinuity of minimizers, while, by assuming it, it is possible to
prove higher integrability of minimizers, which is the first step towards
further regularity. From that moment onwards, a large number of papers devoted
to the mathematical analysis of energy functionals involving variable
exponents appeared in several and different directions, motivated by the fact
that such types of energies describe models (also non variational) coming from
Mathematical Physics that are built using a variable growth exponent. For
instance, we refer to the recent contribution [3], where this theory is
employed in the study of elastic problems, to [21] and [47], where the
variable exponent growth condition is considered in the framework of free
discontinuity problems, i.e. for materials allowing for fractures and damage,
to [36], where a variable exponent model for image restoration has been
studied, from the existence point of view and the $\Gamma-$convergence one, in
the case that the exponent attains the critical value one, to [27] where
power-law approximation of supremal functionals has been studied once more by
means of the $\Gamma-$convergence tool.

Thus, we are in position to state our main results for the first class of
problems. To this end we recall that a Borel measurable function
$g:\mathbb{R}^{d\times N}\rightarrow\mathbb{R}$ is said quasiconvex if

$g(\xi)\leq\int_{Q}g(\xi+\nabla\varphi(x))\,dx$

holds, for every $\xi\in\mathbb{R}^{d\times N}$ and for every $\varphi\in
W^{1,\infty}_{0}(Q;\mathbb{R}^{d})$, where $Q:=\left(-1/2,1/2\right)^{N}$ and
$W^{1,\infty}_{0}(Q;\mathbb{R}^{d})=W^{1,\infty}(Q;\mathbb{R}^{d})\cap
W^{1,1}_{0}(Q;\mathbb{R}^{d})$ (we recall that a finite valued quasiconvex
function is locally Lipschitz, hence continuous and locally bounded; thus the
class of test functions can be replaced by $C^{\infty}_{c}(Q;\mathbb{R}^{d})$
or $C^{1}_{0}(Q;\mathbb{R}^{d})$, as in [45]). In Section 3 we show the
following lower semicontinuity. We stress that the $L^{\infty}$ convergence on
the variable exponents is a natural assumption in this context. Indeed, such
uniformly converging sequences as in Theorems 1.1 and 1.2 can arise by means
of a change of variables as in (1.6) from a single exponent $p$ satisfying a
uniform log-Hölder continuity assumption.

######  Theorem 1.1.

Let $\Omega\subseteq{R}^{N}$ be an open set. Let $W:\Omega\times{R}^{d\times
N}\rightarrow[0,+\infty)$ be a function such that

  * -

$W(\cdot,\xi)$ is $\mathcal{L}^{N}(\Omega)$-measurable for every
$\xi\in\mathbb{R}^{d\times N}$;

  * -

$W(x,\cdot)$ is quasiconvex for $\mathcal{L}^{N}$-a.e. $x\in\Omega$;

  * -

there exists $C_{2}>0$ such that

$W(x,\xi)\leq\,C_{2}(|\xi|+1),\qquad\hbox{for $\mathcal{L}^{N}$-a.e.
$x\in\Omega$, for every $\xi\in\mathbb{R}^{d\times N}.$}$ (1.9)

Let $p_{0}\in L^{\infty}(\Omega)$ such that $p_{0}\in\mathcal{P}_{b}^{log}(A)$
for every $A\Subset\Omega$. Let $\\{p_{k}\\}\subseteq L^{\infty}(\Omega)$ and
$\\{v_{k}\\}\subset W^{1,1}(\Omega;\mathbb{R}^{d})$ such that $\
p_{k}\rightarrow p_{0}$ in $L^{\infty}(\Omega)$ and $v_{k}\rightharpoonup v$
in $L^{1}(\Omega,{R}^{d})$ as $k\rightarrow+\infty$. If $\\{v_{k}\\}$
satisfies

$\sup_{k\in{{N}}}\int_{\Omega}|\nabla v_{k}(x)|^{p_{k}(x)}dx<+\infty,$ (1.10)

then

$v\in W^{1,p_{0}(\cdot)}_{\rm loc}(\Omega;\mathbb{R}^{d}),\,|\nabla v|\in
L^{p_{0}(\cdot)}(\Omega),$ (1.11)

and

$\int_{\Omega}W^{p_{0}(x)}(x,\nabla
v(x))dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx.$

######  Theorem 1.2.

Let $\Omega\subseteq{R}^{N}$ be an open set. Let $W:{R}^{d\times
N}\rightarrow[0,+\infty)$ be a quasiconvex function. Assume that there exists
a constant $C_{2}>0$ such that

$W(\xi)\leq\,C_{2}(|\xi|+1),\qquad\hbox{for every $\xi\in\mathbb{R}^{d\times
N}$}.$ (1.12)

Let $p_{0}$, $\\{p_{k}\\}$, $v$ and $\\{v_{k}\\}$ be as in Theorem 1.1. Let
$a_{0}\in L^{\infty}(\Omega)$ be such that $a_{0}\geq 0$
$\mathcal{L}^{N}$-a.e. in $\Omega$ and assume that $a_{k}\rightarrow a_{0}$ in
$L^{\infty}(\Omega)$ as $k\rightarrow+\infty$. Then $v$ satisfies (1.11) and

$\int_{\Omega}a_{0}(x)W^{p_{0}(x)}(\nabla
v(x))dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{k}(x)W^{p_{k}(x)}(\nabla
v_{k}(x))dx.$

Note that, in the previous semicontinuity theorems, assumption (1.10) can be
dropped when the energy density $W$ satisfies a suitable coercivity
assumption, see Remark 3.4 (2).

The asymptotic analysis (via $\Gamma$-convergence) as $\varepsilon\rightarrow
0^{+}$ of the functionals appearing in (1.4), in the $3d-2d$ dimensional
reduction hyperelastic setting, besides our study can be carried out
analogously for any $nd-md$ dimension reduction, ($m<n,m,n\in\mathbb{N}$).
With this aim, here and in the sequel, we consider $\Omega_{\varepsilon}$ and
$\Omega$ as in (1.2) and (1.3), respectively. The standard scaling argument in
the $x_{3}$ direction introduced above, allows us to rephrase the problem in
the fixed domain $\Omega$, setting $v(x)\equiv
v(x_{\alpha},x_{3}):=u(x_{\alpha},\varepsilon x_{3})$, and taking
$p_{\varepsilon}$ as in (1.6). Moreover, assuming that the sample is clamped
on its lateral boundary, we can make precise the functional space
$X_{\varepsilon}(\Omega)$ appearing in (1.4). Indeed, denoting the lateral
boundary of $\Omega$ by
$\displaystyle\partial_{L}\Omega:=\partial\omega\times\left(-1/2,1/2\right)$,
the functional space $X_{\varepsilon}(\Omega;\mathbb{R}^{3})$ can be
specialized as

$W^{1,p_{\varepsilon}(\cdot)}_{L}(\Omega;\mathbb{R}^{3}):=W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})\cap
W^{1,p_{\varepsilon}(\cdot)}(\Omega;\mathbb{R}^{3}),$

where, according to Definition 2.3 below,
$W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})$ is the closure in
$W^{1,p^{-}}(\Omega)$ of the subspace

$\\{v\in W_{loc}^{1,p^{-}}({R}^{3};\mathbb{R}^{3}):v(x)\equiv 0\hbox{ in a
neighbourhood of }\partial_{L}\Omega\\}.$

Consequently, the functional
$\mathcal{F}_{\varepsilon}:L^{1}(\Omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$
in (1.4) can be rewritten as

$\displaystyle\mathcal{F}_{\varepsilon}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}f(x_{\alpha},\varepsilon
x_{3},p_{\varepsilon}(x),\nabla_{\alpha}u(x),\tfrac{1}{\varepsilon}\nabla_{3}u(x))dx,&\hbox{
if }u\in W^{1,p_{\varepsilon}(\cdot)}_{L}(\Omega;\mathbb{R}^{3}),\\\ \\\
+\infty&\hbox{ otherwise in $L^{1}(\Omega;\mathbb{R}^{3})$}\end{array}\right.$
(1.16)

where $f_{\varepsilon}$ appearing in (1.4) is now defined by (1.5) and
$p_{\varepsilon}$ represents the growth condition of $f_{\varepsilon}$
according to the rescaling (1.6). In the following theorems we aim to
represent the $\Gamma$-limit of the family $\\{\mathcal{F}_{\varepsilon}\\}$
(with respect to the $L^{1}$\- convergence of the deformation fields
$u_{\varepsilon}$) under a suitable convergence assumption of
$p_{\varepsilon}.$

First we consider the case when $f=f(x_{\alpha},y,q,\xi)$ is a function
defined on $\omega\times(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}$, convex in the gradient variable $\xi$ and depending on the variable $q$
(which assumes the value $p_{\varepsilon}(x)$).

The first result in the dimension reduction framework is the following:

######  Theorem 1.3.

Let $\omega\subset\mathbb{R}^{2}$ be a bounded, connected, open set with
Lipschitz boundary. Let $\displaystyle
f:\omega\times(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}\rightarrow[0,+\infty)$ be a function such that

  * -

$f(x_{\alpha},\cdot,\cdot,\cdot)$ is continuous for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$,

  * -

$f(\cdot,y,q,\xi)$ is measurable for every
$\displaystyle(y,q,\xi)\in(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}$,

  * -

$f(x_{\alpha},y,q,\cdot)$ is convex for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$ and for every
$\displaystyle(y,q)\in(-1/2,1/2)\times[1,+\infty)$.

Assume that there exist $0<C_{1}\leq C_{2}$ such that

$C_{1}|\xi|^{q}-\frac{1}{C_{1}}\leq f(x_{\alpha},y,q,\xi)\leq
C_{2}(|\xi|^{q}+1)\qquad$ (1.17)

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ and for every
$\displaystyle(y,q,\xi)\in(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}$.

Let $p\in\mathcal{P}_{b}(\Omega)$ with $\Omega=\omega\times(-1/2,1/2)$, let
$p_{\varepsilon}$ be as in (1.6) and let $\\{\mathcal{F}_{\varepsilon}\\}$ be
the family of functionals in (1.16). If there exists
$p_{0}\in\mathcal{P}^{log}_{b}(\omega)$ such that $p_{\varepsilon}\rightarrow
p_{0}\hbox{ in }L^{1}(\Omega)$ as $\varepsilon\rightarrow 0^{+}$, then

$\Gamma(L^{1})\hbox{-}\lim_{\varepsilon\rightarrow
0^{+}}{\mathcal{F}}_{\varepsilon}={\mathcal{F}}$

where $\mathcal{F}:L^{1}(\Omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$ is the
functional defined by

${\mathcal{F}}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}{f}_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},&\hbox{
if }u\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3}),\\\ \\\ +\infty&\hbox{
otherwise in }L^{1}(\Omega;\mathbb{R}^{3}),\end{array}\right.$ (1.18)

with $\displaystyle f_{0}:\omega\times(-1/2,1/2)\times{R}^{3\times
2}\rightarrow{R}$ defined by

$f_{0}(x_{\alpha},y,q,\xi_{\alpha}):=\inf_{\xi_{3}\in\mathbb{R}^{3}}f(x_{\alpha},y,q,\xi_{\alpha},\xi_{3}).$
(1.19)

In the following results, the convexity assumption on $f$ with respect to the
gradient variable is dropped, but a special structure is imposed on $f$ in
(1.5) and a stronger convergence assumption on $\\{p_{\varepsilon}\\}$ is
required. More precisely, we neglect the dependence of $f$ on the transverse
variable and the dependence on the variable exponent $p_{\varepsilon}$ is
explicitly of power-law type. We denote this particular class of functionals
as

$\mathcal{I}_{\varepsilon}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}W^{p_{\varepsilon}(x)}\left(x_{\alpha},\nabla_{\alpha}u(x),\tfrac{1}{\varepsilon}\nabla_{3}u(x)\right)dx&\hbox{
if }u\in W_{L}^{1,p_{\varepsilon}(\cdot)}(\Omega;\mathbb{R}^{3}),\\\ \\\
+\infty&\hbox{ otherwise in }L^{1}(\Omega;\mathbb{R}^{3}).\end{array}\right.$
(1.20)

######  Theorem 1.4.

Let $\omega\subset\mathbb{R}^{2}$ be a bounded, connected, open set with
Lipschitz boundary. Let $W:\omega\times{R}^{3\times 3}\rightarrow[0,+\infty)$
be a function such that

  * -

$W(\cdot,\xi)$ is measurable for every $\xi\in\mathbb{R}^{3\times 3}$;

  * -

$W(x_{\alpha},\cdot)$ is continuous for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$;

  * -

there exist $0<C_{1}\leq C_{2}$ such that

$C_{1}|\xi|-\frac{1}{C_{1}}\leq\,W(x_{\alpha},\xi)\leq\,C_{2}(|\xi|+1)\quad\hbox{
for }\mathcal{L}^{2}\hbox{-a.e. }x_{\alpha}\in\omega,\hbox{ for every
}\xi\in{R}^{3\times 3}.$ (1.21)

Assume that

$|W(x_{\alpha},\xi)-W(x_{\alpha},\eta)|\leq w(x_{\alpha},|\xi-\eta|)\hbox{ for
every }\xi,\eta\in{R}^{3\times 3}\qquad\hbox{for }\mathcal{L}^{2}\hbox{-a.e.
}x_{\alpha}\in\omega,$ (1.22)

where $w:\omega\times[0,+\infty)\rightarrow[0,+\infty)$ is such that

  * -

$w(\cdot,t)$ is measurable for every $t\in[0,+\infty)$,

  * -

$w(x_{\alpha},\cdot)$ is a continuous, increasing function for
${\mathcal{L}}^{2}$ a.e. $x_{\alpha}\in\omega$ satisfying $\lim_{t\rightarrow
0^{+}}w(x_{\alpha},t)=0$.

Let $p\in\mathcal{P}_{b}(\Omega)$ with $\Omega=\omega\times(-1/2,1/2)$, let
$p_{\varepsilon}$ be as in (1.6) and let $\\{\mathcal{I}_{\varepsilon}\\}$ be
the family of functionals in (1.20). If there exists
$p_{0}\in\mathcal{P}^{log}_{b}(\omega)$ such that $p_{\varepsilon}\rightarrow
p_{0}\hbox{ in }L^{\infty}(\Omega)$ as $\varepsilon\rightarrow 0^{+}$, then

$\Gamma(L^{1})\hbox{-}\lim_{\varepsilon\rightarrow
0^{+}}\mathcal{I}_{\varepsilon}=\mathcal{I}$

where $\mathcal{I}:L^{1}(\Omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$ is the
functional given by

$\mathcal{I}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}{Q(W_{0}^{p_{0}(x_{\alpha})}})(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},&\hbox{
if }u\in W_{0}^{1,p_{0}(\cdot)}(\omega;{R}^{3}),\\\ \\\ +\infty&\hbox{
otherwise in }L^{1}(\Omega;\mathbb{R}^{3})\end{array}\right.$ (1.23)

with $W_{0}:\omega\times{R}^{3\times 2}\rightarrow{R}$ defined as

$W_{0}(x_{\alpha},\xi_{\alpha}):=\inf_{\xi_{3}\in\mathbb{R}^{3}}W(x_{\alpha},\xi_{\alpha},\xi_{3}),$

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ and for every
$\xi_{\alpha}\in\mathbb{R}^{3\times 2}$, and
$Q(W_{0}^{p_{0}(x_{\alpha})})(x_{\alpha},\cdot)$ denotes the quasiconvex
envelope of $W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\cdot)$, namely the greatest
quasiconvex function below $W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\cdot)$ for
$\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$.

The third model considered in $3d-2d$-dimensional reduction setting is linked
to the class of functionals considered in Theorem 1.2, i.e. the function $f$
in (1.5) still does not depend on the transverse variable, and the point
dependence on the planar variable $x_{\alpha}$ is of product type, while the
dependence on the variable exponents is of explicit power type. Its proof,
relying on the technical Lemma 3.6, is not a direct application of Theorem
1.2, which, indeed, may provide only a possibly strict lower bound.

More precisely, the family of functionals we consider is the following

$\mathcal{J}_{\varepsilon}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}a(x_{\alpha},\varepsilon
x_{3})W^{p_{\varepsilon}(x)}\left(\nabla_{\alpha}u(x),\tfrac{1}{\varepsilon}\nabla_{3}u(x)\right)dx&\hbox{
if }u\in W_{L}^{1,p_{\varepsilon}(\cdot)}(\Omega;\mathbb{R}^{3}),\\\ \\\
+\infty&\hbox{ otherwise in }L^{1}(\Omega;\mathbb{R}^{3}),\end{array}\right.$
(1.24)

where $a\in L^{\infty}(\Omega)$.

######  Theorem 1.5.

Let $W:{R}^{3\times 3}\rightarrow[0,+\infty)$ be a uniformly continuous
function such that there exist $C_{1}>0$ and $C_{2}\geq 1$ with

$C_{1}|\xi|-\frac{1}{C_{1}}\leq\,W(\xi)\leq\,C_{2}(|\xi|+1)\hbox{ for every
}\xi\in{R}^{3\times 3}.$ (1.25)

Let $a\in L^{\infty}(\Omega)$ be such that

$\displaystyle a^{-}:=\inf_{x\in\Omega}a(x)>0,$ (1.26)

and let

$a_{\varepsilon}(x):=a(x_{\alpha},\varepsilon x_{3}).$

Assume that there exists $a_{0}\in L^{\infty}(\omega)$ such that

$\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}a_{\varepsilon}(x)=\lim_{\varepsilon\rightarrow
0^{+}}a(x_{\alpha},\varepsilon x_{3})=a_{0}(x_{\alpha})\hbox{ in
}L^{\infty}(\Omega).$ (1.27)

Let $p\in\mathcal{P}_{b}(\Omega)$, let $p_{\varepsilon}$ be as in (1.6) and
let $\\{\mathcal{J}_{\varepsilon}\\}$ be the family of functionals in (1.24).
If there exists $p_{0}\in\mathcal{P}^{log}_{b}(\omega)$ such that
$p_{\varepsilon}\rightarrow p_{0}$ in $L^{\infty}(\Omega)$ as
$\varepsilon\rightarrow 0^{+}$, then

$\Gamma(L^{1})\hbox{-}\lim_{\varepsilon\rightarrow
0^{+}}\mathcal{J}_{\varepsilon}=\mathcal{J}$

where $\mathcal{J}:L^{1}(\Omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$ is the
functional given by

$\mathcal{J}(v):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}a_{0}(x_{\alpha})Q(W_{0}^{p_{0}(x_{\alpha})})(\nabla_{\alpha}v(x_{\alpha}))dx_{\alpha},&\hbox{
if }v\in W_{0}^{1,p_{0}(\cdot)}(\omega;{R}^{3}),\\\ \\\ +\infty&\hbox{
otherwise in }L^{1}(\Omega;\mathbb{R}^{3}),\end{array}\right.$ (1.28)

with $W_{0}:\mathbb{R}^{3\times 2}\rightarrow[0,+\infty)$ defined by

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

and $Q(W_{0}^{p_{0}(x_{\alpha})})$ denoting the quasiconvex envelope of
$W_{0}^{p_{0}(x_{\alpha})}.$

We note that, in the dimensional reduction theorems, thanks to the special
structure of the sequence $\\{p_{\varepsilon}\\}$ defined by (1.6), if
$p\in\mathcal{P}_{b}^{log}(\Omega)$, then the family $\\{p_{\varepsilon}\\}$
is equicontinuous on $\overline{\Omega}$ and equibounded since, for every
$\varepsilon>0,$ it holds

$p^{+}_{\varepsilon}:=\mathop{\rm
ess\>sup}_{\omega\times\left(-\frac{1}{2},\frac{1}{2}\right)}p(x_{\alpha},\varepsilon
x_{3})=\mathop{\rm
ess\>sup}_{\omega\times\left(-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\right)}p(x_{\alpha},x_{3})\leq
p^{+}<+\infty.$ (1.30)

and

$p^{-}_{\varepsilon}:=\mathop{\rm
ess\>inf}_{\omega\times\left(-\frac{1}{2},\frac{1}{2}\right)}p(x_{\alpha},\varepsilon
x_{3})=\mathop{\rm
ess\>inf}_{\omega\times\left(-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\right)}p(x_{\alpha},x_{3})\geq
p^{-}>1.$ (1.31)

Hence, thanks to the Ascoli-Arzelà Theorem, the sequence
$p_{\varepsilon}\rightarrow p_{0}$ in $L^{\infty}(\Omega)$ as
$\varepsilon\rightarrow 0^{+}$ where
$p_{0}(x_{\alpha})=p(x_{\alpha},0)\in\mathcal{P}_{b}^{log}(\Omega)$.

It is worth observing that the proofs of Theorems 1.4 and 1.5 do not follow as
mere applications of Theorems 1.1 and 1.2, indeed the representation results
obtained in (1.23) and (1.28) involve densities of the type
$Q(W_{0}^{p_{0}(x_{\alpha})})(\cdot)$ which are, in general, greater than
$(QW_{0})^{p_{0}(x_{\alpha})}(\cdot)$. This is easily seen in the scalar case
(where the quasiconvex envelope coincides with the convex one): for instance,
when $p>1$ is constant and $W=W_{0}$ is a homogeneous function defined as
$W(\xi)=\max\\{|\xi|,|\xi|^{1/p}\\}$ it results

$Q(W^{p})(\xi)=W^{p}(\xi)=\max\\{|\xi|^{p},|\xi|\\}\geq|\xi|^{p}=(QW)^{p}(\xi),\hbox{
for every }\xi\in\mathbb{R},$

where the inequality is strict in $[-1,1]$.

The paper is organized as follows: Section 2 contains notation and preliminary
results regarding the functional spaces and $\Gamma$-convergence. Section 3 is
devoted to the proofs of Theorems 1.1 and 1.2 which follow as corollaries of
the more general result Theorem 3.7. Finally Section 4 contains the proofs of
the dimensional reduction results stated above, namely Theorems 1.3, 1.4 and
1.5, together with a compactness result for energy bounded sequences in
$L^{1}(\Omega),$ namely Proposition 4.1, which motivates the choice of the
topology for our results.

##  2  Notation and preliminary results

In the sequel $\Omega$ indicates a generic open set of $\mathbb{R}^{N},N\geq
1$; by $\mathcal{A}(\Omega)$ we denote the class of open subsets of $\Omega$
and by $\mathcal{A}_{0}(\Omega)$ we denote the subclass of
$\mathcal{A}(\Omega)$ whose elements are well contained in $\Omega$, i.e.
$B\in\mathcal{A}_{0}(\Omega)$ if $B\Subset\Omega$. We denote by
$\mathcal{L}^{N}(\Omega)$ the $N$-dimensional Lebesgue measure of the set
$\Omega$.

###  2.1 Variable exponents Lebesgue spaces

In this section we collect some basic results concerning variable exponent
Lebesgue spaces. For more details we refer to the monograph [23], see also
[40], [24], [25], [26].
Let $\Omega\subset\mathbb{R}^{N}$ be an open set (where $N\geq 1$).

######  Definition 2.1.

For any (Lebesgue) measurable function $p:\Omega\rightarrow[1,+\infty]$ we
define

$p^{-}:=\displaystyle\mathop{\rm ess\>inf}_{x\in\Omega}p(x)\qquad\qquad
p^{+}:=\displaystyle\mathop{\rm ess\>sup}_{x\in\Omega}p(x).$

Such function $p$ is called variable exponent on $\Omega$. If $p^{+}<+\infty$
then we call $p$ a bounded variable exponent.
We denote by $\mathcal{P}(\Omega)$ the class of variable exponents and with
$\mathcal{P}_{b}(\Omega)$ the class of variable exponents satisfying (1.7).

In the sequel we consider the case $p^{+}<+\infty$. In this case, the variable
exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ can be defined as

$L^{p(\cdot)}(\Omega):=\left\\{u:\Omega\rightarrow{R}\,\,\,\textnormal{measurable
such that}\,\,\int_{\Omega}|u(x)|^{p(x)}\,dx<+\infty\right\\}.$

Let us note that in the case $p^{+}=+\infty$ the space above defined may even
fail to be a vector space (see [17] Section 2) and a different definition of
the variable Lebesgue spaces has been given in order to preserve the vectorial
structure of the space (we refer to [23], Definition 3.2.1). In addition, if
$p^{+}<+\infty,$ then $L^{p(\cdot)}(\Omega)$ is a Banach space endowed with
the Luxemburg norm

$\|u\|_{p(\cdot)}:=\inf\left\\{\lambda>0:\,\,\,\int_{\Omega}\left|\frac{u(x)}{\lambda}\right|^{p(x)}\,dx\leq\,1\right\\}$

(see Theorem 3.2.7 in [23]). Moreover, if $p^{+}<+\infty$, the space
$L^{p(\cdot)}(\Omega)$ is separable and
$\mathcal{C}^{\infty}_{0}(\Omega)\hbox{ is dense in }L^{p(\cdot)}(\Omega)$,
while, if $1<p^{-}\leq p^{+}<+\infty$, the space $L^{p(\cdot)}(\Omega)$ is
reflexive and uniformly convex (see Theorem 3.4.12, Theorem 3.4.7, Theorem
3.4.9 and Theorem 3.4.12 in [23]). For any variable exponent $p$, we define
$p^{\prime}$ by setting

$\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)}=1,$

with the convention that, if $p(x)=+\infty$ then $p^{\prime}(x)=1$. The
function $p^{\prime}$ is called the dual variable exponent of $p$.

The following result holds (for more details, see [23, Lemma 3.2.20] in the
case $\varphi(t)=|t|^{p}$).

######  Theorem 2.2.

(Hölder’s inequality) Let $p,q,s$ be measurable exponents such that

$\frac{1}{s(x)}=\frac{1}{p(x)}+\frac{1}{q(x)}\qquad\hbox{for
}\mathcal{L}^{N}\hbox{-a.e. }x\in\Omega.$

Then, for all $f\in L^{p(\cdot)}(\Omega)$ and $g\in L^{q(\cdot)}(\Omega)$, it
holds

$\|fg\|_{s(\cdot)}\leq\,\left(\left(\frac{s}{p}\right)^{+}+\left(\frac{s}{q}\right)^{+}\right)\,\|f\|_{p(\cdot)}\,\|g\|_{q(\cdot)}$

where, in the case $s=p=q=\infty$, we use the convention
$\frac{s}{p}=\frac{s}{q}=1.$ In particular, in the case $s=1$, it holds

$\left|\int_{\Omega}f\,g\,dx\right|\leq
2\|f\|_{p(\cdot)}\,\|g\|_{p^{\prime}(\cdot)}\ .$

We introduce the modular of the space $L^{p(\cdot)}(\Omega)$ which is the
mapping $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\rightarrow\mathbb{R}$ defined by

$\rho_{p(\cdot)}(u):=\int_{\Omega}|u(x)|^{p(x)}\,dx.$

Thanks to Lemma 3.2.4 in [23], for every $u\in L^{p(\cdot)}(\Omega)$

$\displaystyle\|u\|_{p(\cdot)}\leq 1\Longleftrightarrow\rho_{p(\cdot)}(u)\leq
1$ $\displaystyle\|u\|_{p(\cdot)}\leq
1\Longrightarrow\rho_{p(\cdot)}(u)\leq\|u\|_{p(\cdot)}\hbox{ and
}\|u\|_{p(\cdot)}>1\Longrightarrow\|u\|_{p(\cdot)}\leq\,\rho_{p(\cdot)}(u).$
(2.1)

The following further results hold in the special case $p^{+}<+\infty$. By
Lemma 3.2.5 in [23], for every $u\in L^{p(\cdot)}(\Omega)$ it holds

$\min\left\\{\big{(}\rho_{p(\cdot)}(u)\big{)}^{\frac{1}{p^{-}}},\big{(}\rho_{p(\cdot)}(u)\big{)}^{\frac{1}{p^{+}}}\right\\}\leq\|u\|_{p(\cdot)}\leq\max\left\\{\big{(}\rho_{p(\cdot)}(u)\big{)}^{\frac{1}{p^{-}}},\big{(}\rho_{p(\cdot)}(u)\big{)}^{\frac{1}{p^{+}}}\right\\}.$
(2.2)

In particular,

$\min\left\\{\big{(}{\cal L}^{N}(\Omega)\big{)}^{\frac{1}{p^{-}}},\big{(}{\cal
L}^{N}(\Omega)\big{)}^{\frac{1}{p^{+}}}\right\\}\leq\|1\|_{p(\cdot)}\leq\max\left\\{\big{(}{\cal
L}^{N}(\Omega)\big{)}^{\frac{1}{p^{-}}},\big{(}{\cal
L}^{N}(\Omega)\big{)}^{\frac{1}{p^{+}}}\right\\}.$ (2.3)

Moroever, from (2.2), taking into account (2.1), it follows that for every
$u\in L^{p(\cdot)}(\Omega)$

$\displaystyle\|u\|_{p(\cdot)}>1\Longrightarrow\|u\|_{p(\cdot)}^{p^{-}}\leq\,\rho_{p(\cdot)}(u)\leq\,\|u\|^{p^{+}}_{p(\cdot)}\hbox{
and
}\|u\|_{p(\cdot)}<1\Longrightarrow\|u\|_{p(\cdot)}^{p^{+}}\leq\,\rho_{p(\cdot)}(u)\leq\,\|u\|^{p^{-}}_{p(\cdot)}.$

Finally, by [23, Corollary 3.3.4], if $0<\mathcal{L}^{N}(\Omega)<+\infty$ and
$p$ and $q$ are variable exponents such that $q\leq p$ $\mathcal{L}^{N}$ a.e.
in $\Omega$, then the embedding $L^{p(\cdot)}(\Omega)\hookrightarrow
L^{q(\cdot)}(\Omega)$ is continuous. In view of [23, Theorem 3.3.1, part(a)]
and (2.3), the embedding constant $C_{p,q,\Omega}$ satisfies

$\displaystyle C_{p,q,\Omega}\leq
2\min\left\\{\mathcal{L}^{N}(\Omega)+1,\max\Big{\\{}{\mathcal{L}^{N}(\Omega)}^{{(\frac{1}{q}-\frac{1}{p})}^{+}},{\mathcal{L}^{N}(\Omega)}^{{(\frac{1}{q}-\frac{1}{p})}^{-}}\Big{\\}}\right\\}\leq
2(\mathcal{L}^{N}(\Omega)+1).$ (2.4)

In particular, if $q(x)\equiv p^{-}$

$C_{p,p^{-},\Omega}\leq
2\min\left\\{\mathcal{L}^{N}(\Omega),{\mathcal{L}^{N}(\Omega)}^{\frac{1}{p^{-}}-\frac{1}{p^{+}}}\right\\}+2.$

###  2.2 Trace operator in Sobolev spaces

In view of introducing some important results concerning variable exponents
Sobolev spaces, in this section we recall some useful remarks concerning the
trace operator.

Let $\Omega$ be a bounded, connected open set with Lipschitz boundary and
$1\leq r\leq\infty$. Hence, set

$s=\begin{cases}\frac{(N-1)r}{N-r}&\hbox{ if }r<N\\\ q\in[1,+\infty)&\hbox{ if
}r=N\\\ +\infty&\hbox{ if }r>N,\end{cases}$

it is well defined the linear and continuous “trace operator”
$T_{\partial\Omega}:W^{1,r}(\Omega)\rightarrow L^{s}(\partial\Omega)$, where
the measure on $\partial\Omega$ is ${\mathcal{H}}^{N-1}$, such that

(i) when $1\leq r\leq N$, it holds

$T_{\partial\Omega}(u)(x)=u(x)\hbox{ for $\mathcal{H}^{N-1}$ a.e.
}x\in\partial\Omega,\hbox{ for every }u\in W^{1,r}(\Omega)\cap
C(\overline{\Omega});$

(ii) when $r>N$, it holds

$T_{\partial\Omega}(u)(x)=u(x)\hbox{ for every }x\in\partial\Omega,\hbox{ for
every }u\in W^{1,r}(\Omega).$

(see [14, Theorem 4.3.12]).

######  Definition 2.3.

Let $\mathcal{B}(\partial\Omega)$ be the class of Borel subsets of
$\partial\Omega$. For every set $\Gamma\in\mathcal{B}(\partial\Omega)$ and
$1\leq r<+\infty$, we define $W^{1,r}_{0,\Gamma}(\Omega)$ to be the closure in
$W^{1,r}(\Omega)$ of the subspace

$\\{u\in W_{loc}^{1,r}({R}^{N}):u\equiv 0\hbox{ on a neighbourhood of
}\Gamma\\}.$

Hence $W^{1,r}_{0,\Gamma}(\Omega)$ is closed once we endow it with the
weak-$W^{1,r}(\Omega)$ topology and

$W^{1,r}_{0,\Gamma}(\Omega)\subseteq\\{u\in
W^{1,r}(\Omega):T_{\partial\Omega}u\equiv 0\hbox{ for $\mathcal{H}^{N-1}$-a.e.
}x\in\Gamma\\}.$

Thanks to [14, Proposition 4.3.13] it holds

$W^{1,r}_{0,\partial\Omega}(\Omega)=W^{1,r}_{0}(\Omega)=\\{u\in
W^{1,r}(\Omega):T_{\partial\Omega}u\equiv 0\\}.$

Finally, by [14, Theorem 4.3.18], for $1\leq r<+\infty$, the following
Poincarè inequality holds

$\|u\|_{L^{r^{*}}(\Omega)}\leq C(\Omega,r,N,\Gamma)\|\nabla
u\|_{L^{r}(\Omega)}\;\;\forall u\in W^{1,r}_{0,\Gamma}(\Omega),$ (2.5)

where

$r^{*}=\left\\{\begin{array}[]{lll}\\!\\!\\!\\!\\!\\!&\displaystyle\frac{Nr}{N-r}&\qquad\textnormal{if
$r\in[1,N)$}\\\\[5.69054pt]
\\!\\!\\!\\!\\!\\!&\displaystyle+\infty&\qquad\textnormal{otherwise.}\end{array}\right.$

###  2.3 Variable exponents Sobolev spaces

In this subsection we recall the definition of variable exponents Sobolev
spaces. For more details we refer to [17] (see also [23], Definition 8.1.2).

######  Definition 2.4.

Let $k,d\in{{N}}$, $k\geq 0$, and let $p\in\mathcal{P}(\Omega).$ We define

$W^{k,p(\cdot)}(\Omega;{R}^{d}):=\\{u:\Omega\rightarrow{R}^{d}:u,\partial_{\alpha}u\in
L^{p(\cdot)}(\Omega;{R}^{d})\quad\forall\alpha\hbox{ multi-index such that
$|\alpha|\leq\,k$}\\},$

where

$L^{p(\cdot)}(\Omega;{R}^{d}):=\\{u:\Omega\rightarrow{R}^{d}:\ |u|\in
L^{p(\cdot)}(\Omega)\\}.$

We define the semimodular on $W^{k,p(\cdot)}(\Omega)$ by

$\rho_{W^{k,p(\cdot)}(\Omega)}(u):=\sum_{0\leq|\alpha|\leq
k}\rho_{L^{p(\cdot)}(\Omega)}(|\partial_{\alpha}u|)$

which induces a norm by

$\|u\|_{W^{k,p(\cdot)}(\Omega)}:=\inf\left\\{\lambda>0:\,\,\rho_{W^{k,p(\cdot)}(\Omega)}\left(\frac{u}{\lambda}\right)\leq\,1\right\\}.$

For $k\in\mathbb{N}\setminus\\{0\\},$ the space $W^{k,p(\cdot)}(\Omega)$ is
called Sobolev space and its elements are called Sobolev functions. Clearly
$W^{0,p(\cdot)}(\Omega)=L^{p(\cdot)}(\Omega)$. The space
$W^{k,p(\cdot)}(\Omega)$ is a Banach space, which is separable if
$1\leq{p^{-}}\leq p^{+}<+\infty,$ and reflexive and uniformly convex if
$1<{p^{-}}\leq p^{+}<+\infty$ (see [23, Theorem 8.1.6]). We recall that the
Sobolev conjugate exponent $p^{*}:\Omega\rightarrow(1,+\infty]$ is defined as

$p^{*}(x):=\begin{cases}\frac{Np(x)}{N-p(x)}\hbox{ if }p(x)<N,\\\
+\infty\quad\,\hbox{ otherwise.}\end{cases}$

Finally, we define $W^{1,p(\cdot)}_{0}(\Omega)$ as the closure of the set of
$W^{1,p(\cdot)}(\Omega)$-functions with compact support.

Now we complement Definition 2.1 with the following one.

######  Definition 2.5.

We denote by $\mathcal{P}^{log}(\Omega)$ the class of variable exponents $p$
satisfying (1.8) and by
$\mathcal{P}^{log}_{b}(\Omega)=\mathcal{P}^{log}(\Omega)\cap\mathcal{P}_{b}(\Omega)$.

######  Remark 2.6.

We recall that, when $\Omega$ is a bounded open set, (1.8) implies the so
called decay condition

$\displaystyle\exists
p_{\infty}\in\mathbb{R},\gamma_{\infty}>0:|p(x)-p_{\infty}|\leq\frac{\gamma_{\infty}}{\log(e+|x|)}\hbox{
for every }x\in\Omega,$

(see [22, Remark 2.4]). This ensures that our class
$\mathcal{P}^{log}(\Omega)$ coincides with the class of variable exponents
denoted in the same way in [23, Definition 4.1.4].

In the sequel we frequently use the following results.

######  Remark 2.7.

Let $\Omega$ be a bounded open set. Hence

  1. (1)

if $p\in\mathcal{P}^{log}(\Omega)$ satisfies $p^{+}<+\infty$, then
$C_{0}^{\infty}(\Omega)$ is dense in $W^{1,p(\cdot)}_{0}(\Omega)$ (see [23,
Corollary 11.2.4]);

  2. (2)

if $p\in\mathcal{P}^{log}(\Omega)$, then the following Poincaré inequality
holds

$\|u\|_{L^{p(\cdot)}(\Omega)}\leq cdiam(\Omega)\|\nabla
u\|_{L^{p(\cdot)}(\Omega)}\ \forall u\in W_{0}^{1,p(\cdot)}(\Omega)$ (2.6)

Finally, we report here the proof of the compact embedding
$W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d})\hookrightarrow
L^{p(\cdot)}(\Omega;\mathbb{R}^{d})$ by using arguments similar to those
employed in [46, Proposition 3.3] (see, also, [28, Theorem 1.3]).

######  Proposition 2.8.

Let $\Omega\subset\mathbb{R}^{N}$ be a Lipschitz bounded open set and let
$p\in C(\overline{\Omega})$ such that $p\geq 1$. Then

$\\{u\in L^{1}(\Omega;{R}^{d}):\,\nabla u\in L^{p(\cdot)}(\Omega;{R}^{d\times
N})\\}=W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d}).$ (2.7)

Moreover, the embedding $W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d})\hookrightarrow
L^{p(\cdot)}(\Omega;\mathbb{R}^{d})$ is compact and on
$W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d})$ the norm

$|||u|||=\|u\|_{L^{1}(\Omega;\mathbb{R}^{d})}+\|\nabla
u\|_{L^{p(\cdot)}(\Omega;\mathbb{R}^{d\times{{N}}})}$

is equivalent to the norm $\|\cdot\|_{W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d})}$.

###### Proof.

Without loss of generality, we can assume $d=1$. Identity (2.7) follows by
[17, Lemma 2.4]. In order to show that the embedding
$W^{1,p(\cdot)}(\Omega)\hookrightarrow L^{p(\cdot)}(\Omega)$ is compact, first
of all we construct a suitable covering of $\overline{\Omega}.$ Since $p$ is
uniformly continuous on $\overline{\Omega}$, for every fixed
$0<\sigma<\frac{(p^{-})^{2}}{N+p^{-}}$ there exists $\rho>0$ such that

$\sup_{B_{r}\cap\Omega}p-\inf_{B_{r}\cap\Omega}p<\sigma,$ (2.8)

for every ball $B_{r}$ di radius $r<\rho$. Being $\overline{\Omega}$ a compact
set with Lipschitz boundary, we can find a finite number of open balls
$(B_{j})_{j\in F}$ and $(\widetilde{B}_{{j}})_{j\in F}$, such that
$\overline{B_{j}}\subseteq\widetilde{B}_{j}$, $B_{j}$ and $\widetilde{B}_{j}$
have the same centers and

$\begin{array}[]{lll}&\overline{\Omega}\subseteq\bigcup_{j\in F}B_{j},\\\
&\Omega\cap\widetilde{B}_{j}\hbox{ has Lipschitz continuous boundary,}\\\
&\hbox{for every }j\hbox{ the radius }r_{j}\hbox{ of }\widetilde{B}_{j}\hbox{
satisfies }r_{j}<\rho.\end{array}$ (2.9)

For every $j\in F$, we set

$p_{j}^{+}=\sup_{\widetilde{B}_{j}\cap\Omega}p,\quad
p_{j}^{-}=\inf_{\widetilde{B}_{j}\cap\Omega}p.$ (2.10)

We claim that, for every $j\in F$, there exists $\delta_{j}>0$ such that the
embedding

$W^{1,p_{0,j}^{-}-\delta_{j}}(\Omega)\hookrightarrow
L^{p_{0,j}^{+}}(\Omega)\hbox{ holds and is compact}.$ (2.11)

Indeed, fix $j\in F$. We distinguish two cases:

  * •

if $p_{j}^{-}>N$, then there exists $\delta_{j}>0$ such that
$p_{j}^{-}-\delta_{j}>N$. Hence the claim follows thanks to the Rellich-
Kondrachov theorem;

  * •

if $p_{j}^{-}\leq N$, since $p^{-}\leq p_{j}^{-}\leq p_{j}^{+}$ and the
function $x\rightarrow\frac{x^{2}}{N+x}$ is increasing on $[0,+\infty)$, we
get that

$\sigma<\frac{(p^{-})^{2}}{N+p^{-}}\leq\frac{(p_{j}^{+})^{2}}{N+p_{j}^{+}}.$

Thanks to the choice of $\sigma$ and by applying (2.8) with $r=r_{j}<\rho$,
the above inequality implies

$p_{j}^{-}>p_{j}^{+}-\sigma>p_{j}^{+}-\frac{(p_{j}^{+})^{2}}{N+p_{j}^{+}}=\frac{Np_{j}^{+}}{N+p_{j}^{+}}.$

Then, there exists $\delta_{j}>0$ such that

$p_{j}^{-}-\delta_{j}>\frac{Np_{j}^{+}}{N+p_{j}^{+}}.$

Hence, taking into account that $p_{j}^{-}-\delta_{j}<p_{j}^{-}\leq N$, we
obtain that

$p_{j}^{+}<\frac{(p_{j}^{-}-\delta_{j})N}{N-p_{j}^{-}+\delta_{j}}=(p_{j}^{-}-\delta_{j})^{*}$

and, by applying again the Rellich-Kondrachov theorem, we get that
$W^{1,p_{j}^{-}-\delta_{j}}(\Omega)$ is compactly embedded into
$L^{p_{j}^{+}}(\Omega).$

Now, for every $j\in F$, let $\varphi_{j}\in C^{1}(\Omega)$ be such that

$0\leq\varphi_{j}\leq 1\,,\quad\varphi_{j}=1\hbox{ on
}B_{j}\cap\Omega\,,\quad\varphi_{j}=0\hbox{ on
}\Omega\setminus\widetilde{B}_{j}$

and let $\sup_{j\in F}||\nabla\varphi_{j}||_{L^{\infty}(\Omega)}=C$. Since
$p_{j}^{-}-\delta_{j}<p(\cdot)$ on $\Omega\cap\widetilde{B}_{j}$, using the
continuity of the embedding
$L^{p(\cdot)}(\Omega\cap\widetilde{B}_{j})\hookrightarrow
L^{p^{-}_{j}-\delta_{j}}(\Omega\cap\widetilde{B}_{j})$, we get that there
exists a constant $C_{1}=C_{1}(\Omega)>0$ such that for all $u\in
W^{1,p(\cdot)}(\Omega)$ it holds

$||u||_{L^{p(\cdot)}(\Omega\cap
B_{j})}\leq||\varphi_{j}u||_{L^{p(\cdot)}(\Omega\cap\tilde{B}_{j})}\leq
C_{1}||\varphi_{j}u||_{L^{p^{+}_{j}}(\Omega\cap\tilde{B}_{j})}=C_{1}||\varphi_{j}u||_{L^{p^{+}_{j}}(\Omega)}.$

Hence, set $E_{j}:=\\{v\in L^{p^{+}_{j}}(\Omega):\ v=\varphi_{j}u,u\in
W^{1,p(\cdot)}(\Omega)\\}$ endowed with the norm
$\|\cdot\|_{L^{p^{+}_{j}}(\Omega)}$, for every $j\in F$ it holds

$E_{j}\ni\varphi_{j}u\longmapsto u\in L^{p(\cdot)}(\Omega)\hbox{ holds and is
continuous}.$ (2.12)

Moreover, for every $u\in W^{1,p(\cdot)}(\Omega)$,

$||\varphi_{j}u||_{L^{p^{-}_{j}-\delta_{j}}(\Omega)}=||\varphi_{j}u||_{L^{p^{-}_{j}-\delta_{j}}(\Omega\cap\tilde{B}_{j})}\leq
C_{1}||\varphi_{j}u||_{L^{p(\cdot)}(\Omega\cap\tilde{B}_{j})}\leq
C_{1}||u||_{L^{p(\cdot)}(\Omega)}$

and

$\displaystyle||D(\varphi_{j}u)||_{L^{p^{-}_{j}-\delta_{j}}(\Omega)}$
$\displaystyle=||D(\varphi_{j}u)||_{L^{p^{-}_{j}-\delta_{j}}(\Omega\cap\tilde{B}_{j})}$
$\displaystyle\leq
C_{1}(||\varphi_{j}Du||_{L^{p(\cdot)}(\Omega\cap\tilde{B}_{j})}+||uD\varphi_{j}||_{L^{p(\cdot)}(\Omega\cap\tilde{B}_{j})})$
$\displaystyle\leq C_{2}||u||_{W^{1,p(\cdot)}(\Omega)}$

i.e.

$W^{1,p(\cdot)}(\Omega)\ni u\longmapsto\varphi_{j}u\in
W^{1,p^{-}_{j}(\cdot)-\delta_{j}}(\Omega)\hbox{ holds and is continuous
}\forall j\in F.$ (2.13)

By combining (2.13), (2.11) and (2.12), it easily follows that the embedding
$W^{1,p(\cdot)}(\Omega)\hookrightarrow L^{p(\cdot)}(\Omega)$ is compact.

Finally, it is easy to show that $W^{1,p(\cdot)}(\Omega;{R}^{d})$ is complete
with respect to the norm $|||\cdot|||$. Since $|||\cdot|||\leq
C||\cdot||_{W^{1,p(\cdot)}(\Omega;\mathbb{R}^{d})}$, by a classical result of
Functional Analysis, we get that the two norms are equivalent on
$W^{1,p(\cdot)}(\Omega;{R}^{d})$.

We conclude this subsection by recalling the following theorem which is of
crucial importance for the results contained in Section 4. It is a consequence
of [23, Theorem 11.2.7] and properties of sets with Lipschitz boundary in [23,
Section 9.5].

######  Theorem 2.9.

Let $\Omega$ be a bounded open set with Lipschitz boundary and let
$p\in\mathcal{P}_{b}^{log}(\Omega)$. If $u\in W^{1,1}_{0}(\Omega)\cap
W^{1,p(\cdot)}(\Omega)$ then $u\in W^{1,p(\cdot)}_{0}(\Omega)$.

###  2.4 $\Gamma$-convergence

For an introduction to $\Gamma$-convergence, we refer to [19]. We recall the
sequential characterization of the $\Gamma$-limit when $X$ is a metric space.

######  Proposition 2.10 ([19] Proposition 8.1).

Let $X$ be a metric space and let
$\varphi_{k}:X\rightarrow{R}\cup\\{\pm\infty\\}$ for every $k\in{{N}}$. Then
$\\{\varphi_{k}\\}$ $\Gamma$-converges to $\varphi$ with respect to the strong
topology of $X$ (and we write
$\displaystyle\Gamma(X)\hbox{-}\lim_{k\rightarrow+\infty}\varphi_{k}=\varphi$)
if and only if

(i)

($\Gamma$-liminf inequality) for every $x\in X$ and for every sequence
$\\{x_{k}\\}$ converging to $x$, it is

$\varphi(x)\leq\liminf_{k\rightarrow+\infty}\varphi_{k}(x_{k});$

(ii)

($\Gamma$-limsup inequality) for every $x\in X,$ there exists a sequence
$\\{x_{k}\\}$ converging to $x\in X$ such that

$\varphi(x)=\lim_{k\rightarrow+\infty}\varphi_{k}(x_{k}).$

We recall that the $\Gamma\hbox{-}\lim_{k\rightarrow+\infty}\varphi_{k}$ is
lower semicontinuous on $X$ (see [19] Proposition 6.8).

######  Definition 2.11.

We say that a family $\\{\varphi_{\varepsilon}\\}$ $\Gamma$-converges to
$\varphi$, with respect to the topology considered on $X$ as
$\varepsilon\rightarrow 0^{+}$, if $\\{\varphi_{{\varepsilon}_{k}}\\}$
$\Gamma$-converges to $\varphi$ for all sequences $\\{{\varepsilon}_{k}\\}$ of
positive numbers converging to $0$ as $k\rightarrow+\infty$.

##  3 Some lower semicontinuity results

In this section we provide some lower semicontinuity results for different
classes of integral functionals of the type

$F(u,p)=\int_{\Omega}\varphi(x,p(x),\nabla u(x))\,dx$ (3.1)

where the variable $p$ may stand for a variable growth exponent and
$\Omega\subset\mathbb{R}^{N}$ is a open set. We discuss the two cases:
convexity and quasiconvexity of $\varphi(x,p,\cdot),$ in order to point out
the different sets of assumptions.

Before proceeding, we recall the following definition.

######  Definition 3.1.

A function $\varphi:O\times{R}^{m}\times{R}^{s}\rightarrow(-\infty,\infty]$ is
said to be a normal integrand if

  * -

$\varphi$ is ${\cal L}^{N}(O)\otimes{\cal
B}(\mathbb{R}^{m}\times\mathbb{R}^{s})$-measurable;

  * -

$\varphi(x,\cdot,\cdot)$ is lower semicontinuous for ${\mathcal{L}}^{N}$\-
a.e. $x\in O$.

Given a closed set $M\subseteq{R}^{m}$, a function $\varphi:O\times
M\times{R}^{s}\rightarrow{R}$ is said to be a Carathéodory integrand if

  * -

$\varphi(\cdot,z,\xi)$ is $\mathcal{L}^{N}(O)$-measurable for every
$(z,\xi)\in M\times{R}^{m}$;

  * -

$\varphi(x,\cdot,\cdot)$ is continuous for ${\cal L}^{N}$ a.e. $x\in O$.

###  3.1 The convex case

If the function $\varphi$ in (3.1) is convex in the last variable, then the
functional $F$ is sequentially lower semicontinuous along sequences
$u_{k}\rightharpoonup u$ in $W^{1,1}(\Omega,{R}^{d})$ and $p_{k}\rightarrow
p_{0}$ in $L^{1}(\Omega;\mathbb{R}^{m})$, in view of the following result,
shown by De Giorgi and Ioffe (see [6, Theorem 5.8]).

######  Theorem 3.2.

Let $O$ be a bounded open subset of $\mathbb{R}^{N}$ and let
$\varphi:O\times\mathbb{R}^{m}\times{R}^{s}\rightarrow[0,+\infty]$ be a normal
integrand such that the map $\xi\in\mathbb{R}^{s}\rightarrow\varphi(x,z,\xi)$
is convex for ${\cal L}^{N}$\- a.e. $x\in O$ and every $z\in\mathbb{R}^{m}$.
Let $\\{z_{k}\\}\subset L^{1}(O,{R}^{m})$ and $\\{U_{k}\\}\subset
L^{1}(O;\mathbb{R}^{s})$. If $z_{k}\rightarrow z_{0}$ in $L^{1}(O;{R}^{m})$
and $U_{k}\rightharpoonup U_{0}$ in $L^{1}(O;\mathbb{R}^{s})$ then it holds

$\int_{O}\varphi(x,z_{0}(x),U_{0}(x))dx\leq\liminf_{k\rightarrow+\infty}\int_{O}\varphi(x,z_{k}(x),U_{k}(x))dx.$

As an application of Theorem 3.2 with $m=1$ and $s=d\times N$, we give the
following result, that is crucial in the proof of Theorem 1.1.

######  Corollary 3.3.

Let $\\{p_{k}\\}\subset\mathcal{P}(\Omega)\cap L^{1}(\Omega)$ and
$\\{v_{k}\\}\subset W^{1,1}(\Omega;\mathbb{R}^{d})$. If $p_{k}\rightarrow
p_{0}$ in $L^{1}(\Omega)$ and $v_{k}\rightharpoonup v$ in
$W^{1,1}_{loc}(\Omega;\mathbb{R}^{d})$, then

$\int_{\Omega}|\nabla
v(x)|^{p_{0}(x)}\,dx\leq\,\liminf_{k\rightarrow+\infty}\int_{\Omega}|\nabla
v_{k}(x)|^{p_{k}(x)}\,dx.$ (3.2)

In particular, if $\\{v_{k}\\}$ satisfies the equiboundedness assumption
(1.10), then $\nabla v\in L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{d\times N}).$
Finally, if $\\{v_{k}\\}$ satisfies (1.10), $\Omega$ is a Lipschitz bounded
open set, $p_{0}\in C(\overline{\Omega})$ and $v\in
L^{1}(\Omega;\mathbb{R}^{d})$, then $v\in
W^{1,p_{0}(\cdot)}(\Omega;\mathbb{R}^{d})$.

###### Proof.

In order to show (3.2), it is sufficient to apply Theorem 3.2 on a sequence of
open sets $\\{O_{n}\\}$ such that $O_{n}\Subset\Omega$ and
$\cup_{n}O_{n}=\Omega$, with $\varphi:\Omega\times{R}^{d\times
N}\rightarrow[0,+\infty]$ given by

$\varphi(x,z,U)=\left\\{\begin{array}[]{ll}|U|^{z}&\hbox{ if
}z\in[1,+\infty)\\\ +\infty&\hbox{ otherwise},\end{array}\right.$

and

$z_{k}=p_{k},\quad U_{k}=\nabla u_{k}.$

For every $n\in{{N}}$, we obtain that

$\int_{O_{n}}|\nabla
v(x)|^{p_{0}(x)}\,dx\leq\,\liminf_{k\rightarrow+\infty}\int_{O_{n}}|\nabla
v_{k}(x)|^{p_{k}(x)}\,dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}|\nabla
v_{k}(x)|^{p_{k}(x)}dx$

and (3.2) follows by sending $n\rightarrow+\infty.$ Finally, the additional
assumption (1.10) combined with inequality (3.2) implies that $\nabla v\in
L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{d\times N}).$ The last part of the theorem
follows by Proposition 2.8.

######  Remark 3.4.

Let $\\{p_{k}\\}\subset\mathcal{P}_{b}(\Omega)\cap L^{1}(\Omega)$ be such
that, for some $k_{0}\in{{N}}$, it holds

$1<\inf_{k\geq k_{0}}p_{k}^{-}.$ (3.3)

  1. (1)

If $\\{v_{k}\\}\subset W^{1,1}(\Omega;\mathbb{R}^{d})$ is equibounded in the
sense of (1.10) and $v_{k}\rightharpoonup v$ in $L^{1}(\Omega;\mathbb{R}^{d})$
then $v_{k}\rightharpoonup v$ in $W^{1,1}(\Omega;\mathbb{R}^{d})$.

Indeed, exploiting (3.3), we can take $1<r<\inf_{k\geq k_{0}}p_{k}^{-}$.
Hence, by (2.4), it follows that

$\displaystyle\sup_{k\geq k_{0}}\|\nabla
v_{k}\|_{L^{r}(\Omega;\mathbb{R}^{d\times N})}$ $\displaystyle\leq$
$\displaystyle 2(\mathcal{L}^{N}(\Omega)+1)\sup_{k\geq k_{0}}\|\nabla
v_{k}\|_{L^{p_{k}(\cdot)}(\Omega;\mathbb{R}^{d\times N})}$ $\displaystyle\leq$
$\displaystyle 2(\mathcal{L}^{N}(\Omega)+1)\sup_{k\geq
k_{0}}\left\\{\big{(}\rho_{p_{k}(\cdot)}(\nabla
v_{k})\big{)}^{\frac{1}{p_{k}^{-}}},\big{(}\rho_{p_{k}(\cdot)}(\nabla
v_{k})\big{)}^{\frac{1}{p_{k}^{+}}}\right\\}<+\infty.$

Since $r>1$, we obtain that there exists a not relabeled subsequence
$\\{v_{k}\\}$ such that $\nabla v_{k}\rightharpoonup V$ in
$L^{r}(\Omega;\mathbb{R}^{d\times N})$. This easily implies that $V=\nabla v$
and that $v_{k}\rightharpoonup v$ in $W^{1,1}(\Omega;\mathbb{R}^{d})$. This
fact is used in the proof of Theorem 1.1.

  2. (2)

Let $\varphi:\Omega\times{R}\times\mathbb{R}^{d\times N}\rightarrow{R}$ be a
$\mathcal{L}(\Omega)\otimes\mathcal{B}({R}\times{R}^{d\times N})$-measurable
function satisfying the growth condition

$\varphi(x,p,\xi)\geq\,b(x)+C|\xi|^{p}\,\quad\hbox{ for
}{\mathcal{L}}^{N}\hbox{-}\hbox{a.e. }x\in\Omega,\,\forall\xi\in{R}^{d\times
N},\forall p\geq 1,$ (3.4)

where $C>0$ and $b\in L^{1}(\Omega)$. Assume that $p_{k}\rightarrow p_{0}$ in
$L^{1}(\Omega)$ with $p_{0}\in\mathcal{P}(\Omega)\cap L^{1}(\Omega)$ and let
$\\{v_{k}\\}\subset W^{1,1}(\Omega;\mathbb{R}^{d})$ be such that

$\sup_{k}F(v_{k},p_{k})<+\infty,$ (3.5)

where $F$ is the functional defined by (3.1). If $v_{k}\rightharpoonup v$ in
$L^{1}(\Omega;\mathbb{R}^{d})$ then $v_{k}\rightharpoonup v$ in
$W^{1,1}(\Omega;\mathbb{R}^{d})$ and $\nabla v\in
L^{p_{0}{(\cdot)}}(\Omega;\mathbb{R}^{d\times N}).$ Indeed, (3.4) and (3.5)
imply (1.10) and then it is sufficient to use the Part(1) of this remark,
combined with Corollary 3.3.

In particular, the coercivity assumption (3.4) is satisfied by integrands of
the form $\varphi(x,p,\xi)=W^{p}(x,\xi)$ with $p\geq 1$ and

$W(x,\xi)\geq b(x)+C|\xi|>0\quad\hbox{ for }\mathcal{L}^{N}\hbox{-}\hbox{a.e.
}x\in\Omega,\,\forall\xi\in{R}^{d\times N},$

where $C>0$ and $b(x)\in L^{\infty}(\Omega)$.

###  3.2 The quasiconvex case

The following special case of [45, Theorem 4.12] holds (see also [4, Theorem
7.4]). It is important in order to show Theorems 1.1 and 1.2. For the reader’s
convenience, after the statement, we provide an outline of the main steps
necessary to deduce this application from the results contained in [45].

######  Theorem 3.5.

Let $p_{0}\in C(\Omega)$ be such that $p_{0}\in\mathcal{P}_{b}^{log}(A)$ for
every $A\Subset\Omega$. Let $\varphi:\Omega\times\mathbb{R}^{d\times
N}\rightarrow[0,+\infty)$ be a function such that

  * -

$\varphi(\cdot,\xi)$ is $\mathcal{L}^{N}(\Omega)$-measurable for every
$\xi\in\mathbb{R}^{d\times N}$;

  * -

$\varphi(x,\cdot)$ is quasiconvex for $\mathcal{L}^{N}$-a.e. $x\in\Omega$;

  * -

there exists $C>1$ and $b(x)\in L^{1}_{\rm loc}(\Omega)$ with $b(x)\geq\,0$
such that

$0\leq\,\varphi(x,\xi)\leq\,b(x)+C|\xi|^{p_{0}(x)}.$

Then, for every sequence $\\{v_{k}\\}\subset W^{1,1}(\Omega;\mathbb{R}^{d})$
with $v_{k}\rightarrow v$ in $L^{1}(\Omega;\mathbb{R}^{d})$ and such that

$\sup_{k}\int_{\Omega}|\nabla v_{k}(x)|^{p_{0}(x)}\,dx<+\infty,$ (3.6)

we have that $v\in W^{1,{p_{0}(\cdot)}}_{loc}(\Omega;\mathbb{R}^{d})$,
$|\nabla v|^{{p_{0}(\cdot)}}\in L^{1}(\Omega)$ and

$\int_{\Omega}\varphi(x,\nabla
v(x))\,dx\leq\,\liminf_{k\rightarrow+\infty}\int_{\Omega}\varphi(x,\nabla
v_{k}(x))\,dx.$ (3.7)

###### Sketch of the proof.

We observe that, by applying Remark 3.4(1) with $p_{k}\equiv p_{0}$, we have
that the sequence $\\{v_{k}\\}$ weakly converges to $v$ in
$W^{1,1}_{loc}(\Omega;{R}^{d})$. Hence, by applying Corollary 3.3, taking into
account (3.6), we get that $v\in W^{1,p_{0}(\cdot)}_{\rm
loc}(\Omega;\mathbb{R}^{d})$ and $|\nabla v|^{p_{0}(\cdot)}\in L^{1}(\Omega)$.
Hence, the lower semicontinuity result given by (3.7) follows as an
application of [45, Theorem 4.12] recalling that, for every $A\Subset\Omega$,

  * •

if $p_{0}\in C(A)$ and $p_{0}>1$ on $A$, then the function
$\psi(x,\xi)=|\xi|^{p_{0}(x)}$ enjoys the Sobolev type property and the
Rellich type property on $A$ required by [45, Theorem 4.12] (see [45,
Definition 3.2 and Example 3.4], and [17, Lemmas 2.4 and 2.8]);

  * •

if $p_{0}\in\mathcal{P}^{log}(A)$ then $\psi$ verifies the maximal property on
$A$, see [45, Proposition 5.2 and Definition 4.4]. Hence $\psi$ satisfies also
the density property on $A$, see [45, Definition 4.5 and Proposition 4.6].

Hence, [45, Theorem 4.12] entails that for every $A\Subset\Omega$

$\int_{A}\varphi(x,\nabla
v(x))\,dx\leq\,\liminf_{k\rightarrow+\infty}\int_{A}\varphi(x,\nabla
v_{k}(x))\,dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}\varphi(x,\nabla
v_{k}(x))\,dx.$

By passing to the sup on $A\Subset\Omega$, we get the desired conclusion
(3.7).

By using the arguments in [24, Lemma 4.1] (see also [43, Lemma 5.1]), we show
the following key result.

######  Lemma 3.6.

Let $\Omega\subset\mathbb{R}^{N}$ an open set with finite Lebesgue measure.
Let $W:\Omega\times{R}^{d\times N}\rightarrow[0,+\infty)$ be a Borel function.
Let $a_{0}\in L^{\infty}(\Omega)$ be such that $a_{0}\geq 0$
$\mathcal{L}^{N}$-a.e. in $\Omega$. Let $p_{0}\in\mathcal{P}_{b}(\Omega)$ and
let $\\{p_{k}\\}\subset L^{\infty}(\Omega)$ be such that $p_{k}\rightarrow
p_{0}$ in $L^{\infty}(\Omega)$. Then, for every
$0<\delta<\min\\{1/2,p_{0}^{-}-1\\}$, there exists $k_{0}\in{{N}}$ such that,
for every $k\geq k_{0}$ and for every $U\in L^{1}(\Omega,{R}^{d\times N})$, it
holds

$\int_{\Omega}a_{0}(x)W^{p_{0}(x)-{\delta}}(x,U(x))dx\leq
2{\delta}\mathcal{L}^{N}(\Omega)+(2{\delta})^{-2{\delta}}||a_{0}||_{L^{\infty}(\Omega)}\int_{\Omega}a_{0}(x)W^{p_{k}(x)}(x,U(x))dx.$
(3.8)

Proof. Let $0<\delta<\min\\{1/2,p_{0}^{-}-1\\}$. Thanks to the uniform
convergence of $\\{p_{k}\\}$ to $p_{0}$, there exists $k_{0}\in{{N}}$ and a
negligible measurable set $N\subseteq\Omega$ such that

$1<p_{0}(x)-{\delta}<p_{k}(x)<p_{0}(x)+{\delta}\quad\hbox{ for every }x\in\
\Omega\setminus N,$

for every $k\geq k_{0}$. Hence, for every $x\in\Omega\setminus N$ and for
every $U\in L^{1}(\Omega,{R}^{d\times N})$, if $2{\delta}<W(x,U(x))<1$, then

$(2{\delta})^{2{\delta}}<(2{\delta})^{p_{k}(x)-p_{0}(x)+{\delta}}<W^{p_{k}(x)-p_{0}(x)+{\delta}}(x,U(x))<1$

that implies

$1<W^{p_{0}(x)-p_{k}(x)-{\delta}}(x,U(x))<(2{\delta})^{-2{\delta}},$

while, if $W(x,U(x))\leq 2\delta$, then

$W^{p_{0}(x)-\delta}(x,U(x))\leq(2\delta)^{p_{0}(x)-\delta}<2\delta,$

being $2\delta<1$ and $p_{0}(x)-\delta>1.$ Hence, for every $k\geq k_{0}$ and
for every $U\in L^{1}(\Omega,{R}^{d\times N})$, it holds

$\displaystyle\int_{\Omega}a_{0}(x)W^{p_{0}(x)-{\delta}}(x,U(x))dx$
$\displaystyle=$ $\displaystyle\int_{\\{W(x,U(x))\leq
2{\delta}\\}}a_{0}(x)W^{p_{0}(x)-{\delta}}(x,U(x))dx+\int_{\\{2{\delta}<W(x,U(x))<1\\}}a_{0}(x)W^{p_{0}(x)-{\delta}}(x,U(x))dx$
$\displaystyle+\int_{\\{W(x,U(x))\geq
1\\}}a_{0}(x)W^{p_{0}(x)-{\delta}}(x,U(x))dx$ $\displaystyle\leq$
$\displaystyle
2\delta||a_{0}||_{L^{\infty}(\Omega)}\mathcal{L}^{N}(\Omega)+(2{\delta})^{-2{\delta}}\bigg{(}\int_{\\{2{\delta}<W(x,U(x))<1\\}}W^{p_{k}(x)}(x,U(x))dx+\int_{\\{W(x,U(x))\geq
1\\}}W^{p_{k}(x)}(x,U(x))dx\bigg{)}$ $\displaystyle\leq$ $\displaystyle
2\delta||a_{0}||_{L^{\infty}(\Omega)}\mathcal{L}^{N}(\Omega)+(2{\delta})^{-2{\delta}}\int_{\Omega}W^{p_{k}(x)}(x,U(x))dx.$

which gives the desired conclusion.

Now, we are in position to show the following general result which allows us
to deduce Theorems 1.1 and 1.2 as immediate corollaries.

######  Theorem 3.7.

Let $\Omega$, $W$, $p_{0}$, $\\{p_{k}\\}$, $v$, $\\{v_{k}\\}$ be as in Theorem
1.1 and let $a_{0}$, $\\{a_{k}\\}$ be as in Theorem 1.2. Then $v$ satisfies
(1.11) and

$\int_{\Omega}a_{0}(x)W^{p_{0}(x)}(x,\nabla
v(x))dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{k}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx.$

Proof. First of all, we note that, for every $A\Subset\Omega$, since
$\inf_{A}p_{0}>1$ and $p_{k}$ uniformly converges to $p_{0}$ as
$k\rightarrow\infty$, we have that there exists $k_{0}=k_{0}(A)\in{{N}}$ such
that

$\inf_{k\geq k_{0}}\mathop{\rm ess\>inf}_{A}p_{k}(x)>1.$

In view of Remark 3.4 (1), this implies that $v_{k}\rightharpoonup v$ in
$W^{1,1}(A;{R}^{d})$ for every $A\Subset\Omega$. By applying Corollary 3.3 in
view of (1.10), we get that $v\in W^{1,p_{0}(\cdot)}_{\rm
loc}(\Omega;\mathbb{R}^{d})$ and $\nabla v\in L^{p_{0}(\cdot)}(\Omega)$. Now
we claim that for every $A\Subset\Omega$ it holds

$\int_{A}a_{0}(x)W^{p_{0}(x)}(x,\nabla
v(x))dx\leq\liminf_{k\rightarrow+\infty}\int_{A}a_{k}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx.$ (3.9)

Hence, by passing to the sup in (3.9) with respect to $A\Subset\Omega$, we get
the desired conclusion.

In order to show (3.9), for fixed $A\Subset\Omega$, we note that

$\displaystyle\liminf_{k\rightarrow+\infty}\int_{A}a_{k}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx$ $\displaystyle=$
$\displaystyle\liminf_{k\rightarrow+\infty}\left(\int_{A}(a_{k}(x)-a_{0}(x))W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx+\int_{A}a_{0}(x)W^{p_{k}(x)}(x,\nabla v_{k}(x))dx\right)$
$\displaystyle\geq$
$\displaystyle\liminf_{k\rightarrow+\infty}\left(-||a_{k}-a_{0}||_{L^{\infty}(A)}\int_{A}W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx+\int_{A}a_{0}(x)W^{p_{k}(x)}(x,\nabla v_{k}(x))dx\right).$

On the other hand, (1.10) and (1.12) ensure

$\liminf_{k\rightarrow+\infty}\int_{A}W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx<+\infty$

and, since $a_{k}\rightarrow a_{0}$ in $L^{\infty}(\Omega)$, this implies

$\lim_{k\rightarrow+\infty}||a_{k}-a_{0}||_{L^{\infty}(A)}\int_{A}W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx=0.$

Hence

$\liminf_{k\rightarrow+\infty}\int_{A}a_{k}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx\geq\liminf_{k\rightarrow+\infty}\int_{A}a_{0}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx.$ (3.10)

Since $\\{p_{k}\\}$ uniformly converges to $p_{0}$ on $A$ and $p_{0}>1$ on
$A$, by applying Lemma 3.6, there exists $k_{0}=k_{0}(A)\in{{N}}$ such that,
for every $k\geq k_{0}$, it holds

$\int_{A}W^{p_{0}(x)-{\delta}}(x,U(x))dx\leq
2{\delta}\mathcal{L}^{N}(A)+(2{\delta})^{-2{\delta}}\int_{A}W^{p_{k}(x)}(x,U(x))dx\qquad\forall\,U\in
L^{1}(A,{R}^{d\times N}).$ (3.11)

By applying (3.11) with $U=\nabla v_{k}$ and then passing to the limit when
$k\rightarrow\infty$, we obtain that

$\liminf_{k\rightarrow+\infty}\int_{A}a_{0}(x)W^{p_{0}(x)-\delta}(x,\nabla
v_{k}(x))dx\leq(2\delta)^{-2\delta}\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{0}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx+2\delta||a||_{L^{\infty}(A)}\mathcal{L}^{N}(A).$

Taking into account the growth condition (1.12), we can apply Theorem 3.5 with
the variable exponent $p_{0}(\cdot)-\delta\in P_{b}^{log}(A)$ to obtain that

$\displaystyle\int_{A}a_{0}(x)W^{p_{0}(x)-\delta}(x,\nabla
v(x))dx\leq(2\delta)^{-2\delta}\liminf_{k\rightarrow+\infty}\int_{A}a_{0}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx+2\delta||a_{0}||_{L^{\infty}(A)}\mathcal{L}^{N}(A).$

By sending $\delta\rightarrow 0$ and by using Fatou’s Lemma in the above
inequality, we finally get the desired conclusion

$\displaystyle\int_{A}a_{0}(x)W^{p_{0}(x)}(x,\nabla v(x))dx$
$\displaystyle\leq\liminf_{k\rightarrow\infty}\int_{A}a_{0}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx$
$\displaystyle\leq\liminf_{k\rightarrow\infty}\int_{A}a_{k}(x)W^{p_{k}(x)}(x,\nabla
v_{k}(x))dx,$

where the last inequality follows by (3.10).

##  4 Dimensional reduction

The aim of this section consists proving the $\Gamma$-convergence results
stated in Theorems 1.3, 1.4, and 1.5 concerning the dimension reduction
problems defined in
$\displaystyle\Omega_{\varepsilon}:=\omega\times(-\varepsilon/2,\varepsilon/2)$
where $\omega\subseteq{R}^{2}$ is a bounded, Lipschitz open set. To this end,
in view of Definition 2.11, in this section
$\\{\varepsilon_{k}\\}\subseteq(0,1/2)$ is any sequence converging to $0$.

We start proving a compactness result for energy bounded sequences, which,
indeed, motivates the choice of the topology in our $\Gamma$-convergence
results. Furthermore, according to the regularity of the variable exponent,
modeling the point-dependent growth of the hyperelastic energy, we obtain
different results for the limiting deformation fields. Finally, the proofs of
Theorems 1.3, 1.4 and 1.5 are provided showing that, for every infinitesimal
sequence $\\{\varepsilon_{k}\\},$ the $\Gamma-$limit of
$\\{\mathcal{F}_{\varepsilon_{k}}\\}$ exists and admits the same integral
representation, when $p\in\mathcal{P}_{b}^{log}(\Omega)$.

###  4.1 Compactness of bounded energy sequences

Let $\\{\mathcal{F}_{\varepsilon}\\}$ be the family of integral energies in
(1.16). The following compactness result for energy bounded sequences, relies
on Poincaré inequality (2.5), Corollary 3.3 and Proposition 2.8.

######  Proposition 4.1.

Let $f:\omega\times(-1/2,1/2)\times[1,\infty)\times{R}^{3\times
3}\rightarrow{R}$ be such that

  * -

$f$ is ${\cal L}^{2}(\omega)\otimes{\cal
B}((-1/2,1/2)\times[1,\infty)\times{R}^{3\times 3})$-measurable;

  * -

$f$ satisfies (1.17).

Let $p\in\mathcal{P}_{b}(\Omega)$, let $\varepsilon_{k}\rightarrow 0^{+}$ and
let $\\{p_{{\varepsilon_{k}}}\\}$ be as in (1.6). Suppose that
$\\{u_{\varepsilon_{k}}\\}\subset L^{1}(\Omega;\mathbb{R}^{3})$ is a sequence
such that

$\sup_{k\in\mathbb{N}}\mathcal{F}_{\varepsilon_{k}}(u_{\varepsilon_{k}})<+\infty.$
(4.1)

Then

  1. (1)

up to a not relabelled subsequence, $\\{u_{\varepsilon_{k}}\\}$ converges
weakly in $W^{1,p^{-}}(\Omega;\mathbb{R}^{3}),$ (where $p^{-}$ is given by
(1.7)), to a function $u_{0}\in
W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})$ (cf. Definition
2.3), which, with an abuse of notation, we can identify with a function
$u_{0}\in W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3})$.

  2. (2)

if $p_{0}\in C(\overline{\omega})$ and $p_{\varepsilon_{k}}\rightarrow
p_{0}\hbox{ in }L_{loc}^{1}(\Omega)$, then $u_{0}\in
W^{1,p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})\cap
W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})$ and it can be
identified with $u_{0}\in W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})\cap
W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3})$;

  3. (3)

if $p_{0}\in\mathcal{P}_{b}^{log}(\omega)$ and $p_{\varepsilon_{k}}\rightarrow
p_{0}\hbox{ in }L_{loc}^{1}(\Omega)$, then $u_{0}$ can be identified with a
function $u_{0}\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$;

  4. (4)

if $p_{0}\in C(\overline{\omega})$ and $p_{\varepsilon_{k}}\rightarrow
p_{0}\hbox{ in }L^{\infty}(\Omega)$, then $u_{\varepsilon_{k}}\rightarrow
u_{0}$ in $L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3}).$

######  Remark 4.2.

Note that, if $p$ satisfies the assumptions (1.7) and (1.8) on $\Omega$, then
$\\{p_{{\varepsilon_{k}}}\\}$ is equicontinuous on $\overline{\Omega}$ and
equibounded (thanks to (1.30) and (1.31)). By applying Ascoli-Arzelà Theorem,
it follows that
$\|p_{{{\varepsilon_{k}}}}-p_{0}\|_{L^{\infty}(\Omega)}\rightarrow 0$ with
$p_{0}(x_{\alpha},x_{3})=p(x_{\alpha},0).$

###### Proof.

  1. (1)

Thanks to the growth condition (1.17), the assumption (4.1) implies that
$u_{\varepsilon_{k}}\in
W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})\cap
W^{1,p_{{\varepsilon_{k}}}(\cdot)}(\Omega;\mathbb{R}^{3})$ for every
$k\in{{N}}$ and that there exists a constant $C\geq 1$ such that

$\displaystyle\sup_{k\in\mathbb{N}}\int_{\Omega}|\nabla_{\alpha}u_{\varepsilon_{k}},\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}|^{p_{\varepsilon_{k}}(x)}dx\leq
C.$ (4.2)

By exploiting (1.31), both the sequences $\\{|\nabla u_{\varepsilon_{k}}|\\}$
and
$\left\\{\left|\frac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}\right|\right\\}$
are bounded in $L^{p^{-}}(\Omega)$ and, by using the Poincarè inequality (2.5)
with $\Gamma:=\partial\omega\times(-1/2,1/2)$, it turns out that
$(u_{\varepsilon_{k}})$ is bounded in $W^{1,p^{-}}(\Omega;\mathbb{R}^{3})$,
hence, $\\{u_{\varepsilon_{k}}\\}$ admits a not relabelled subsequence which
is weakly converging in $W^{1,p^{-}}(\Omega;\mathbb{R}^{3})$ to $u_{0}\in
W^{1,p^{-}}_{0,\partial_{L}\Omega}(\Omega;\mathbb{R}^{3})$, being the latter
space weakly closed. On the other hand, $\displaystyle\frac{\partial
u_{0}}{\partial x_{3}}=0$ $\mathcal{L}^{3}$-a.e. in $\Omega$, hence, with an
abuse of notation, we can identify $u_{0}$ with a function in
$W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3})$.

  2. (2)

The statement follows by Part (1) and Corollary 3.3.

  3. (3)

In view of Theorem 2.9, under the additional assumption,
$p_{0}\in\mathcal{P}_{b}^{log}(\omega)$, the space
$W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3})\cap
W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$ coincides with
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$.

  4. (4)

Now, under the additional hypothesis that

$p_{\varepsilon_{k}}\rightarrow p_{0}\hbox{ in }L^{\infty}(\Omega),$ (4.3)

we show that $u_{\varepsilon_{k}}\rightarrow u_{0}$ in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3}).$ Since $p_{0}\in
C(\overline{\Omega})$ and $\partial\Omega$ is Lipschitz continuous, we can
find a finite number of open balls $(B_{i})_{i\in F}$ and
$(\tilde{B}_{i})_{i\in F}$, such that $\bar{B}_{i}\subset\subset\tilde{B}_{i}$
and (2.9) hold. By following the same argument in Proposition 2.8, for every
$j\in F$ there exists $\delta_{j}>0$ such that (2.11) holds with
$p_{0,j}^{+},p_{0,j}^{-}$ given by (2.10).

Let $\varphi_{j}\in C^{1}(\Omega)$ be such that $0\leq\varphi_{j}\leq 1$,
$\varphi_{j}=1$ on $B_{j}$, ${\rm supp}\,\varphi_{j}\subset\tilde{B}_{j}$ and
$|\nabla\varphi_{j}|\leq C$. Now we choose $0<\delta<\min_{j\in F}\delta_{j}$.
Since $||p_{\varepsilon_{k}}-p_{0}||_{L^{\infty}(\Omega)}\rightarrow 0$, there
exists $k_{0}\in\mathbb{N}$ such that $p_{\varepsilon_{k}}(x)>p_{0}(x)-\delta$
for $k\geq k_{0}$ and for ${\mathcal{L}}^{3}$-a.e. $x\in\Omega$. This,
together with Hölder’s inequality, implies that for $k\geq k_{0}$ the
embedding

$W^{1,p_{\varepsilon_{k}}(\cdot)}(\Omega;\mathbb{R}^{3})\hookrightarrow
W^{1,p_{0}(\cdot)-\delta_{j}}(\Omega;\mathbb{R}^{3})\hbox{ is continuous
}\forall j\in F.$ (4.4)

Moreover, by reasoning as in the proof of (2.13), the embedding

$W^{1,p_{0}(\cdot)-\delta_{j}}(\Omega;\mathbb{R}^{3})\ni
u\mapsto\varphi_{j}u\in
W^{1,p^{-}_{0,j}(\cdot)-\delta_{j}}(\Omega;\mathbb{R}^{3})\hbox{ holds and is
continuous }\forall j\in F.$ (4.5)

By (4.4) , (4.5) and (2.12), it easily follows that, $\forall j\in F$ and for
$k\geq k_{0},$ the embedding

$W^{1,p_{\varepsilon_{k}}(\cdot)}(\Omega;\mathbb{R}^{3})\ni
u\hookrightarrow\varphi_{j}u\in L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})\hbox{
is compact.}$

Hence, for every $j\in F$, the sequence $\\{u_{\varepsilon_{k}}\\}$ admits a
subsequence $\\{u_{\varepsilon^{j}_{k}}\\}$ such that
$\\{\varphi_{j}u_{\varepsilon^{j}_{k}}\\}$ converges strongly in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$ to a function $u_{0}^{j}$ for every
$j\in F$. Since $p^{-}\leq p_{0}(\cdot)$ in $\Omega$ then
$\varphi_{j}u_{\varepsilon^{j}_{k}}\rightarrow u_{0}^{j}$ as
$k\rightarrow+\infty$ in $L^{p^{-}}(\Omega;\mathbb{R}^{3})$. Then
$u_{0}^{j}=u_{0}$ for every $j\in F$ that implies
$u_{\varepsilon^{j}_{k}}\rightarrow u_{0}$ in $L^{p_{0}(\cdot)}(\Omega\cap
B_{j};\mathbb{R}^{3}).$ Passing to subsequences repeatedly, we obtain that
$u_{\varepsilon_{k}}\rightarrow u_{0}$ in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3}).$

With the aim at proving Theorems 1.3 1.4 and 1.5, we start with a preliminary
lemma.

######  Lemma 4.3.

Let $f:\omega\times(-1/2,1/2)\times[1,\infty)\times{R}^{3\times
3}\rightarrow{R}$ be such that

  * ($1_{f}$)

$f$ is ${\cal L}^{2}(\omega)\otimes{\cal
B}((-1/2,1/2)\times[1,\infty)\times{R}^{3\times 3})$-measurable.

Then

  * ($1_{f_{0}}$)

the function $\displaystyle f_{0}:\omega\times(-1/2,1/2)\times{R}^{3\times
2}$, defined by (1.19), is ${\cal L}^{2}(\omega)\otimes{\cal
B}((-1/2,1/2)\times[1,\infty)\times{R}^{3\times 2})$-measurable.

Furthermore, assuming that

  * ($2_{f}$)

$f(x_{\alpha},\cdot,\cdot,\cdot)$ is lower semicontinuous for
${\mathcal{L}}^{2}$\- a.e. $x_{\alpha}\in\omega$;

  * ($3_{f}$)

there exists $C_{1}>0$ such that for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$ and for every
$\displaystyle(y,q,\xi)\in(-1/2,1/2)\times[1,\infty)\times\mathbb{R}^{3\times
3}$, it holds:

$f(x_{\alpha},y,q,\xi)\geq C_{1}|\xi|^{q}-\frac{1}{C_{1}}$

then,

  * ($2_{f_{0}}$)

$f_{0}(x_{\alpha},\cdot,\cdot,\cdot)$ is lower semicontinuous on
$(-1/2,1/2)\times[1,\infty)\times{R}^{3\times 2}$ for ${\mathcal{L}}^{2}$-a.e.
$x_{\alpha}\in\omega$;

  * ($3_{f_{0}}$)

for every
$\displaystyle(y,q,\xi_{\alpha})\in(-1/2,1/2)\times[1,\infty)\times\mathbb{R}^{3\times
2}$ it holds

$f_{0}(x_{\alpha},y,q,\xi_{\alpha})\geq
C_{1}|\xi_{\alpha}|^{q}-\frac{1}{C_{1}}.$ (4.6)

Moreover, if

  * $(4_{f})$

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ the function
$f(x_{\alpha},y,q,\cdot)$ is convex on ${R}^{3}\times{R}^{3}$ for every
$(y,q)\in(-1/2,1/2)\times[1,\infty)$,

then

  * ($4_{f_{0}}$)

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ the function
$f_{0}(x_{\alpha},y,q,\cdot)$ is convex on ${R}^{3}\times{R}^{2}$ for every
$(y,q)\in(-1/2,1/2)\times[1,\infty)$.

Finally, if $f$ satisfies (1.17), then

$C_{1}|\xi_{\alpha}|^{q}-\frac{1}{C_{1}}\leq
f_{0}(x_{\alpha},y,q,\xi_{\alpha})\leq C_{2}(|\xi_{\alpha}|^{q}+1),$ (4.7)

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega,$ and for every
$(y,q,\xi_{\alpha})\in(-1/2,1/2)\times[1,\infty)\times\mathbb{R}^{3\times 2}$.

###### Proof.

For simplicity, in the sequel we use $f(x,y,q,\xi_{\alpha},\xi_{3})$ in place
of $f(x,y,q,(\xi_{\alpha},\xi_{3}))$. ($1_{f_{0}}$) follows by the fact that,
by definition, $f_{0}$ is the infimum of the family of functions
$f(\cdot,\cdot,\cdot,\cdot,\xi_{3})$ which are ${\cal
L}^{2}(\omega)\otimes{\cal B}((-1/2,1/2)\times[1,\infty)\times{R}^{3\times
2})$-measurable.

The proof of $(2_{f_{0}})-(3_{f_{0}})$ and (4.7) follows along the lines the
arguments of [44, Proposition 1]. The property $(4_{f_{0}})$ easily follows by
the very definition (1.19) of $f_{0}$.

###  4.2 The case of convex densities

######  Proposition 4.4.

Let $f:\omega\times(-1/2,1/2)\times[1,\infty)\times{R}^{3\times
3}\rightarrow{R}$ be a function satisfying $(1_{f})$, $(2_{f})$ and $(4_{f})$
of Lemma 4.3. Let $p\in\mathcal{P}_{b}(\Omega)$, let
$\\{p_{\varepsilon_{k}}\\}$ be as in (1.6) and let
${\mathcal{F}}_{\varepsilon_{k}}$ be as in (1.16). Let
$\varepsilon_{k}\rightarrow 0^{+}$ and assume that
$p_{\varepsilon_{k}}\rightarrow p_{0}\hbox{ in }L^{1}(\Omega)$ with
$p_{0}=p_{0}(x_{\alpha})\in C(\overline{\Omega})$. Let
$\\{u_{\varepsilon_{k}}\\}\subseteq{L^{1}(\Omega;\mathbb{R}^{3})}$ be a
sequence satisfying (4.2) and such that $u_{\varepsilon_{k}}\rightarrow u$ in
$L^{1}(\Omega;\mathbb{R}^{3}).$ Then

${\mathcal{F}}_{p^{-}}(u)\leq\liminf_{k\rightarrow+\infty}{\mathcal{F}}_{\varepsilon_{k}}(u_{\varepsilon_{k}}),$

where $\mathcal{F}_{p^{-}}:L^{1}(\Omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$
be the functional defined by

${\mathcal{F}}_{p^{-}}(v):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}{f}_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}v(x_{\alpha}))dx_{\alpha},&\hbox{
if }v\in W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})\cap
W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3}),\\\ +\infty&\hbox{ otherwise in
}L^{1}(\Omega;\mathbb{R}^{3}),\end{array}\right.$ (4.8)

with $f_{0}$ given by (1.19).

###### Proof.

Without loss of generality, we can assume that
$\liminf_{k\rightarrow\infty}{\mathcal{F}}_{\varepsilon_{k}}(u_{\varepsilon_{k}})<+\infty$,
and that, up to a not relabelled subsequence, such a liminf is a limit. Then,
arguing as in Proposition 4.1 (see Part (1) and (3)) in view of (4.2),
$u_{\varepsilon_{k}}$ weakly converges to $u$ in
$W^{1,p^{-}}(\Omega;\mathbb{R}^{3})$, and $u$ can be identified with an
element in $W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})\cap
W^{1,p^{-}}_{0}(\omega;\mathbb{R}^{3})$. Moreover, up to a subsequence,

$\displaystyle\frac{1}{\varepsilon_{k_{h}}}\nabla_{3}u_{\varepsilon_{k_{h}}}\rightharpoonup\bar{w}\in
L^{p^{-}}(\Omega;\mathbb{R}^{3}),$ (4.9)

where the latter function might depend on the selected subsequence. Now we
observe that the assumptions on $f$ guarantee that the function
$\varphi:\Omega\times\mathbb{R}^{2}\times{R}^{3\times
3}\rightarrow[0,+\infty]$, given by

$\varphi(x,y,q,\xi)=\left\\{\begin{array}[]{ll}f(x_{\alpha},y,q,\xi)&\hbox{ if
}(y,q)\in(-1/2,1/2)\times[1,+\infty)\\\ +\infty&\hbox{
otherwise},\end{array}\right.$

with $O=\Omega$, $m=2$ and $d=3\times 3$, satisfies all the assumptions of
Theorem 3.2. Since

$z_{k}(x)=(\varepsilon_{k}x_{3},p_{\varepsilon_{k}}(x))\rightarrow(0,p_{0}(x_{\alpha}))\hbox{
in }L^{1}(\Omega;\mathbb{R}^{2})$

and, thanks to (4.9), (up to a not relabeled subsequence),

$U_{k}(x)=\left(\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x)\right)\rightharpoonup(\nabla
u(x_{\alpha}),\bar{w}(x))\hbox{ in }L^{1}(\Omega,{R}^{3\times 3}),$

it follows that

$\displaystyle\lim_{\varepsilon_{k}\rightarrow
0^{+}}\int_{\Omega}f(x_{\alpha},\varepsilon_{k}x_{3},p_{\varepsilon_{k}}(x),\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx$
$\displaystyle\geq$
$\displaystyle\int_{\Omega}f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}u(x_{\alpha}),\bar{w}(x))dx$
$\displaystyle\geq$
$\displaystyle\int_{\Omega}f_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha}dx_{3}$
$\displaystyle=$
$\displaystyle\int_{\omega}f_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},$

which concludes the proof, in view of the measurability property ($1_{f_{0}}$)
satisfied by $f_{0}$ thanks to Lemma 4.3.

######  Proposition 4.5.

Let $\displaystyle
f:\omega\times(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}\rightarrow[0,+\infty)$ be a function such that

  * -

$f(x_{\alpha},\cdot,\cdot,\cdot)$ is continuous for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$,

  * -

$f(\cdot,y,q,\xi)$ is measurable for every
$\displaystyle(y,q,\xi)\in(-1/2,1/2)\times[1,+\infty)\times\mathbb{R}^{3\times
3}$.

Morever assume that (1.17) holds. Let $p\in\mathcal{P}_{b}(\Omega)$, let
$\\{p_{\varepsilon_{k}}\\}$ be as in (1.6). Let $\varepsilon_{k}\rightarrow
0^{+}$ and assume that $p_{\varepsilon_{k}}\rightarrow p_{0}\hbox{ in
}L^{1}(\Omega)$ with $p_{0}\in\mathcal{P}^{log}_{b}(\omega)$. Then it holds

$\Gamma(L^{p_{0}(\cdot)})\hbox{-}\limsup_{k\rightarrow+\infty}{\mathcal{F}}_{{\varepsilon_{k}}}\leq{\mathcal{F}}$
(4.10)

where $\mathcal{F}_{{{\varepsilon_{k}}}}$ and ${\mathcal{F}}$ are defined,
respectively, by (1.16) and (1.18).

###### Proof.

Define

$\displaystyle
F:=\Gamma(L^{p_{0}(\cdot)})\hbox{-}{\limsup_{k\rightarrow+\infty}}\,\mathcal{F}_{{{\varepsilon_{k}}}}.$
(4.11)

By the very definition of $\mathcal{F}$, we can restrict to prove that
$F\leq\mathcal{F}$ on $W_{0}^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3}).$ First,
we consider the case $u,w\in\mathcal{C}^{\infty}_{0}(\omega;\mathbb{R}^{3})$.
Set

$u_{\varepsilon_{k}}(x):=u(x_{\alpha})+\varepsilon_{k}\,x_{3}\,w(x_{\alpha}),$

we have that, as $k\rightarrow+\infty$,

$u_{{{\varepsilon_{k}}}}(x_{\alpha},x_{3})\rightarrow
u(x_{\alpha})\qquad\textnormal{in $L^{\infty}(\Omega;\mathbb{R}^{3})$}$

and

$\left(\nabla_{\alpha}u_{{\varepsilon_{k}}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x)\right)=\left(\nabla
u(x_{\alpha})+{{\varepsilon_{k}}}\,x_{3}\nabla
w(x_{\alpha}),w(x_{\alpha})\right)\rightarrow(\nabla
u(x_{\alpha}),w(x_{\alpha}))\hbox{ in }L^{\infty}(\Omega;\mathbb{R}^{3\times
3}).$

Thus, in particular, there exists $M>0$ such that

$\sup_{k\in\mathbb{N}}\|\left(\nabla u+{{\varepsilon_{k}}}\,x_{3}\nabla
w,w\right)\|_{L^{\infty}(\Omega;\mathbb{R}^{3\times 3})}\leq M.$

Being $f(x_{\alpha},\cdot,\cdot,\cdot,\cdot)$ continuous for
$\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$, by the above convergences we get
that, as $k\rightarrow+\infty$,

$f\left(x_{\alpha},{{\varepsilon_{k}}}x_{3},{p_{{{\varepsilon_{k}}}}(x)},\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{{{\varepsilon_{k}}}}(x)\right)\rightarrow
f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla u(x_{\alpha}),w(x_{\alpha}))$

for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ and for every
$x_{3}\in(-1/2,1/2)$. Moreover, by (1.17) and (1.30), it holds

$\displaystyle\mathop{\rm ess\>sup}_{x\in\Omega}f\left(x_{\alpha},\varepsilon
x_{3},p_{{{\varepsilon_{k}}}}(x),\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{{{\varepsilon_{k}}}}(x)\right)$
$\displaystyle\leq\,C_{2}\left(\mathop{\rm
ess\>sup}_{x\in\Omega}\left|\nabla_{\alpha}u_{{{\varepsilon_{k}}}},\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{{{\varepsilon_{k}}}}\right|^{p_{\varepsilon_{k}}(x)}+1\right)$
$\displaystyle=C_{2}\big{(}\mathop{\rm
ess\>sup}_{x\in\Omega}\left|\left(\nabla u+{{\varepsilon_{k}}}\,x_{3}\nabla
w,w\right)\right|^{p_{{{\varepsilon_{k}}}}(x)}+1\big{)}$ $\displaystyle\leq
C_{2}\big{(}(1+M)^{p^{+}}+1\big{)}.$

Hence, by applying the Dominated Convergence Theorem, by (1.17) it follows
that

$\displaystyle
F(u)\leq\,\limsup_{k\rightarrow+\infty}\mathcal{F}_{{{\varepsilon_{k}}}}(u_{\varepsilon_{k}})$
$\displaystyle=\limsup_{k\rightarrow+\infty}\int_{\Omega}f\left(x_{\alpha},{{\varepsilon_{k}}}x_{3},{p_{{{\varepsilon_{k}}}}(x)},\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{{{\varepsilon_{k}}}}(x)\right)dx$
$\displaystyle=\,\int_{\omega}f(x_{\alpha},0,{p_{0}(x_{\alpha})},\nabla
u(x_{\alpha}),w(x_{\alpha}))\,dx_{\alpha}<+\infty.$

Being $w\in\mathcal{C}^{\infty}_{0}(\omega;\mathbb{R}^{3})$ arbitrary, we get

$F(u)\leq\,\inf_{w\in\mathcal{C}^{\infty}_{0}(\omega;\mathbb{R}^{3})}\int_{\omega}f(x_{\alpha},0,{p_{0}(x_{\alpha})},\nabla
u(x_{\alpha}),w(x_{\alpha}))\,dx_{\alpha},\qquad\forall\,u\in\mathcal{C}^{\infty}_{0}(\Omega;\mathbb{R}^{3}).$

Since $\mathcal{C}^{\infty}_{0}(\omega;\mathbb{R}^{3})$ is dense in
$L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$, arguing in components and taking
into account the growth condition (1.17), by applying Vitali-Lebesgue
Dominated Convergence Theorem, we get

$F(u)\leq\,\inf_{w\in
L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})}\int_{\omega}f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla
u(x_{\alpha}),w(x_{\alpha}))\,dx_{\alpha},\qquad\forall\,u\in\mathcal{C}^{\infty}_{0}(\Omega;\mathbb{R}^{3}).$
(4.12)

Following the same argument in [44, page 558], there exists a measurable
function $w_{0}=w_{0}(x_{\alpha})$ such that

$f_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla
u(x_{\alpha}))=\inf_{\xi_{3}\in\mathbb{R}^{3}}[f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla
u(x_{\alpha}),\xi_{3})]=f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla
u(x_{\alpha}),w_{0}(x_{\alpha}))$

for ${\mathcal{L}^{2}}$-a.e. $x_{\alpha}\in\omega.$ Thanks to (1.17), such a
function $w_{0}$ belongs to $L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$. Hence,
by using (4.12) and by recalling the definition of $\mathcal{F}$, we get that
for every $u\in C^{\infty}_{0}(\omega;\mathbb{R}^{3})$ it holds

$F(u)\leq\int_{\omega}f(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla_{\alpha}u(x_{\alpha}),w_{0}(x_{\alpha}))dx_{\alpha}=\,\int_{\omega}f_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\nabla
u(x_{\alpha}))\,dx_{\alpha}=\mathcal{F}(u).$ (4.13)

In view of the coercivity assumption on $f$ (see (1.17)), we can apply Lemma
4.3 to obtain that the function $f_{0}(x_{\alpha},\cdot,\cdot,\cdot)$ is lower
semicontinuous on $(-1/2,1/2)\times[1,\infty)\times{R}^{3\times 2}$ for
${\mathcal{L}^{2}}$-a.e. $x_{\alpha}\in\omega$; being also the supremum of
continuous function, we obtain that $f_{0}(x_{\alpha},\cdot,\cdot,\cdot)$ is
continuous on $(-1/2,1/2)\times[1,\infty)\times{R}^{3\times 2}$ for
${\mathcal{L}^{2}}$\- a.e. $x_{\alpha}\in\omega$. Moreover, taking into
account (4.7),

$f_{0}(x_{\alpha},0,p_{0}(x_{\alpha}),\xi_{\alpha})\leq
C_{2}(|\xi_{\alpha}|^{p_{0}(x_{\alpha})}+1),\qquad\hbox{for
${\mathcal{L}^{2}}$-a.e.
}x_{\alpha}\in\omega,\forall\,\xi_{\alpha}\in\mathbb{R}^{3\times 2}.$

Therefore $\mathcal{F}$ is strongly continuous in
$W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$. Moreover, by the very definition
(see (4.11)), the functional $F$ is lower semicontinuous with respect to the
strong convergence in $L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$. Hence, by
applying the density of $C_{0}^{\infty}(\omega;\mathbb{R}^{3})$ in
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ (see Remark 2.7, part (1)),
(4.13) implies that

$F(u)\leq\,{\mathcal{F}}(u)\qquad\forall u\in
W_{0}^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3}).$

By the very definitions of $F$ and ${\mathcal{F}}$, this entails (4.10).

###### Proof of Theorem 1.3.

By Theorem 2.9, when $p_{0}\in\mathcal{P}^{log}_{b}(\omega)$, the functionals
$\mathcal{F}$ in (1.18) and $\mathcal{F}_{p^{-}}$ in (4.8) coincide. Hence the
result follows by applying Propositions 4.4 and 4.5, noticing that for every
$\\{u_{\varepsilon_{k}}\\}\subseteq{L^{1}(\Omega;\mathbb{R}^{3})}$ satisfying
$\displaystyle\liminf_{k\rightarrow+\infty}{\mathcal{F}}_{\varepsilon_{k}}(u_{\varepsilon_{k}})<+\infty$,
the growth condition in (1.17), implies (4.2).

######  Corollary 4.6.

Under the same assumptions and with the same notation of Theorem 1.3 it
results that

$\Gamma(L^{p_{0}(\cdot)})\hbox{-}\lim_{k\rightarrow+\infty}\mathcal{F}_{{{\varepsilon_{k}}}}=\mathcal{F}.$

###### Proof.

The result is a direct consequence of Theorem 1.3 together with Proposition
4.10, since the first ensures that $\mathcal{F}$ is a lower bound for the
liminf of ${\mathcal{F}_{{{\varepsilon_{k}}}}}(u_{{{\varepsilon_{k}}}})$ for
every $u_{{{\varepsilon_{k}}}}\rightarrow u$ in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$ while the upper bound is achieved
using the recovery sequence constructed in Proposition 4.10 which is
converging in $L^{p_{0}(\cdot)}(\Omega;{R}^{3})$.

######  Remark 4.7.

  * (i)

A careful inspection of the result of Corollary 4.6 shows that it holds also
if we replace, in the definition of the functional
$\mathcal{F}_{\varepsilon_{k}},$ the null boundary datum on
$\partial_{L}\Omega$ by any other function
$u_{0}\in\cap_{{{\varepsilon_{k}}}>0}W^{1,p_{{{\varepsilon_{k}}}}(\cdot)}(\Omega;\mathbb{R}^{3})\cap
W^{1,p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$ satisfying $u_{0}\equiv
u_{0}(x_{\alpha}).$

  * (ii)

The $\Gamma$-convergence result with respect to the $L^{p_{0}(\cdot)}$-strong
convergence, stated in Corollary 4.6, appears as a natural convergence under
the assumption (4.3) on $\\{p_{{\varepsilon_{k}}}\\}$.

###  4.3 The nonconvex case

This subsection is devoted to the proofs of Theorems 1.4 and 1.5. With this
aim, we recall a relaxation result which extends, in the case of variable
exponents, to integral functionals with trace constraints on the boundary the
more general results proven in [45] in the uncostrained case. For the readers’
convenience we state it in our framework, which deals with functionals with
growth of the type $|\xi|^{p(x)}$, referring to [48, Theorem 5.1] for the
result stated in full generality.

######  Theorem 4.8.

Let $U$ be a bounded open subset of $\mathbb{R}^{N}$ with Lipschitz boundary,
let $p\in\mathcal{P}_{b}(U)$ and consider a Carathéodory integrand
$\varphi:U\times\mathbb{R}^{d}\times\mathbb{R}^{d\times
N}\rightarrow\mathbb{R}$ satisfying the growth condition

$C_{1}+C_{2}|\xi|^{p(x)}\leq\varphi(x,\xi)\leq C_{3}+C_{4}|\xi|^{p(x)},$
(4.14)

for a.e. $x\in U$, every $u\in\mathbb{R}^{d}$ and $\xi\in\mathbb{R}^{d\times
N}$, with $0<C_{2}<C_{4}$. Let
$J:L^{1}(U;\mathbb{R}^{d})\rightarrow[0,+\infty]$ be the functional defined by

$J(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\Omega}\varphi(x,\nabla
u(x))dx&\hbox{ if }\varphi(\cdot,\nabla u(\cdot))\in L^{1}(U),\\\ \\\
+\infty&\hbox{ otherwise in }L^{1}(U;\mathbb{R}^{d}).\end{array}\right.$

Assume that $u_{0}\in W^{1,p^{-}}(U;\mathbb{R}^{d})$ is such that
$J(u_{0})<\infty$. If there exists a sequence $u_{k}\in
W^{1,p^{-}}(U;\mathbb{R}^{d})\cap C^{1}(U_{k};\mathbb{R}^{d})$, where

$U_{k}:=\\{x\in U:{\rm dist}(x,\partial U)>1/k\\},\quad k\in\mathbb{N},$

such that $u_{k}\lfloor_{\partial U}=u_{0}\lfloor_{\partial U}$,
$u_{k}\rightarrow u_{0}$ in $W^{1,p^{-}}(U;\mathbb{R}^{d})$ and
$J(u_{k})\rightarrow J(u_{0})$, as $k\rightarrow+\infty$, then there exists a
sequence $\tilde{u}_{k}\in W^{1,p^{-}}(\Omega;\mathbb{R}^{d})\cap
W^{1,\infty}(U_{k};\mathbb{R}^{d})$ such that $\tilde{u}_{k}\lfloor_{\partial
U}=u_{0}\lfloor_{\partial U}$, $\tilde{u}_{k}\rightharpoonup u_{0}$ in
$W^{1,p^{-}}(U;\mathbb{R}^{d})$ and

$\lim_{k\rightarrow\infty}J(\tilde{u}_{k})=\int_{U}Q\varphi(x,\nabla
u_{0}(x))dx,$

where $Q\varphi$ denotes the quasiconvex envelope of $\varphi(x,\cdot)$.

Now, exploiting the same techniques of Theorem 1.1, we are in position to show
the lower bound inequality.

######  Proposition 4.9.

Let $\\{\mathcal{I}_{\varepsilon_{k}}\\}$ and $\mathcal{I}$ be the functionals
defined by (1.20) and (1.23), respectively. Under the assumptions of Theorem
1.4,

$\mathcal{I}(u)\leq\liminf_{k\rightarrow+\infty}\mathcal{I}_{\varepsilon_{k}}(u_{\varepsilon_{k}}),$

for every $L^{1}(\Omega;\mathbb{R}^{3})\ni u_{\varepsilon_{k}}\rightarrow u$
in $L^{1}(\Omega;\mathbb{R}^{3})$ as ${k\rightarrow+\infty}.$

###### Proof.

Without loss of generality, assume that
$\displaystyle\liminf_{k\rightarrow+\infty}\mathcal{I}_{\varepsilon_{k}}(u_{\varepsilon_{k}})<+\infty$.
By exploiting the assumption $p_{0}\in\mathcal{P}_{b}^{log}(\omega)$,
Proposition 4.1 ensures that $u$ can be identified with a function in
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ and
$u_{\varepsilon_{k}}\rightarrow u$ in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$ and $u\rightharpoonup u$ in
$W^{1,p^{-}}(\Omega;\mathbb{R}^{3}).$ Taking into account that $p_{0}^{-}>1$,
let $0<\delta<\min\\{1/2,p_{0}^{-}-1\\}$ and let $k_{0}\in{{N}}$ be such that

$1<p_{0}(x)-\delta<p_{\varepsilon_{k}}(x)<p_{0}(x)+\delta\quad\hbox{ for
$\mathcal{L}^{3}$ \hbox{-} a.e. }x\in\Omega$

for every $k\geq k_{0}$. By using (3.8) we have that

$2\delta|\Omega|+(2\delta)^{-2\delta}\int_{\Omega}W^{p_{\varepsilon_{k}}(x)}(x_{\alpha},\nabla_{\alpha}u_{{\varepsilon_{k}}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\geq\int_{\Omega}W^{p_{0}(x_{\alpha})-\delta}(x_{\alpha},\nabla_{\alpha}u_{{\varepsilon_{k}}}(x),\tfrac{1}{{\varepsilon_{k}}}\nabla_{3}u_{\varepsilon_{k}}(x))dx.$

Hence, by passing to the liminf as ${k\rightarrow+\infty}$ and by applying the
very definition of $W_{0}$ and $Q(W_{0}^{p_{0}(\cdot)})$, we get that

$\displaystyle
2\delta|\Omega|+(2\delta)^{-2\delta}\liminf_{k\rightarrow+\infty}\int_{\Omega}W^{p_{\varepsilon_{k}}(x)}(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\geq$
(4.15)
$\displaystyle\liminf_{k\rightarrow+\infty}\int_{\Omega}W^{p_{0}(x_{\alpha})-\delta}(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\geq$
$\displaystyle\liminf_{k\rightarrow+\infty}\int_{\Omega}W_{0}^{p_{0}(x_{\alpha})-\delta}(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x))dx\geq$
$\displaystyle\liminf_{k\rightarrow+\infty}\int_{\Omega}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x))dx.$

Then, by Theorem 3.5, we have

$\displaystyle\liminf_{k\rightarrow+\infty}\int_{\Omega}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x))dx$
$\displaystyle\geq\int_{\Omega}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha}dy$
$\displaystyle=\int_{\omega}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha}.$

Hence, by combining (4.15) and the above inequality, as $\delta\rightarrow
0^{+}$, we obtain

$\liminf_{k\rightarrow+\infty}\int_{\Omega}W^{p_{\varepsilon_{k}}(x)}(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\geq\liminf_{\delta\rightarrow
0^{+}}\int_{\omega}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha}.$

Now we claim that,

$|W_{0}(x_{\alpha},\xi_{\alpha})-W_{0}(x_{\alpha},\eta_{\alpha})|\leq
w(x_{\alpha},|\xi_{\alpha}-\eta_{\alpha}|)\hbox{ for every
}\xi_{\alpha},\eta_{\alpha}\in{R}^{3\times 2},\hbox{ for
}\mathcal{L}^{2}\hbox{-a.e. }x_{\alpha}\in\omega.$ (4.16)

Indeed, let $N\subseteq\omega$ be such that (1.21), (1.9) and (1.22) hold for
every $x_{\alpha}\in\omega\setminus N$ and for every $\xi,\eta\in{R}^{3\times
3}.$ Hence, thanks to the coercivity assumption (1.21), for a.e.
$x_{\alpha}\in\omega\setminus N$ and for every $\xi_{\alpha}\in{R}^{3\times
2}$ there exists $\bar{\xi}_{3}\in{R}^{3}$ such that
$W_{0}(x_{\alpha},\xi_{\alpha})=W(x_{\alpha},\xi_{\alpha},\bar{\xi}_{3}).$
Therefore, for every $\eta_{\alpha}\in{R}^{3\times 2}$,

$\displaystyle W_{0}(x_{\alpha},\eta_{\alpha})\leq
W(x_{\alpha},\eta_{\alpha},\bar{\xi}_{3})\leq
w(x_{\alpha},|(\xi_{\alpha},\bar{\xi}_{3})-(\eta_{\alpha},\bar{\xi}_{3}|)+W(x_{\alpha},\xi_{\alpha},\bar{\xi}_{3})=$
$\displaystyle
w(x_{\alpha},|\xi_{\alpha}-\eta_{\alpha},|)+W(x_{\alpha},\xi_{\alpha},\bar{\xi}_{3})\leq
w(x_{\alpha},|\xi_{\alpha}-\eta_{\alpha}|)+W_{0}(x_{\alpha},\xi_{\alpha}).$

By changing the role of $\xi_{\alpha}$ and $\eta_{\alpha}$ we get (4.16),
which implies that the map $(x_{\alpha},q,\xi)\mapsto
W_{0}^{q}(x_{\alpha},\xi)$ is a Carathéodory integrand. In particular, this in
turn yields that for every $q\geq 1$ and $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$ the function $W_{0}^{q}(x_{\alpha},\cdot)$ is Borel
measurable and locally bounded; moreover, it is bounded from below by the
right-hand side of (4.6), hence, its quasiconvex envelope
$Q(W_{0}^{q})(x_{\alpha},\cdot)$ satisfies the assumptions of [18, Theorem
6.9] and can be represented as

$Q(W_{0}^{q})(x_{\alpha},\xi_{\alpha})=\inf\left\\{\frac{1}{\mathcal{L}^{2}(D)}\int_{D}W_{0}^{q}(x_{\alpha},\xi_{\alpha}+\nabla_{\alpha}\psi(y_{\alpha}))dy_{\alpha}:\psi\in
W^{1,\infty}_{0}(D;\mathbb{R}^{3})\right\\}$

with $D$ any bounded open subset of $\mathbb{R}^{2}$.

Hence, by using [18, part 3(b) in Proposition 9.5], the map
$(x_{\alpha},q,\xi)\mapsto Q(W_{0}^{q})(x_{\alpha},\xi)$ is a Carathéodory
integrand too. Therefore, for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$ and
for every $\xi_{\alpha}\in{R}^{3\times 2}$ it holds

$\lim_{\delta\rightarrow
0^{+}}Q(W^{p_{0}(x_{\alpha})-\delta}_{0})(x_{\alpha},\xi_{\alpha})=Q(W^{p_{0}(x_{\alpha})}_{0})(x_{\alpha},\xi_{\alpha})$
(4.17)

and, by using Fatou’s Lemma, we finally get

$\liminf_{\varepsilon_{k}\rightarrow
0^{+}}\int_{\Omega}W^{p_{\varepsilon_{k}}(x)}(x_{\alpha},\nabla_{\alpha}u_{\varepsilon_{k}}(x),\tfrac{1}{\varepsilon_{k}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\geq\int_{\omega}Q(W^{p_{0}(x_{\alpha})}_{0})(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha}.$

Now we are in position to prove the upper bound inequality for Theorem 1.4.

######  Proposition 4.10.

Under the assumptions of Theorem 1.4, it results that

$\Gamma(L^{1})\hbox{-}\limsup_{k\rightarrow+\infty}\mathcal{I}_{{{\varepsilon_{k}}}}\leq\mathcal{I},$

where $\mathcal{I}_{\varepsilon}$ and $\mathcal{I}$ are defined by (1.20) and
(1.23), respectively.

###### Proof.

First of all, we observe that the functional ${\mathcal{I}}_{\varepsilon_{k}}$
coincides with the functional ${\mathcal{F}}_{\varepsilon_{k}}$ in (1.16) when
$f:\omega\times(-1/2,1/2)\times[1,\infty)\times{R}^{3\times 3}\rightarrow{R}$
is given by $f(x_{\alpha},y,q,\xi)=W^{q}(x_{\alpha},\xi)$. Moreover, the
assumptions on $W$ ensure that for $\mathcal{L}^{2}$-a.e.
$x_{\alpha}\in\omega$ and $\xi_{\alpha}\in{R}^{3\times 2}$, it holds

$\inf_{\xi_{3}\in\mathbb{R}^{3}}(W^{p_{0}(x_{\alpha})}(x_{\alpha},\xi_{\alpha},\xi_{3}))=W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\xi_{\alpha})$
(4.18)

and the infimum above is attained.

Hence, by applying Proposition 4.5, we have that

$I:=\Gamma(L^{p_{0}(\cdot)})\hbox{-}\limsup_{k\rightarrow+\infty}{\mathcal{I}}_{\varepsilon_{k}}\leq{\mathcal{I}}_{1}$
(4.19)

where

${\mathcal{I}}_{1}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}{W}^{p_{0}(x_{\alpha})}_{0}(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},&\hbox{
if }u\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3}),\\\ \\\ +\infty&\hbox{
otherwise in }L^{1}(\Omega;\mathbb{R}^{3}).\end{array}\right.$

We introduce

$J(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}{W}^{p_{0}(x_{\alpha})}_{0}(x_{\alpha},\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},&\hbox{
if }{W}^{p_{0}(\cdot)}_{0}(\cdot,\nabla_{\alpha}u(\cdot))\in L^{1}(\omega)\\\
\\\ +\infty&\hbox{ otherwise in
}L^{1}(\Omega;\mathbb{R}^{3}).\end{array}\right.$

Thanks to (1.9) and by applying Lemma 4.3, $W_{0}:\omega\times{R}^{3\times
2}\rightarrow{R}$ is a Carathéodory integrand satisfying

$C_{1}|\xi_{\alpha}|-\frac{1}{C_{1}}\leq\,W_{0}(x_{\alpha},\xi_{\alpha})\leq\,C_{2}(|\xi_{\alpha}|+1)\qquad\hbox{for
$\mathcal{L}^{2}$-a.e. }x_{\alpha}\in\omega,\hbox{ for every
}\xi\in{R}^{3\times 3}.$ (4.20)

This, clearly, implies that also the map $(x_{\alpha},\xi_{\alpha})\rightarrow
W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\xi_{\alpha})$, is a Carathéodory
integrand satisfying the growth condition (4.14).

In order to show that $I\leq\mathcal{I}$, without loss of generality, we can
consider the case when $\mathcal{I}(u)<+\infty$. Then $u\in
W_{0}^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$ and, thanks to (4.14),
$J(u)<+\infty.$ Moreover, being $p_{0}\in\mathcal{P}^{\log}_{b}(\omega)$, Part
(1) in Remark 2.7 entails that $C_{0}^{\infty}(\omega;\mathbb{R}^{3})$ is
dense in $W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$. Since $J$ is
continuous with respect to the convergence in
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$, we get that $J$ satisfies the
assumptions of Theorem 4.8 on $U=\omega$. Hence, we can infer the existence of
a sequence $\tilde{u}_{{\varepsilon_{k}}}\in
W_{0}^{1,p_{0}^{-}}(\omega;\mathbb{R}^{3})\cap
W^{1,\infty}(\omega_{{\varepsilon_{k}}};\mathbb{R}^{3})$, such that
$\tilde{u}_{{\varepsilon_{k}}}\rightharpoonup u$ in
$W^{1,p_{0}^{-}}(\omega;\mathbb{R}^{3})$ and

$\lim_{k\rightarrow+\infty}\int_{\omega}W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\nabla\tilde{u}_{{\varepsilon_{k}}}(x_{\alpha}))dx_{\alpha}=\int_{\omega}Q(W_{0}^{p_{0}(x_{\alpha})})(x_{\alpha},\nabla
u(x_{\alpha}))dx_{\alpha}=\mathcal{I}(u)<+\infty.$ (4.21)

In particular, thanks to the coercivity of $W_{0}^{p_{0}(\cdot)}$, (4.21)
implies that there exists $k_{0}\in{{N}}$ such that
$\nabla\tilde{u}_{{\varepsilon_{k}}}\in
L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3\times 2})$ for every $k\geq k_{0}$. By
exploiting Proposition 2.8, this implies that $u_{{\varepsilon_{k}}}\in
W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$. Finally, by Theorem 2.9, we obtain
that $u_{{\varepsilon_{k}}}\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$
for $k\geq k_{0}$. Hence
$\mathcal{I}_{1}(\tilde{u}_{{\varepsilon_{k}}})=J(\tilde{u}_{{\varepsilon_{k}}})$
for every $k\geq k_{0}$. Moreover, by using (4.20) and the Poincaré inequality
in $W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ (see (2.6)), it is easy to
obtain that $\tilde{u}_{{\varepsilon_{k}}}\rightharpoonup u$ in
$W^{1,p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$. Therefore, by Proposition 2.8,
$\tilde{u}_{{\varepsilon_{k}}}\rightarrow u$ in
$L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$ and then
$\tilde{u}_{{\varepsilon_{k}}}\rightarrow u$ in
$L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$. Since $I$ is lower semicontinuous
with respect to such a convergence, by (4.19) and (4.21), we get that

$I(u)\leq\liminf_{k\rightarrow+\infty}I(\tilde{u}_{{\varepsilon_{k}}})\leq\lim_{k\rightarrow+\infty}\mathcal{I}_{1}(\tilde{u}_{{\varepsilon_{k}}})=\lim_{k\rightarrow+\infty}J(\tilde{u}_{{\varepsilon_{k}}})=\int_{\omega}Q(W_{0}^{p_{0}(x_{\alpha})})(x_{\alpha},\nabla\tilde{u}(x_{\alpha}))dx_{\alpha}=\mathcal{I}(u).$

###### Proof of Theorem 1.4.

It follows from Propositions 4.9 and 4.10.

Finally we prove Theorem 1.5, splitting the proof in a double inequality. We
underline that the adopted arguments are very similar to those exploited in
the proof of Theorem 1.4, relying for the lower bound on techniques stemmed
from Theorem 1.2. Indeed we provide for the reader’s convenience only the
proof of the main differences. Arguing as in Theorem 3.7, it is possible to
show that the result generalizes to the case when $W$ depends also on
$x_{\alpha}$; however we prefer to focus just on the model behaviour of an
energy density of product-type between the spatial variable and a gradient
dependence.

######  Proposition 4.11.

Let $\\{\mathcal{J}_{\varepsilon_{k}}\\}$ and $\mathcal{J}$ be the functionals
defined by in (1.24) and (1.28), respectively. Under the assumptions of
Theorem 1.5,

$\mathcal{J}(u)\leq\liminf_{k\rightarrow+\infty}\mathcal{J}_{\varepsilon_{k}}(u_{\varepsilon_{k}}),$

for every $L^{1}(\Omega;\mathbb{R}^{3})\ni u_{\varepsilon_{k}}\rightarrow u$
in $L^{1}(\Omega;\mathbb{R}^{3})$ as $\varepsilon_{k}\rightarrow 0^{+}.$

Proof. Without loss of generality, we can assume that
$\displaystyle\liminf_{k\rightarrow+\infty}\mathcal{J}_{\varepsilon_{k}}(u_{\varepsilon_{k}})<+\infty$.
Thus, by (1.26)

$\liminf_{k\rightarrow+\infty}\int_{\Omega}W^{p_{{{\varepsilon_{k}}}}(x)}(\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{\varepsilon_{k}}(x))dx\leq\frac{1}{a^{-}}\liminf_{{k\rightarrow+\infty}}\mathcal{J}_{{\varepsilon_{k}}}(u_{{\varepsilon_{k}}})<+\infty.$

In turn, thanks to (1.25), (4.2) holds. Hence, in view of our assumptions on
$p_{\varepsilon_{k}}$ and $p_{0}$, by (1), (3) and (4) in Proposition 4.1, we
have that, up to a not relabelled subsequence
$u_{\varepsilon_{k}}\rightharpoonup u$ in $W^{1,p^{-}}(\Omega;\mathbb{R}^{3})$
and strongly in $L^{p_{0}(\cdot)}(\Omega;\mathbb{R}^{3})$ with $u\in
W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$.

Furthermore, $a_{\varepsilon_{k}}$, $p_{\varepsilon_{k}}$and
$u_{\varepsilon_{k}}$ satisfy the assumptions of Theorem 1.2, hence we can
mimick the arguments therein. More precisely, by reasoning as in the first
part of Theorem 1.2 (see (3.10)) we have that

$\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{{{\varepsilon_{k}}}}(x)W^{p_{{{\varepsilon_{k}}}}(x)}(\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{{{\varepsilon_{k}}}}(x))dx\geq\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{0}(x_{\alpha})W_{0}^{p_{{{\varepsilon_{k}}}}(x)}(\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x))dx,$

where $W_{0}$ is the function appearing in (1.29). Moreover, by applying the
same argument of Lemma 3.6 and Theorem 3.5, for every
$0<\delta<\min\\{1/2,p_{0}^{-}-1\\}$ it holds

$\displaystyle(2\delta)^{-2\delta}\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{0}(x_{\alpha})W_{0}^{p_{\varepsilon_{k}}(x)}(\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x))dx+2\delta||a||_{L^{\infty}(\Omega)}{\mathcal{L}}^{3}(\Omega)$
$\displaystyle\geq\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{0}(x_{\alpha})W_{0}^{p_{0}(x_{\alpha})-\delta}(\nabla_{\alpha}u_{\varepsilon_{k}}(x))dx$
$\displaystyle\geq\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{0}(x_{\alpha})Q(W_{0}^{p_{0}(x)-\delta})(\nabla_{\alpha}u_{\varepsilon_{k}}(x))dx$
(4.22)
$\displaystyle\geq\int_{\Omega}a(x_{\alpha})Q(W_{0}^{p_{0}(x_{\alpha})-\delta})(\nabla_{\alpha}u(x_{\alpha}))dx.$

Now, arguing as in Proposition 4.9 (see (4.17)) i.e. exploiting the fact that
quasiconvex envelope $Q(W_{0}^{q})$ satisfies the assumptions of [18, Theorem
6.9] , it follows that

$\lim_{\delta\rightarrow
0}Q(W_{0}^{p_{0}(x_{\alpha})-\delta})(\xi_{\alpha})=Q(W_{0}^{p_{0}(x_{\alpha})})(\xi_{\alpha})$
(4.23)

Hence, combining (4.22) and (4.23) and sending $\delta\rightarrow 0$, Fatou’s
lemma gives

$\displaystyle\liminf_{k\rightarrow+\infty}\int_{\Omega}a_{{{\varepsilon_{k}}}}(x)W^{p_{{{\varepsilon_{k}}}}(x)}(\nabla_{\alpha}u_{{{\varepsilon_{k}}}}(x),\tfrac{1}{{{\varepsilon_{k}}}}\nabla_{3}u_{{{\varepsilon_{k}}}}(x))dx\geq\int_{\omega}a_{0}(x_{\alpha})Q(W_{0}^{p_{0}(x_{\alpha})})(\nabla_{\alpha}u(x_{\alpha}))dx_{\alpha},$

for every $u\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$, which in view
of the definition of $\mathcal{J}$ in (1.28) concludes the proof.

######  Proposition 4.12.

Under the assumptions of Theorem 1.5, it results that

$\Gamma(L^{1})\hbox{-}\limsup_{k\rightarrow+\infty}\mathcal{J}_{{{\varepsilon_{k}}}}\leq\mathcal{J},$

where $\\{\mathcal{J}_{\varepsilon_{k}}\\}$ and $\mathcal{J}$ are the
functionals defined by (1.24) and (1.28), respectively.

###### Proof.

Without loss of generality, we can restrict to consider the case when
$\mathcal{J}(u)<+\infty$.

Taking into account (1.27), up to some minor modifications, we can reason as
in the first part of Proposition 4.5 and obtain that

$I(u)\leq\,\inf_{w\in
L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})}\int_{\omega}a_{0}(x_{\alpha})W^{p_{0}(x_{\alpha})}(\nabla
u(x_{\alpha}),w(x_{\alpha}))\,dx_{\alpha}\qquad\forall
u\in\mathcal{C}^{\infty}_{0}(\omega;\mathbb{R}^{3})$ (4.24)

where

$I:=\Gamma(L^{p_{0}(\cdot)})\hbox{-}{\limsup_{k\rightarrow+\infty}}\mathcal{J}_{{{\varepsilon_{k}}}}.$

Following the same argument in [44, page 558], there exists a measurable
function $w_{0}=w_{0}(x_{\alpha})$ such that

$W_{0}(\nabla u(x_{\alpha}))=\inf_{\xi_{3}\in\mathbb{R}^{3}}[W(\nabla
u(x_{\alpha}),\xi_{3})]=W(\nabla u(x_{\alpha}),w_{0}(x_{\alpha}))\qquad\mbox{
for }\mathcal{L}^{2}-\hbox{ a.e. }x_{\alpha}\in\omega.$

In particular, by using (4.18), we deduce that

$\displaystyle
a_{0}(x_{\alpha})W_{0}^{p_{0}(x_{\alpha})}(\nabla_{\alpha}u(x_{\alpha}))=a_{0}(x_{\alpha})W^{p_{0}(x_{\alpha})}(\nabla_{\alpha}u(x_{\alpha}),w_{0}(x_{\alpha}))=\inf_{\xi_{3}\in\mathbb{R}^{3}}[a_{0}(x_{\alpha})W^{p_{0}(x_{\alpha})}(\nabla_{\alpha}u(x_{\alpha}),\xi_{3})].$

Moreover, thanks to (1.25), such a function $w_{0}$ belongs to
$L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$. Hence, by using (4.24), we get that

$I(u)\leq\mathcal{J}_{1}(u)\qquad\forall u\in
C^{\infty}_{0}(\omega;\mathbb{R}^{3})$

where $\mathcal{J}_{1}:L^{1}(\omega;\mathbb{R}^{3})\rightarrow[0,+\infty]$ is
the functional defined by

${\mathcal{J}}_{1}(u):=\left\\{\begin{array}[]{ll}\displaystyle\int_{\omega}a_{0}(x_{\alpha})W^{p_{0}(x_{\alpha})}_{0}(\nabla_{\alpha}u(x_{\alpha}))\,dx_{\alpha}&\hbox{
if }u\in W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})\\\ \\\ +\infty&\hbox{
otherwise in }L^{1}(\omega;\mathbb{R}^{3}).\end{array}\right.$

Since $a_{0}\in L^{\infty}(\omega)$ and thanks to the growth condition

$W_{0}(\xi_{\alpha})\leq C_{2}(|\xi_{\alpha}|+1)\hbox{ for every
}\xi_{\alpha}\in\mathbb{R}^{3\times 2},$

inherited by (1.12), ${\mathcal{J}}_{1}$ results strongly continuous on
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$.
By using the density of $C_{0}^{\infty}(\omega;\mathbb{R}^{3})$ in
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ (see Part (1) in Remark 2.7)
and the lower semicontinuity of $I$ with respect to the strong convergence in
$W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ we obtain that

$I(u)\leq\,\mathcal{J}_{1}(u)\quad\forall u\in L^{1}(\omega;\mathbb{R}^{3}).$
(4.25)

Now, by reasoning as in the proof of Proposition (4.10), we can show that that
there exists a sequence $\tilde{u}_{{\varepsilon_{k}}}\in
W^{1,p_{0}(\cdot)}_{0}(\omega;\mathbb{R}^{3})$ such that
$\tilde{u}_{{\varepsilon_{k}}}\rightharpoonup u$ in
$L^{p_{0}(\cdot)}(\omega;\mathbb{R}^{3})$ and

$\lim_{k\rightarrow+\infty}\mathcal{J}_{1}(\tilde{u}_{{\varepsilon_{k}}})=\lim_{k\rightarrow+\infty}\int_{\omega}a_{0}(x_{\alpha})W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\nabla\tilde{u}_{{\varepsilon_{k}}}(x_{\alpha}))dx_{\alpha}=\int_{\omega}Q(a_{0}(x_{\alpha})W_{0}^{p_{0}(x_{\alpha})})(x_{\alpha},\nabla
u(x_{\alpha}))dx_{\alpha}.$ (4.26)

Since $I$ is lower semicontinuous with respect to the
$L^{p_{0}(\cdot)}$-convergence and
$Q\left(a_{0}(x_{\alpha})W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\cdot)\right)=a_{0}(x_{\alpha})Q\left(W_{0}^{p_{0}(x_{\alpha})}(x_{\alpha},\cdot)\right),$
for $\mathcal{L}^{2}$-a.e. $x_{\alpha}\in\omega$, (4.26) combined with (4.25)
implies

$I(u)\leq\liminf_{k\rightarrow+\infty}I(\tilde{u}_{{\varepsilon_{k}}})\leq\lim_{k\rightarrow+\infty}\mathcal{J}_{1}(\tilde{u}_{{\varepsilon_{k}}})=\lim_{k\rightarrow+\infty}J(\tilde{u}_{{\varepsilon_{k}}})=\int_{\omega}Q(W_{0}^{p_{0}(x_{\alpha})})(x_{\alpha},\nabla\tilde{u}(x_{\alpha}))dx_{\alpha}=\mathcal{I}(u).$

and the proof is over.

###### Proof of Theorem 1.5.

It follows from Propositions 4.11 and 4.12.

Analogously to Corollary 4.6, the following results hold. The proof is
omitted, being very similar to the one presented in the convex case.

######  Corollary 4.13.

Let $\mathcal{I}_{\varepsilon_{k}},\mathcal{I},\mathcal{J}_{\varepsilon_{k}}$
and $\mathcal{J}$ be the functionals in (1.20), (1.23),(1.24), and (1.28),
respectively. Under the same assumptions and with the same notation of
Theorems 1.4 and 1.5 it results that

$\Gamma(L^{p_{0}(\cdot)})\hbox{-}\lim_{k\rightarrow+\infty}\mathcal{I}_{\varepsilon_{k}}=\mathcal{I},$

and

$\Gamma(L^{p_{0}(\cdot)})\hbox{-}\lim_{k\rightarrow+\infty}\mathcal{J}_{\varepsilon_{k}}=\mathcal{J}.$

Acknowledgments. The authors are members of GNAMPA-INdAM, whose support is
gratefully acknowledged, through the projects “Prospettive nella scienza dei
materiali: modelli variazionali, analisi asintotica e omogeneizzazione”
(2023), “Analisi variazionale di modelli non-locali nelle scienze applicate”
(2020) and “Problemi ellittici e parabolici con singolarità: esistenza,
regolarità ed omogeneizzazione” (2020). The work of ME has been partially
supported by PRIN 2020 “Mathematics for industry 4.0 (Math4I4)” . FP thanks
Dipartimento di Scienze di Base ed Applicate per l’Ingegneria at Sapienza -
Università di Roma for its hospitality. EZ is grateful to Dipartimento di
Scienze Agrarie, Alimentari e Agro-ambientali at Università di Pisa, whose
hospitality is gratefully acknowledged. FP and EZ are indebted with
Dipartimento di Scienze Fisiche Informatiche e Matematiche at University of
Modena and Reggio Emilia, whose support is kindly acknowledged. The work of EZ
is also supported by Sapienza - University of Rome through the projects
Progetti di ricerca medi, (2021), coordinator S. Carillo e Progetti di ricerca
piccoli, (2022), coordinator E. Zappale.

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