# The $p-$Laplace “Signature” for Quasilinear Inverse Problems with Large
Boundary Data

Antonio Corbo Esposito1, Luisa Faella1, Gianpaolo Piscitelli2, Ravi Prakash3,
Antonello Tamburrino1,4

###### Abstract.

AbstractAbstract 111
1Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”,
Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n.
43, 03043 Cassino (FR), Italy.
2Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
Studi di Napoli Federico II, Via Cinthia n. 26, Complesso Universitario Monte
Sant’Angelo, 81026 Napoli, Italy.
3Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Concepción, Avenida Esteban Iturra s/n, Bairro Universitario,
Casilla 160 C, Concepción, Chile.
4Department of Electrical and Computer Engineering, Michigan State University,
East Lansing, MI-48824, USA.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is inspired by an imaging problem encountered in the framework of
Electrical Resistance Tomography involving two different materials, one or
both of which are nonlinear. Tomography with nonlinear materials is in the
early stages of developments, although breakthroughs are expected in the not-
too-distant future.

We consider nonlinear constitutive relationships which, at a given point in
the space, present a behaviour for large arguments that is described by
monomials of order $p$ and $q$.

The original contribution this work makes is that the nonlinear problem can be
approximated by a weighted $p-$Laplace problem. From the perspective of
tomography, this is a significant result because it highlights the central
role played by the $p-$Laplacian in inverse problems with nonlinear materials.
Moreover, when $p=2$, this provides a powerful bridge to bring all the imaging
methods and algorithms developed for linear materials into the arena of
problems with nonlinear materials.

The main result of this work is that for “large” Dirichlet data in the
presence of two materials of different order (i) one material can be replaced
by either a perfect electric conductor or a perfect electric insulator and
(ii) the other material can be replaced by a material giving rise to a
weighted $p-$Laplace problem.

MSC 2020: 35J62, 78A46, 35R30.

Key words and phrases. Elliptic PDE, Quasilinear PDE, Nonlinear problems,
Linear approximation, Asymptotic behaviour, Imaging, Electrical Resistance
Tomography, Inverse problem.

AbstractAbstract 111
1Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”,
Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n.
43, 03043 Cassino (FR), Italy.
2Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
Studi di Napoli Federico II, Via Cinthia n. 26, Complesso Universitario Monte
Sant’Angelo, 81026 Napoli, Italy.
3Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Concepción, Avenida Esteban Iturra s/n, Bairro Universitario,
Casilla 160 C, Concepción, Chile.
4Department of Electrical and Computer Engineering, Michigan State University,
East Lansing, MI-48824, USA.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

This paper treats a nonlinear imaging problem in Electrical Resistance
Tomography (ERT). The aim is to retrieve the nonlinear electrical conductivity
$\sigma$, from boundary data in stationary conditions (steady currents):

$\begin{cases}\operatorname{div}\Big{(}\sigma(x,|\nabla u(x)|)\nabla
u(x)\Big{)}=0\ \text{in }\Omega\vspace{0.2cm}\\\
u(x)=f(x)\qquad\qquad\qquad\quad\ \text{on }\partial\Omega,\end{cases}$ (1.1)

where $f$ is the applied boundary potential, $u$ is the electric scalar
potential and $\Omega\subset\mathbb{R}^{n}$, $n\geq 2,$ is an open bounded
domain with Lipschitz boundary which represents the region occupied by the
conducting material. Retrieving the nonlinear electrical conductivity $\sigma$
from boundary data, i.e. from the
$\Lambda_{\sigma}:f\mapsto\sigma\partial_{n}u$ operator, the so-called
Dirichlet-to-Neumann (DtN) operator, is the nonlinear variant of the Calderón
problem [1, 2]. In the literature, there are very few contributions on the
subject of imaging in the presence of nonlinear materials. As quoted in [3], “
… the mathematical analysis for inverse problems governed by nonlinear
Maxwell’s equations is still in the early stages of development.”. In this
framework, we mention the work made for $p-$Laplace type nonlinearities [4, 5,
6, 7, 8, 9, 10], the work by Sun [11, 12] for weak nonlinearities, the work by
Cârstea and Kar [13] which treated a nonlinear problem (linear plus a
nonlinear term) and the work by Corbo Esposito et al. [14]. The latter treat a
general nonlinearity within the framework of the Monotonicity Principle
Method. It can be expected that as new methods and algorithms become
available, the demand for nondestructive evaluation and imaging of nonlinear
materials will eventually and significantly rise. From the physical
standpoint, in steady current operations, the electric field is given by the
electrical scalar potential as ${\bf E}(x)=-\nabla u(x)$, and the electrical
current density ${\bf J}(x)$ depends on the electric field as follows:

${\bf J}(x)=\sigma(x,|{\bf E}(x)|){\bf E}(x)\quad\forall x\in\Omega.$ (1.2)

Equation (1.2) represents a constitutive relationship which is nonlinear,
local, isotropic and memoryless. From a general perspective, nonlinear
electrical conductivities can be found in many materials like semiconducting
and ceramic materials [15, 16, 17], superconductors [18, 19] and in biological
tissues [20, 21]. Other than steady currents, problem (1.1) is common to other
physical settings. Remaining in the framework of electromagnetism, both
nonlinear electrostatic phenomena (see [22] and references therein, see [23])
and nonlinear magnetostatic2221In magnetostatics, it is possible to introduce
a magnetic scalar potential for treating simply connected and source free
regions.1 phenomena (see [24]) can be modelled as in (1.1). Specifically, we
address ERT with a nonlinear electrical conductivity when the boundary data is
“large”, in the sense that the boundary data is $\lambda f(x)$ where constant
$\lambda$ approaches infinity. This contribution is companion of [25], where
ERT with nonlinear conductivities was treated in the “small” boundary data
limit ($\lambda\to 0$) and it completes the “philosophy” of limiting problems.
Indeed, when the electrical conductivity is unknown, no other limiting cases
can be conceived, other than the small and large boundary data cases. During
the analysis of this class of problems, we recognize the limiting behaviour of
the normalized solution defined as $v^{\lambda}=u^{\lambda}/\lambda$, being
$u^{\lambda}$ the solution of (1.1) when the boundary data is $\lambda f$. To
be more specific, we assume that the electrical conductivity $\sigma$ has a
different behaviour for $E\to+\infty$ in region $A\subset\subset\Omega$ and
$B=\Omega\backslash A$, i.e.

$\displaystyle\sigma_{B}(x,E)\sim\beta(x)E^{p-2}\quad\text{for a.e.}\ x\in B,$
(1.3) $\displaystyle\sigma_{A}(x,E)\sim\alpha(x)E^{q-2}\quad\text{for a.e.}\
x\in A,$ (1.4)

where $\sigma_{B}$ and $\sigma_{A}$ are the restriction of $\sigma$ to $B$ and
$A$, respectively, $E=\left|\mathbf{E}\right|=\left|\nabla u\right|$,
$\alpha(x)$ and $\beta(x)$ are proper functions and $p$ and $q$ are the
asymptotic growth exponents (see Figure 1).

Figure 1. The electrical conductivity has different asymptotic growth
exponents $q$ and $p$ in regions $A$ and $B$, respectively. Region $B$
surrounds $A$ and $\partial\Omega\subset\partial B$.

In this contribution, we prove that the normalized solution $v^{\lambda}$
approaches a limit for $\lambda\to+\infty$ which is the solution of a weighted
$p-$Laplace problem in region $B$. Moreover, in solving this weighted
$p-$Laplace problem in region $B$, the boundary condition at the interface
between $B$ and $A$ is either that of a Dirichlet boundary condition (constant
scalar potential for $q>p$) or that of a Neumann boundary condition (vanishing
normal component of the scalar potential for $q<p$). From the physical
standpoint, region $A$ is seen as either a perfect electric conductor ($q>p$)
or a perfect electric insulator ($q<p$). Finally, the case $p=q$ (that is the
case when $A=\emptyset$) has been treated in [14, 26]. It is worth noting that
understanding the behaviour of the solution in region $B$ is of paramount
importance because the imaging system collects data, i.e. it measures a
discrete approximation of $\Lambda_{\sigma}$, onto $\partial\Omega$, which is
the outer boundary of $B$. In other terms, the behaviour in $B$ determines the
measured data. These results make it possible to approximate a nonlinear
problem with a weighted $p-$Laplace one in region $B$, together with a proper
boundary condition on $\partial A$, corresponding to either a perfect electric
conductor or a perfect electric insulator in $A$ (see Section 3.3 for the
underlying assumptions). They are quite interesting from both a mathematical
and an engineering point of view because they reduce the original quasilinear
problem in $\Omega$ to the canonical problem of the weighted $p-$Laplace
equation only in region $B$. From the perspective of inverse problems, these
results allow to apply to (1.1) the results developed for the nonlinear
$p-$Laplace variant of the Calderón problem, initially posed by Salo and Zhong
[4], and studied in [5, 6, 7, 8, 9, 10]. Moreover, the case $p=2$ is of
paramount importance because the problem in region $B$ reduces to a linear one
and, therefore, we have a powerful bridge to apply all the imaging methods and
algorithms developed for linear materials to nonlinear ones.

Figure 2. Description of two possible applications. Left: inverse obstacle
problem where the interface ($\partial A$) between two phases is unknown. $A$
and $B$ are the regions occupied by the inner material and the outer material,
respectively. Right: nondestructive testing where regions A and B are known,
while the position and shape of region $C$ (a crack) is unknown. Materials in
regions $A$ and $B$ are known, as well.

For instance, a classical class for solving an inverse problem in the presence
of linear materials is that of iterative methods, which is based on the
(iterative) minimization of a proper cost function [27, 28]. An overview of
iterative methods can be found in several specialized textbooks [29, 30, 31,
32]. Other than the popular Gauss-Newton method and its variants (see [33] for
a review), we mention some relevant iterative approaches applied to inverse
problems such as the Quadratic Born approximation [34], Bayesian approaches
[35], the Total Variation regularization [36, 37], the Levenberg-Marquardt
method for nonlinear inverse problems [38], the Level Set method [39, 40], the
Topological Derivative method [41, 42, 43] and the Communication Theory
approach [44]. Inverse problems in the presence of hollow domains, as it is
the case for $B$, has been deeply investigated. Recent results on this subject
includes approaches based on a parameterization of the obstacle [45], on the
level sets method [46], on the concept of shape derivative combined with that
of level sets [47], on the topological derivative method [48], on the quasi-
reversibility method [49], on a shape optimization method [50], and on an
integral equation method [51, 52]. Noniterative methods are an excellent
alternative to iterative ones, because they call for the computation of a
proper function of the space, the indicator function, giving the shape of the
interface between two different materials. The computation of the indicator
function is much less expensive than the computation required by an iterative
method, thus making noniterative methods suitable for real-time operations.
Only a handful of noniterative methods are currently available. These include
the Linear Sampling Method (LSM) by Colton and Kirsch [53], which evolved into
the Factorization Method (FM) proposed by Kirsch [54]. Ikehata proposed the
Enclosure Method (EM) [55, 56] and Devaney applied MUSIC (MUltiple SIgnal
Classification), a well-known algorithm in signal processing, as an imaging
method [57]. Finally, Tamburrino and Rubinacci proposed the Monotonicity
Principle Method (MPM) [58]. Assumptions made in this work are presented in
Section 3.3. They refer to a quite standard framework, apart from (A4) and
(A4’) which correspond to (1.3) and (1.4). These two latter specific
assumptions prescribe the pointwise convergence of the constitutive
relationship for large electric fields $E$. Moreover, they are sharp, as shown
by means of the counterexamples of Section 6. The paper is organized as
follows: in Section 2 we present the ideas underpinning the work; in Section 3
we set the notations and the problem, together with the required assumptions;
in Section 4 we study the fundamental inequality for large Dirichlet data; in
Section 5 we discuss the limiting case for large Dirichlet data; in Section 6
we provide the counterexamples proving that the specific assumptions are sharp
and, finally, in Section 7 we provide some conclusions.

##  2\. Underlying ideas and expected results

In this section we present the main ideas underpinning this work. The key is
the “educated guess” that when the boundary data is “large”, we expect the
electric field $\mathbf{E}=-\nabla{u}$ to be large a.e. in $\Omega$ and,
therefore, its behaviour is expected to be governed by the asymptotic
behaviour of the constitutive relationship $\sigma=\sigma\left(x,E\right)$ in
(1.1). Let $A\subset\subset\Omega$ and $B:=\Omega\setminus\overline{A}$, we
assume that there exist two constants $q$ and $p$, and two functions $\beta$
and $\alpha$ which capture the behaviour of $\sigma$, as $E\to+\infty$, in $B$
and $A$, according to (1.3) and (1.4), respectively. Analysis of nonlinear
problems is fascinating because of the wide variety of different cases. Some
representative cases are shown in Figure 3.

Figure 3. Relationship between the electrical conductivities for regions $A$
and $B$, when considering their asymptotic approximations:
$\sigma_{B}(\cdot,E)\sim\beta(\cdot)E^{p-2}$ (solid line) and
$\sigma_{A}(\cdot,E)\sim\alpha(\cdot)E^{q-2}$ (dashed line). Similar
configurations are obtained when the order relation between $p$ and $q$ is
reversed. Only the behaviour for large $E$ is shown.

When, for instance, $q<p$ it can be reasonably expected that either (i) region
$A$ is a perfect electric insulator or (ii) region $B$ is a perfect electric
conductor, because $\sigma_{B}$ would be dominant if compared to $\sigma_{A}$.
When $\partial\Omega$ is contained in $\partial B$, that is
$A\subset\subset\Omega$, the ambiguity between (i) and (ii) can be resolved by
looking at the boundary data. Specifically, if the boundary data is
nonconstant, then region $B$ cannot be assimilated to a PEC because the
potential $u$ would be constant and, therefore, noncompatible with the
boundary data. Region $A$ can therefore reliably be assimilated to a PEI.
Moreover, the limiting problem where the conductor in region $A$ is replaced
by a PEI can reliably be modelled by a weighted $p-$Laplace problem in region
$B$, with a boundary condition on $\partial A$ given by a vanishing normal
component of $\mathbf{J}$. In other words, $u\sim u_{p}$ in $B$, where $u_{p}$
is the solution of the weighted $p-$Laplace problem arising from the
electrical conductivity $\beta(x)E^{p-2}$ in $B$. The latter observation is
also inspiring because it properly defines the concept of “large” boundary
data and the limiting problem. Specifically, it is well known that the
operator mapping the boundary data $f$ into the solution of a weighted
$p-$Laplace problem is a homogeneous operator of degree 1, i.e. the solution
corresponding to $\lambda f(x)$ is equal to $\lambda u_{p}(x)$, where $u_{p}$
is the solution corresponding to the boundary data $f$. Thus, the term
“problem for large boundary data” means (1.1) where the boundary data is
$\lambda f$ and $\lambda\to+\infty$. Moreover, this suggests the need to study
the convergent properties of the normalized solution $v^{\lambda}$, defined as
the ratio $u^{\lambda}/\lambda$, where $u^{\lambda}$ is the solution of (1.1)
corresponding to the Dirichlet data $\lambda f(x)$. Indeed, if $u^{\lambda}$
can be approximated by the solution of the weighted $p-$Laplace problem, then
the normalized solution $v^{\lambda}$ converges in $B$, i.e. it is expected to
be constant w.r.t. $\lambda$, as $\lambda$ approaches $+\infty$. We term this
limit as $v^{\infty}$ and we expect it to be equal to $u_{p}$, i.e. the
solution of the weighted $p-$Laplace problem with boundary data $f$. From the
formal point of view, when $q<p$, we will indeed prove that the normalized
solution $v^{\lambda}$ weakly converges to $v_{\Omega}$, which is termed
$v_{B}$ in $B$ and $v_{A}$ in $A$. The function $v_{B}\in W^{1,p}(B)$ is the
solution of a weighted $p-$Laplace problem in region $B$:

$\begin{cases}\operatorname{div}\Big{(}\beta(x)|\nabla{v_{B}}(x)|^{p-2}\nabla
v_{B}(x)\Big{)}=0&\text{in }B\vspace{0.2cm}\\\
\beta(x)|\nabla{v_{B}}(x)|^{p-2}\partial_{\nu}{v_{B}}(x)=0&\text{on }\partial
A\vspace{0.2cm}\\\ {v_{B}}(x)=f(x)&\text{on }\partial\Omega;\vspace{0.2cm}\\\
\end{cases}$ (2.1)

the function $v_{A}\in W^{1,q}(A)$ is the solution of a weighted $q-$Laplace
problem in region $A$:

$\begin{cases}\operatorname{div}\Big{(}\alpha(x)|\nabla{v_{A}}(x)|^{q-2}\nabla{v_{A}}(x)\Big{)}=0&\text{in
}A\vspace{0.2cm}\\\ {v_{A}}(x)={v_{B}}(x)&\text{on }\partial A.\end{cases}$
(2.2)

From the physical standpoint, problem (2.1) corresponds to stationary currents
where the electrical conductivity in $B$ is $\sigma(x,E)=\beta(x)E^{p-2}$, and
region $A$ is replaced by a perfectly insulating material (PEI). Problem (2.2)
corresponds to stationary currents in region $A$, with boundary data given by
the solution of (2.1), and electrical conductivity equal to
$\sigma(x,E)=\alpha(x)E^{q-2}$. It is worth noting that any solutions of
problems (2.1) and (2.2) satisfy the minimum problems (5.1) and (5.2),
respectively, described in Section 5. Similarly, when $p<q$, it turns out that
$v^{\lambda}$ converges to $w\in W^{1,p}(\Omega)$ for $\lambda\to+\infty$,
where $w$ is constant in each connected component of $A$ and is the solution
of:

$\begin{cases}\operatorname{div}\Big{(}\beta(x)|\nabla w(x)|^{p-2}\nabla
w(x)\Big{)}=0\ &\text{in }B\vspace{0.2cm}\\\ {|\nabla w|=0}&\text{a.e. in
}A\\\ \int_{\partial A}\sigma(x,|\nabla w(x)|)\partial_{\nu}w(x)dS=0\\\
w(x)=f(x)\qquad\qquad\qquad\qquad\quad&\text{on
}\partial\Omega\vspace{0.2cm}\end{cases}$ (2.3)

In this case, from the physical standpoint, region $A$ can be replaced by a
Perfect Electric Conductor (PEC). It is worth noting that the solution of
problem (2.3) satisfies the minimum problem (5.3), described in Section 5.
Finally, we highlight that in both cases ($q<p$ or $q>p$), the limiting
problems in regions $B$ and $A$ can be solved in cascade (see Figure 4).
Specifically, the problem for region $B$ can be solved first, then the problem
for region $A$ can be solved from the knowledge of the trace on $\partial A$
of the solution evaluated in $B$.

Figure 4. The problem in $B$ can be solved independently of the problem in
$A$. The problem in $A$ can be solved through the knowledge of the trace on
$\partial A$ of the solution in $B$.

##  3\. Framework of the Problem

###  3.1. Notations

Throughout this paper, $\Omega$ denotes the region occupied by the conducting
materials. We assume that $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, is a
bounded domain (i.e. an open and connected set) with Lipschitz boundary and
$A\subset\subset\Omega$ is an open bounded set with Lipschitz boundary and a
finite number of connected components, such that
$B:=\Omega\setminus\overline{A}$ is still a domain. Hereafter we consider
$1<p,q<+\infty$, $p\neq q$. Region $B$ is occupied by a conducting material
with an asymptotic $p-$growth∗333∗ The concept of asymptotic $p-$growth and
asymptotic $q-$growth is formalized in assumption (A4) and (A4’),
respectively.  (for large electric fields) whereas region $A$ is occupied by
the material with an asymptotic $q-$growth (for large electric fields), see
Figure 1. We denote by $dS$ the $(n-1)-$dimensional Hausdorff measure.
Moreover, we set

$L^{\infty}_{+}(\Omega):=\\{\theta\in L^{\infty}(\Omega)\ |\ \theta\geq c_{0}\
\text{a.e. in}\ \Omega,\ \text{for a positive constant}\ c_{0}\\}.$

Furthermore, the Sobolev space $W^{1,p}_{0}(\Omega)$ is the closure set of
$C_{0}^{1}(\Omega)$ with respect to the $W^{1,p}-$norm. The applied boundary
voltage $f$ belongs to the abstract trace space
$B_{p}^{1-\frac{1}{p},p}(\partial\Omega)$, which is a Besov space (refer to
[59, 60]), equipped with the following norm:

$||u||_{B^{1-\frac{1}{p},p}(\partial\Omega)}=||u||_{L^{p}(\partial\Omega)}+|u|_{B^{1-\frac{1}{p},p}(\partial\Omega)}<+\infty,$

where for the precise definition of
$|\cdot|_{B^{1-\frac{1}{p},p}(\partial\Omega)}$ we refer to [60, Def.18.32].
For the sake of brevity, we denote this space by $X^{p}(\partial\Omega)$ and
we point out that its elements can be identified as the functions in
$W^{1,p}(\Omega)$, modulo the equivalence relation
$f\in[g]_{X^{p}(\partial\Omega)}$ if and only if $f-g\in W^{1,p}_{0}(\Omega)$.
Finally, we denote by $X^{p}_{\diamond}(\partial\Omega)$ the set of elements
in $X^{p}(\partial\Omega)$ with zero average on $\partial\Omega$ with respect
to the measure $dS$.

###  3.2. The Scalar Potential and Dirichlet Energy

In terms of the electric scalar potential, that is ${\bf E}(x)=-\nabla u(x)$,
the nonlinear Ohm’s law ${\bf J}(x)=\sigma(x,E(x)){\bf E}(x)$ is

${\bf J}(x)=-\sigma(x,|\nabla u(x)|)\nabla u(x),$

where $\sigma$ is the electrical conductivity, ${\bf E}$ is the electric
field, and ${\bf J}$ is the electric current density. The electric scalar
potential $u$ solves the steady current problem:

$\begin{cases}\operatorname{div}\Big{(}\sigma(x,|\nabla u(x)|)\nabla
u(x)\Big{)}=0\ \text{in }\Omega\vspace{0.2cm}\\\
u(x)=f(x)\qquad\qquad\qquad\quad\ \text{on }\partial\Omega,\end{cases}$ (3.1)

where $f\in X_{\diamond}^{p}(\partial\Omega)$. Problem (3.1) is meant in the
weak sense, that is

$\int_{\Omega}\sigma\left(x,|\nabla u(x)|\right)\nabla
u(x)\cdot\nabla\varphi(x)\ \text{d}x=0\quad\forall\varphi\in
C_{c}^{\infty}(\Omega).$

We observe that the solution $u$ restricted to $B$ belongs to $W^{1,p}(B)$,
whereas $u$ restricted to $A$ belongs to $W^{1,q}(A)$. Therefore, the solution
$u$ as a whole is an element of the larger functional space
$W^{1,p}(\Omega)\cup W^{1,q}(\Omega)$. Furthermore, (i) if $p\leq q$ then
$W^{1,p}(\Omega)\cup W^{1,q}(\Omega)=W^{1,p}(\Omega)$, and (ii) if $p\geq q$
then $W^{1,p}(\Omega)\cup W^{1,q}(\Omega)=W^{1,q}(\Omega)$. The solution $u$
satisfies the boundary condition in the sense that $u-f\in
W_{0}^{1,p}(\Omega)\cup W_{0}^{1,q}(\Omega)$ and we write
$u|_{\partial\Omega}=f$. Moreover, the solution $u$ is variationally
characterized as

$\arg\min\left\\{\mathbb{F}_{\sigma}\left(u\right)\ :\ u\in
W^{1,p}(\Omega)\cup W^{1,q}(\Omega),\ u|_{\partial\Omega}=f\right\\}.$ (3.2)

In (3.2), the functional $\mathbb{F}_{\sigma}\left(u\right)$ is the Dirichlet
Energy

$\mathbb{F}_{\sigma}\left(u\right)=\int_{B}Q_{B}(x,|\nabla u(x)|)\
\text{d}x+\int_{A}Q_{A}(x,|\nabla u(x)|)\ \text{d}x$

where $Q_{B}$ and $Q_{A}$ are the Dirichlet Energy density in $B$ and in $A$,
respectively:

$\displaystyle
Q_{B}\left(x,E\right):=\int_{0}^{E}\sigma_{B}\left(x,\xi\right)\xi\text{d}\xi\quad\text{for
a.e.}\ x\in B\ \text{and}\ \forall E\geq 0,$ $\displaystyle
Q_{A}\left(x,E\right):=\int_{0}^{E}\sigma_{A}\left(x,\xi\right)\xi\text{d}\xi\quad\text{for
a.e.}\ x\in A\ \text{and}\ \forall E\geq 0,$

and $\sigma_{B}$ and $\sigma_{A}$ are the restriction of the electrical
conductivity $\sigma$ in $B$ and $A$.

###  3.3. Requirements on the Dirichlet Energy densities

In this Section, we provide the assumptions on the Dirichlet Energy densities
$Q_{B}$ and $Q_{A}$, to guarantee the well-posedness of the problem and to
prove the main convergence results of this paper. For each individual result,
we will make use of a suitable subset of assumptions, among those listed in
the following. Firstly, we recall the definition of the Carathéodory function.

######  Definition 3.1.

$Q:\Omega\times[0,+\infty)\to\mathbb{R}$ is a Carathéodory function iff:

  * •

$x\in\overline{\Omega}\mapsto Q(x,E)$ is measurable for every
$E\in[0,+\infty)$,

  * •

$E\in[0,+\infty)\mapsto Q(x,E)$ is continuous for almost every $x\in\Omega$.

The existence and the uniqueness of the solution of (3.2) are guaranteed by
the following assumptions on $Q_{B}$ and $Q_{A}$.

  * (A1)

$Q_{B}$ and $Q_{A}$ are Carathéodory functions;

  * (A2)

$E\mapsto Q_{B}(x,E)$ and $E\to Q_{A}(x,E)$ are positive, $C^{1}$, strictly
convex function such that $Q_{B}(x,0)=0$ for a.e. $x\in B$, and $Q_{A}(x,0)=0$
for a.e. $x\in A$.

Since $Q_{B}$ and $Q_{A}$ are positive, convex and vanishing in $0$, they are
both increasing functions in $E$.

  * (A3)

There exist three positive constants $\underline{Q}\leq\overline{Q}$ and
$E_{0}$, such that:

$\begin{split}(i)\
\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}-1\right]\leq
Q_{B}(x,E)\leq\overline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}+1\right]\ \
\text{for a.e.}\ x\in B\ \text{and}\ \forall\ E\geq 0,\\\ (ii)\
\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{q}-1\right]\leq
Q_{A}(x,E)\leq\overline{Q}\left[\left(\frac{E}{E_{0}}\right)^{q}+1\right]\ \
\text{for a.e.}\ x\in A\ \text{and}\ \forall\ E\geq 0.\end{split}$

Assumptions (A1), (A2) and (A3) assure the existence of the solution of
problem (3.2), see Theorem 3.2 below. Assumption (A3) is well-known in
literature (see e.g. assumptions (H4) in [14] and (A2) in [3]). In order to
obtain the convergence results for large Dirichlet boundary data, we introduce
the concept of asymptotic $p-$growth and asymptotic $q-$growth, as in the
following assumption.

  * (A4)

There exists a function $\beta$ such that:

$\begin{split}\lim_{E\to+\infty}\frac{Q_{B}(x,E)}{E^{p}}=\beta(x)\quad\text{for
a.e.}\ x\in B.\end{split}$

We observe that $\beta\in L^{\infty}_{+}(B)$, by (A3). This is a key
assumption for this contribution: it guarantees the convergence of the
solution in region $B$. In Section 6, we provide a counterexample to show that
assumption (A4) is sharp.

  * (A4’)

There exists a function $\alpha$ such that:

$\begin{split}\lim_{E\to+\infty}\frac{Q_{A}(x,E)}{E^{q}}=\alpha(x)\quad\text{for
a.e.}\ x\in A.\end{split}$

We observe that $\alpha\in L^{\infty}_{+}(A)$ by (A3). This assumption is
needed to guarantee the convergence of the solution in region $A$ for $p>q$.
Assumption (A4’) is sharp.

###  3.4. Connection among $\sigma$, ${\bf J}$ and $Q$

This paper is focused on the properties of the Dirichlet Energy density $Q$,
while, in physics and engineering the electrical conductivity $\sigma$ is of
greater interest. From this perspective, assumptions ( Ax) are able to include
a wide class of electrical conductivities (see Figure 5). In other words, the
( Ax)s are not restrictive in practical applications.

Figure 5. Example of a constitutive relationship when $p>2$, $p=2$ and $p<2$:
(a) in terms of electrical conductivity and (b) in terms of current density.
Dashed lines correspond to the upper and lower constraints to either $\sigma$
or ${J}_{\sigma}$. $\underline{\sigma}$ $(\overline{\sigma})$ is related to
$\underline{Q}$ $(\overline{Q})$, $p$ and $E_{0}$.

There is a close connection between $\sigma$, $J_{\sigma}$ and $Q_{\sigma}$.
Indeed, we observe that

$\begin{split}Q_{B}\left(x,E\right)=\int_{0}^{E}{J_{B}}({x,\xi})\
\text{d}{\xi}\quad\text{for a.e.}\ x\in B\ \text{and}\ \forall E>0,\\\
Q_{A}\left(x,E\right)=\int_{0}^{E}{J_{A}}({x,\xi})\
\text{d}{\xi}\quad\text{for a.e.}\ x\in A\ \text{and}\ \forall
E>0,\end{split}$

where $J_{B}$ and $J_{A}$ is the magnitude of the current density in regions
$B$ and $A$, respectively:

$\begin{split}J_{B}(x,E)&=\partial_{E}Q_{B}(x,E)=\sigma_{B}(x,E)E\quad\text{for
a.e.}\ x\in B\ \text{and}\ \forall E>0.\\\
J_{A}(x,E)&=\partial_{E}Q_{A}(x,E)=\sigma_{A}(x,E)E\quad\text{for a.e.}\ x\in
A\ \text{and}\ \forall E>0.\end{split}$ (3.3)

The electrical conductivity $\sigma(x,E)$ is the secant to the graph of the
function $J_{\sigma}(x,E(x))$ and $Q_{\sigma}(x,E(x))$ is the area of the sub-
graph of $J_{\sigma}(x,E(x))$. For a geometric interpretation of the
connections between $\sigma$, $J_{\sigma}$ and $Q_{\sigma}$, see Figure 6.

Figure 6. For any given spatial point in the region $\Omega$, (a) the
electrical conductivity $\sigma(\cdot,E)$ is the secant line to the graph of
the function $J_{\sigma}(\cdot,E)$; (b) $Q_{\sigma}(\cdot,E)$ is the area of
the sub-graph of $J_{\sigma}(\cdot,E)$.

###  3.5. Existence and uniqueness of the solutions

The proof of the existence and uniqueness of the solution for problem (3.2),
relies on standard methods of the Calculus of Variations, when the Dirichlet
Energy density presents the same growth bounds (described in (A3)) in any
point of the domain $\Omega$. The case treated in this work is nonstandard,
because the Dirichlet Energy density presents different growth bounds in $B$
and $A$ and, hence, we provide a proof in the following.

######  Theorem 3.2.

Let $1<p,q<+\infty$, $p\neq q$, $f\in X_{\diamond}^{p}(\partial\Omega)$. If
(A1), (A2), (A3) hold, then there exists a unique solution of problem (3.2).

###### Proof.

Let us consider the case $p>q$; if $E\geq E_{0}$ then $E^{p}\geq
E_{0}^{p-q}E^{q}$. By assumption (A3), for any $E\geq E_{0}$, we have

$\begin{split}&Q_{B}(x,E)\geq\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}{-1}\right]\geq\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{q}{-1}\right]\quad\text{a.e.
in }B,\\\
&Q_{A}(x,E)\geq\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{q}{-1}\right]\qquad\qquad\qquad\qquad\quad\text{a.e.
in }A.\end{split}$

Therefore, setting $Q_{\sigma}=Q_{B}$ in $B$ and $Q_{\sigma}=Q_{A}$ in $A$, we
have

$Q_{\sigma}(x,E)\geq\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{q}{-1}\right]\quad\text{a.e.
in }\Omega,$

$\text{for any }E\geq 0$ (since the inequality is trivial for any $E\geq
E_{0}$). Similarly, when $p<q$, we have

$Q_{\sigma}(x,E)\geq\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}{-1}\right]\quad\text{a.e.
in }\Omega,$

for any $E\geq 0$. Hence, in both cases, the function $Q_{\sigma}(x,\cdot)$ is
coercive for a.e. $x\in\Omega$. Therefore, since $Q_{\sigma}$ is also strictly
convex by assumption, standard methods of the Calculus of Variations [61, Th.
3.30] assure that the solution exists and is unique. ∎

Condition $f\in X_{\diamond}^{p}(\partial\Omega)$ is a necessary assumption
because it guarantees the existence of a function in $f+W_{0}^{1,p}(\Omega)$
such that $\mathbb{F}_{\sigma}(f)<+\infty$ (see [61, Th. 3.30] for details).
Moreover, optimization problems on domains with holes have received a great
deal of interest in recent years, see e.g. [62, 63, 64, 65] and reference
therein.

###  3.6. Normalized solution

Through this paper we study the behaviour of the solution of problem (3.2) for
large Dirichlet boundary data, i.e. the behaviour of $u^{\lambda}$ defined as
the solution of:

$\min_{\begin{subarray}{c}u\in W^{1,p}(\Omega)\cup W^{1,q}(\Omega)\\\
u=\lambda f\ \text{on}\ \partial\Omega\end{subarray}}\mathbb{F}_{\sigma}(u),$
(3.4)

for $\lambda\to+\infty$. To this purpose, as discussed in Section 2, it is
convenient to introduce the normalized solution $v^{\lambda}$ defined as:

$v^{\lambda}=\frac{u^{\lambda}}{\lambda}.$

For any prescribed $f\in X^{p}_{\diamond}(\partial\Omega)$ and $\lambda>0$,
$v^{\lambda}$ is the solution of the following variational problem:

$\min_{\begin{subarray}{c}v\in W^{1,p}(\Omega)\cup W^{1,q}(\Omega)\\\ v=f\
\text{on}\
\partial\Omega\end{subarray}}\mathbb{G}^{\lambda}(v),\quad\mathbb{G}^{\lambda}(v)=\frac{1}{\lambda^{p}}\left(\int_{B}Q_{B}(x,\lambda|\nabla
v(x)|)dx+\int_{A}Q_{A}(x,\lambda|\nabla v(x)|)dx\right).$ (3.5)

The multiplicative factor $1/\lambda^{p}$ is introduced in order to guarantee
that the minimum values of the functional $\mathbb{G}^{\lambda}$ are bounded
for large $\lambda$. The normalized solution makes it possible to “transfer”
parameter $\lambda$ in (3.4) from the boundary data to the functional
$\mathbb{G}^{\lambda}$. Specifically, we will prove that $v^{\lambda}$
converges, under very mild hypotheses, for $\lambda\to+\infty$ to the solution
of a problem where the material in region $A$ is replaced by either a PEC or a
PEI. In the PEC case ($p<q$) the solution $w$ is constant in the region A. In
the PEI case ($p>q$) the limit of $v^{\lambda}$ in $B$ is termed $v_{B}$,
whereas the limit of $v^{\lambda}$ in $A$ is termed $v_{A}$. Moreover (i)
$v_{B}$ and $v_{A}$ arise from a weighted $p-$Laplace and a weighted
$q-$Laplace problem, respectively and (ii) $v_{B}$ provides the boundary data
on $\partial A$ for evaluating $v_{A}$.

##  4\. The fundamental inequality for large Dirichlet data

In this Section we provide the main tool to achieve the convergence results in
the limiting cases for “large” Dirichlet boundary data. Specifically, we will
show that the asymptotic behaviour of the Dirichlet Energy corresponds to a
$p-$Laplace modelled equation [4, 8] in the domain $B$. In the following, we
study the asymptotic behaviour of the Dirichlet Energy in the outer region
$B$. We first analyse the special case when the Dirichlet Energy density
$Q_{B}$ is modelled on the weighted $p-$Laplace case, that is
$Q_{B}(x,E)=\theta(x)E^{p}$ on $B\times[0,+\infty)$; we then give the result
in the more general case.

######  Lemma 4.1.

Let $1<p<+\infty$, $F\subset\mathbb{R}^{n}$ be a bounded domain with Lipschitz
boundary, $\theta\in L^{\infty}_{+}(F)$ and $\\{{h}_{n}\\}_{n\in\mathbb{N}}$
be a sequence weakly convergent to ${h}$ in $W^{1,p}(F)$. Then we have

$\int_{F}\theta(x)|\nabla{h}(x)|^{p}dx\leq\liminf_{n\to+\infty}\int_{F}\theta(x)|\nabla{h}_{n}(x)|^{p}dx.$
(4.1)

###### Proof.

Let us set
$L:=\liminf_{n\to+\infty}\int_{F}\theta(x)|\nabla{h}_{n}(x)|^{p}dx$. If
$L=+\infty$, the inequality (4.1) is trivial; otherwise we consider a
subsequence $\\{n_{j}\\}_{j\in\mathbb{N}}$ such that

$\lim_{{j}\to+\infty}\int_{F}\theta(x)|\nabla{h_{n_{j}}}(x)|^{p}dx=L.$

This means that for any $\varepsilon>0$, there exists $\nu\in\mathbb{N}$ such
that

$L-\varepsilon<\int_{F}\theta(x)|\nabla{h_{n_{j}}}(x)|^{p}dx<L+\varepsilon$
(4.2)

for any ${j}\geq\nu$. Then, by Mazur’s Lemma (refer for example to [66, 67]),
for any $n\in\mathbb{N}$ there exists a function $N:\mathbb{N}\to\mathbb{N}$
and a sequence $\\{\alpha_{n,{l}}\\}_{{l}=n}^{N(n)}$ such that

  1. (M1)

$\alpha_{{n},l}\geq 0$ for any ${l}\in[n,N(n)]$,

  2. (M2)

$\sum_{{l}=n}^{N(n)}\alpha_{n,{l}}=1$,

  3. (M3)

$z_{{n}}:=\sum_{{l}=n}^{N(n)}\alpha_{n,{l}}{h_{n_{l}}}$ $\to{h}$ in
$W^{1,p}(F)$.

Then there exists a subsequence $\\{z_{n_{{k}}}\\}_{{k}\in\mathbb{N}}$ such
that

$\liminf_{n\to+\infty}\int_{F}\theta(x)|\nabla
z_{n}(x)|^{p}dx=\lim_{{k}\to+\infty}\int_{F}\theta(x)|\nabla
z_{n_{{k}}}(x)|^{p}dx$ (4.3)

and another subsequence, still indicated with
$\\{z_{n_{{k}}}\\}_{{k}\in\mathbb{N}}$, such that $\nabla
z_{n_{{k}}}\to\nabla{h}$ a.e. in $F$ [60, Chap. 18]. Therefore, we have

$\begin{split}\mathbb{\int}_{F}\theta(x)|\nabla{h}(x)|^{p}dx&=\int_{F}\theta(x)\lim_{k\to+\infty}|\nabla
z_{n_{k}}(x)|^{p}dx\leq\liminf_{{k}\to+\infty}\int_{F}\theta(x)|\nabla
z_{n_{{k}}}(x)|^{p}dx\\\ &=\lim_{{k}\to+\infty}\int_{F}\theta(x)|\nabla
z_{n_{{k}}}(x)|^{p}dx=\liminf_{n\to+\infty}\int_{F}\theta(x)|\nabla
z_{n}(x)|^{p}dx\\\
&\leq\liminf_{n\to+\infty}\sum_{{l}=n}^{N(n)}\alpha_{n,{l}}\int_{F}\theta(x)|\nabla{h_{n_{l}}}(x)|^{p}dx\\\
&<\liminf_{n\to+\infty}\mathbb{\sum}_{{l}=n}^{N(n)}\alpha_{n,{l}}\left(L+\varepsilon\right)=L+\varepsilon\\\
&=\liminf_{n\to+\infty}\int_{F}\theta(x)|\nabla{h}_{n}(x)|^{p}dx+\varepsilon,\end{split}$

where in the first line the equality follows from the convergence result of
(M3) and the inequality follows from Fatou’s Lemma, in the second line we
applied (4.3), in the third line we applied the convexity of $|\cdot|^{p}$,
and in the fourth line we applied (4.2). Conclusion (4.1) follows from the
arbitrariness of $\varepsilon>0$. ∎

When $Q_{B}$ is factorized, the assumptions (A4) are satisfied. The next step
consists in extending (4.1) from a weighted $p-$Laplace case to the
quasilinear case. In doing this, we restrict the validity of the result to
sequences of the solutions of problem (3.5). The main difficulty in proving
this result lies in evaluating an upper bound of the Lebesgue measure of that
part of $B$ where the solutions $v^{\lambda}$ admit large values of the
gradient (see Figure 7).

Figure 7. The objective of the proof is to show that the set of the points in
$B$ such that the solution $v^{\lambda_{n_{j}}}$ does not satisfy the
fundamental inequality ($C^{c}_{n_{j}}$) and admits large values of the
gradient ($D^{c}_{n_{j}}$) can be made small enough (shaded region).

######  Lemma 4.2.

Let $1<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$, (A1), (A2), (A3),
(A4) hold and let the solutions $v^{\lambda}$ of (3.5) be weakly convergent to
$v$ in $W^{1,p}(B)$, as $\lambda\to+\infty$. Let
$\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ be an increasing sequence such that
$\lambda_{n}\to+\infty$, as $n\to+\infty$, and

$\lim_{n\to+\infty}\frac{1}{\lambda_{n}^{p}}\int_{B}Q_{B}(x,\lambda_{n}|\nabla
v^{\lambda_{n}}(x)|)dx=\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx.$ (4.4)

Then, for any $0<\delta\leq\inf_{B}\beta$ and $\theta>0$, there exists
$F_{\delta,\theta}\subseteq B$ with $|B\setminus F_{\delta,\theta}|<\theta$
such that

$\liminf_{n\to+\infty}\int_{F_{\delta,\theta}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n}}(x)|^{p}dx\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx,$ (4.5)

where $\beta$ is given in (A4).

###### Proof.

Let us fix $0<\delta\leq\inf_{B}\beta$ and $\theta>0$, for any
$n\in\mathbb{N}$, we set

$C_{\delta,n}=C_{n}:=\left\\{x\in B\ :\ (\beta(x)-\delta)(\lambda_{n}|\nabla
v^{\lambda_{n}}(x)|)^{p}\leq Q_{B}(x,\lambda_{n}|\nabla
v^{\lambda_{n}}(x)|)\right\\}.$

Now, for any constant $\ell>0$, we set

$\begin{split}D_{\ell,n}=D_{n}:=\\{x\in B\ :\ \lambda_{n}|\nabla
v^{\lambda_{n}}(x)|\leq\lambda_{n}\ell\\}\\\ D_{\ell,n}^{c}=D^{c}_{n}:=\\{x\in
B\ :\ \lambda_{n}|\nabla v^{\lambda_{n}}(x)|>\lambda_{n}\ell\\}\end{split}$

Now, let us define

$E_{\delta,\ell,n}=E_{n}:=\left\\{x\in B\ :\ (\beta(x)-\delta)E^{p}\leq
Q_{B}(x,E)\ \ \forall E>\lambda_{n}\ell\right\\}.$

Therefore, we have

$\begin{split}&C_{n}\supseteq C_{n}\cap D^{c}_{n}\supseteq E_{n}\cap
D^{c}_{n},\\\ &C_{n}\cup D_{n}\supseteq E_{n}\cup D_{n}=(D^{c}_{n}\setminus
E_{n})^{c},\end{split}$ (4.6)

for any $n\in\mathbb{N}$. Let us observe that $E_{n}$ is increasing with
respect to $n$ and that $\left|\bigcup_{n=1}^{+\infty}E_{n}\right|=|B|$.
Therefore there exists a natural number $n_{1}=n_{1}(\theta)$ such that

$\left|\bigcup_{n=1}^{n_{1}}E_{n}\right|=\left|E_{n_{1}}\right|\geq|B|-\frac{\theta}{2}.$
(4.7)

Passing to the complementary sets in (4.6), we obtain

$C^{c}_{n_{1}}\cap D^{c}_{n_{1}}\subseteq D^{c}_{n_{1}}\setminus
E_{n_{1}}\subseteq B\setminus E_{n_{1}}\quad\text{with}\quad|C_{n_{1}}^{c}\cap
D^{c}_{n_{1}}|\leq\frac{\theta}{2},$

where we have used (4.7). Analogously, we construct a subsequence
$\\{\lambda_{n_{j}}\\}_{j\in\mathbb{N}}$ such that

$|C_{n_{j}}^{c}\cap D^{c}_{n_{j}}|\leq\frac{\theta}{2^{j}}.$

Then, by defining

$F_{\delta,\theta}:=\bigcap_{j=1}^{\infty}(C_{n_{j}}\cup D_{n_{j}}),$

we have

$F^{c}_{\delta,\theta}=\bigcup_{j=1}^{\infty}(C_{n_{j}}\cup
D_{n_{j}})^{c}\quad\text{with}\quad|F^{c}_{\delta,\theta}|\leq\sum_{j=1}^{+\infty}\frac{\theta}{2^{j}}=\theta,$

which implies

$|B|-\theta\leq|F_{\delta,\theta}|\leq|B|.$

Therefore, we have

$\begin{split}&\liminf_{n\to+\infty}\int_{F_{\delta,\theta}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n}}(x)|^{p}dx\leq\liminf_{j\to+\infty}\int_{F_{\delta,\theta}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n_{j}}}(x)|^{p}dx\\\ &\leq\liminf_{j\to+\infty}\int_{C_{n_{j}}\cup
D_{n_{j}}}(\beta(x)-\delta)|\nabla v^{\lambda_{n_{j}}}(x)|^{p}dx\\\
&\leq\liminf_{j\to+\infty}\int_{C_{n_{j}}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n_{j}}}(x)|^{p}dx+\limsup_{j\to+\infty}\int_{D_{n_{j}}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n_{j}}}(x)|^{p}dx\\\
&\leq\liminf_{j\to+\infty}\frac{1}{\lambda_{n_{j}}^{p}}\int_{C_{n_{j}}}Q_{B}(x,|\lambda_{n_{j}}\nabla
v^{\lambda_{n_{j}}}(x)|)dx+\frac{\overline{Q}}{E_{0}^{p}}\ell^{p}|B|\\\
&\leq\lim_{j\to+\infty}\frac{1}{\lambda_{n_{j}}^{p}}\int_{B}Q_{B}(x,|\lambda_{n_{j}}\nabla
v^{\lambda_{n_{j}}}(x)|)dx+\frac{\overline{Q}}{E_{0}^{p}}\ell^{p}|B|\\\
&=\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx+\frac{\overline{Q}}{E_{0}^{p}}\ell^{p}|B|,\end{split}$

where in the second line we exploited $\beta(x)-\delta\geq 0$ a.e. in $B$, in
the fourth line we exploited $\beta\leq\overline{Q}/E_{0}^{p}$, as follows
from (A3) and (A4) and in the last line we exploited (4.4). Therefore, the
conclusion follows from the arbitrariness of $\ell>0$.∎

We are now in a position to prove the previous result in the general case
expressing the asymptotic behaviour of the Dirichlet Energy for the outer
region in terms of a factorized $p-$Laplacian form. Specifically, the
following Proposition provides a fundamental inequality relating the two
cases.

######  Proposition 4.3.

Let $1<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$, (A1), (A2), (A3),
(A4) hold and the solutions $v^{\lambda}$ of (3.5) weakly converge to $v$ in
$W^{1,p}(B)$, as $\lambda\to+\infty$. Then

$\int_{B}\beta(x)|\nabla
v(x)|^{p}dx\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx,$ (4.8)

where $\beta$ is given in (A4).

###### Proof.

We assume that $v$ is nonconstant, otherwise the conclusion is trivial.
Therefore, the integral at the l.h.s. of (4.8) is positive because $\beta\in
L^{\infty}_{+}(B)$. We firstly observe that the measure

$\kappa:F\in{\mathcal{B}(\Omega)}\mapsto\int_{F}\beta(x)|\nabla v(x)|^{p}dx,$

where $\mathcal{B}(\Omega)$ is the class of the borelian sets contained in
$\Omega$, is absolutely continuous with respect to the Lebesgue measure; then,
for any $\varepsilon>0$, there exists $\theta>0$ such that for $|F|<\theta$
implies $\kappa(F)<\frac{\varepsilon}{2}$. Then, for any
$0<\delta\leq\inf_{B}\beta$, the set $F_{\delta,\theta}$, given in Lemma 4.2,
satisfies $|B\setminus F_{\delta,\theta}|<\theta$ and implies that

$\kappa(B\setminus F_{\delta,\theta})=\int_{B\setminus
F_{\delta,\theta}}\beta(x)|\nabla v(x)|^{p}dx<\frac{\varepsilon}{2}.$

Let $\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ be an increasing sequence such that
$\lambda_{n}\to+\infty$ as $n\to+\infty$, with $\lambda_{n}>0\ \forall
n\in\mathbb{N}$ satisfying (4.4) appearing in Lemma 4.2. We have

$\begin{split}&\int_{B}\beta(x)|\nabla
v(x)|^{p}dx-\varepsilon=\int_{B\setminus F_{\delta,\theta}}\beta(x)|\nabla
v(x)|^{p}dx+\int_{F_{\delta,\theta}}\beta(x)|\nabla v(x)|^{p}dx-\varepsilon\\\
&<\int_{F_{\delta,\theta}}\beta(x)|\nabla
v(x)|^{p}dx-\frac{\varepsilon}{2}=\int_{F_{\delta,\theta}}\beta(x)|\nabla
v(x)|^{p}dx-\delta\int_{B}|\nabla v(x)|^{p}dx\\\
&\leq\int_{F_{\delta,\theta}}(\beta(x)-\delta)|\nabla
v(x)|^{p}dx\leq\liminf_{n\to+\infty}\int_{F_{\delta,\theta}}(\beta(x)-\delta)|\nabla
v^{\lambda_{n}}(x)|^{p}dx\\\
&\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,t|\nabla
v^{\lambda}(x)|)dx.\end{split}$

where we applied the absolute continuity of $\kappa$ in the second line, we
set $\delta\leq\min\\{\varepsilon/\left(2\int_{B}|\nabla
v(x)|^{p}dx\right),\inf_{B}\beta(x)\\}$, we applied the lower semicontinuity
of the integral functional in the third line, and we exploited the (4.5) of
Lemma 4.2 in the fourth line. The conclusion follows from letting
$\varepsilon\to 0^{+}$. ∎

##  5\. Limit for large Dirichlet data

In this Section, we treat the limiting case of problem (3.5) for large
boundary data, i.e when $\lambda$ approaches infinity. We distinguish two
cases according to the values of $p$ and $q$, which correspond to the
asymptotic growth exponents of the electrical conductivity in $B$ and $A$,
respectively:

  1. (i)

$1<q<p<+\infty$ (see Section 5.1);

  2. (ii)

$1<p<q<+\infty$ (see Section 5.2).

Specifically, in the first case, we prove that:

  * (i.a)

$v^{\lambda}\rightharpoonup v_{B}$ in $W^{1,p}(B)$, as $\lambda\to+\infty$,
where $v_{B}$ in $B$ is the unique solution of problem (2.1);

  * (i.b)

$v^{\lambda}\rightharpoonup v_{A}$ in $W^{1,q}(A)$, as $\lambda\to+\infty$,
where $v_{A}$ in $A$ is the unique solution of problem (2.2) (under an
additional hypothesis);

For $1<q<p$, the limiting solution is characterized by the following problems:

$\min_{\begin{subarray}{c}v\in W^{1,p}(B)\\\ v=f\ \text{on}\
\partial\Omega\end{subarray}}\mathbb{B}(v),\quad\mathbb{B}(v)=\int_{B}\beta(x)|\nabla
v(x)|^{p}dx,$ (5.1) $\min_{\begin{subarray}{c}v\in W^{1,q}(A)\\\ v=v_{B}\
\text{on}\ \partial
A\end{subarray}}\mathbb{A}(v),\quad\mathbb{A}(v)=\int_{A}\alpha(x)|\nabla
v(x)|^{q}dx.$ (5.2)

Hereafter, we recall that we have denoted by $v_{B}\in W^{1,p}(B)$ and
$v_{A}\in W^{1,q}(A)$ as the unique solutions of (5.1) and (5.2),
respectively.Problems (5.1) and (5.2) are the variational form of problems
(2.1) and (2.2), respectively. Whereas, in the second case, we prove that:

  * (ii.a)

$v^{\lambda}\rightharpoonup w$ in $W^{1,p}(\Omega)$, as $\lambda\to+\infty$,
where $w$ in $B$ is the unique solution of problem (2.3) and it is constant in
each connected component of $A$.

  * (ii.b)

$v^{\lambda}\to w$ in $W^{1,q}(A)$, as $\lambda\to+\infty$, where $w$ is
constant in any connected component of $A$.

In the latter case, the constant value assumed by $w$ in each connected
component of $A$ guarantees the continuity of $w$ across the boundaries with
$B$. For $1<p<q$, the limiting solution is characterized by the following
problem:

$\min_{\begin{subarray}{c}v\in W^{1,p}(\Omega)\\\ |\nabla v|=0\ \text{a.e.
in}\ A\\\ v=f\ \text{on}\
\partial\Omega\end{subarray}}\mathbb{B}(v),\quad\mathbb{B}(v)=\int_{B}\beta(x)|\nabla
v(x)|^{p}dx.$ (5.3)

The unique solution of (5.3) is denoted by ${w\in W^{1,p}(\Omega)}$. Problem
(5.3) is the variational form for problem (2.3).

######  Remark 5.1.

From the physical standpoint, the function ${w}\in W^{1,p}(\Omega)$ solution
of problem (5.3), corresponds to the solution of a weighted $p-$Laplace
problem in $B$ with region $A$ filled by a PEC.

For the sequel, we need an upper bound for both the Dirichlet Energy density
and the $L^{p}-$norm of the gradient of the solution $v^{\lambda}$ (in the
outer region) of problem (3.5), as provided by the following Lemma.

######  Lemma 5.2.

Let $1<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$, (A1), (A2) and
(A3) hold and ${w}\in W^{1,p}(\Omega)$ be the function achieving the minimum
of the variational problem (5.3). Then, the solution $v^{\lambda}$ of (3.5)
satisfies

$\int_{B}|\nabla
v^{\lambda}(x)|^{p}dx\leq\overline{Q}/\underline{Q}\left(\int_{B}|\nabla{w}(x)|^{p}dx+\frac{E^{p}_{0}}{\lambda^{p}}|B|\right)+\frac{{E_{0}^{p}}}{\lambda^{p}}|B|$
(5.4)

and

$\begin{split}\mathbb{G}^{\lambda}(v^{\lambda})\leq\frac{\overline{Q}}{E_{0}^{p}}\int_{B}|\nabla{w}(x)|^{p}dx+\frac{\overline{Q}}{\lambda^{p}}|B|.\end{split}$
(5.5)

###### Proof.

Let us consider the two auxiliary functions

$\begin{split}Q_{1}(E)=\underline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}-{1}\right]\quad\forall
E>0,\\\
Q_{2}(E)=\overline{Q}\left[\left(\frac{E}{E_{0}}\right)^{p}+{1}\right]\quad\forall
E>0.\end{split}$

In terms of $Q_{1}$ and $Q_{2}$, the assumption (A3) reads as

$Q_{1}(E)\leq Q_{B}(x,E)\leq Q_{2}(E)\quad\text{for a.e.}\ x\in B\ \text{and}\
\forall E>0.$ (5.6)

Therefore, we have

$\begin{split}\int_{B}|\nabla
v^{\lambda}(x)|^{p}dx-\frac{{E_{0}^{p}}}{\lambda^{p}}|B|&=\frac{E_{0}^{p}}{\underline{Q}}\frac{1}{\lambda^{p}}\int_{B}Q_{1}(\lambda|\nabla
v^{\lambda}(x)|)dx\\\
&\leq\frac{E_{0}^{p}}{\underline{Q}}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx\leq\frac{E_{0}^{p}}{\underline{Q}}\mathbb{G}^{\lambda}(v^{\lambda})\\\
&\leq\frac{E_{0}^{p}}{\underline{Q}}\mathbb{G}^{\lambda}({w})=\frac{E_{0}^{p}}{\underline{Q}}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla{w}(x)|)dx\\\
&\leq\frac{E_{0}^{p}}{\underline{Q}}\frac{1}{\lambda^{p}}\int_{B}Q_{2}(\lambda|\nabla{w}(x)|)dx\\\
&=\frac{\overline{Q}}{\underline{Q}}\int_{B}|\nabla{w}(x)|^{p}dx+\frac{\overline{Q}}{\underline{Q}}\frac{{E_{0}^{p}}}{\lambda^{p}}|B|,\end{split}$
(5.7)

where in the first line we applied the definition of $Q_{1}$ when
$E=\lambda\nabla v^{\lambda}$, in the second line we exploited the leftmost
inequality (5.6), in the third line we applied $w$ as the test function for
functional (3.5) and exploited that $|\nabla w|=0$ a.e. in $A$, in the fourth
line we exploited the rightmost inequality (5.6), and in the fifth line we
applied the definition of $Q_{2}$ when $E=\lambda\nabla{w}$. This concludes
the proof of (5.4). Inequality (5.5) is a byproduct of (5.7), obtained by
comparing the last term of the second line with the final term. ∎

###  5.1. First case: $\mathbf{q<p}$

For any fixed $f\in X_{\diamond}^{p}(\partial\Omega)$, we study the problem as
$\lambda$ approaches infinity. Since $q<p$, we have $W^{1,p}(\Omega)\cup
W^{1,q}(\Omega)=W^{1,q}(\Omega)$ and, hence, the variational problem (3.5)
particularizes as

$\min_{\begin{subarray}{c}v\in W^{1,q}(\Omega)\\\ v=f\ \text{on}\
\partial\Omega\end{subarray}}\mathbb{G}^{\lambda}(v).$ (5.8)

Here, we first show that $v^{\lambda}$ is weakly convergent in $W^{1,p}(B)$ to
the solution of problem (5.1). Then, we prove that $v^{\lambda}$ is weakly
convergent in $W^{1,q}(A)$ to the solution of problem (5.2), under an
additional assumption.

######  Theorem 5.3.

Let $1<q<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$ and $v^{\lambda}$
be the solution of (5.8). If (A1), (A2), (A3) and (A4) hold, then

  * $v^{\lambda}\rightharpoonup v_{B}$ in $W^{1,p}(B)$, as $\lambda\to+\infty$,

where $v_{B}\in W^{1,p}(B)$ is the unique solution of (5.1).

###### Proof.

Let $w\in W^{1,p}(\Omega)$ be the minimizer of problem (5.3). From (5.4) of
Lemma 5.2, we have

$\int_{B}|\nabla
v^{\lambda}(x)|^{p}dx\leq{\overline{Q}}/{\underline{Q}}\left(\int_{B}|{w}(x)|^{p}dx+\frac{{E_{0}^{p}}}{\lambda^{p}}|B|\right)+\frac{{E_{0}^{p}}}{\lambda^{p}}|B|<\infty.$
(5.9)

Since the last two integral terms tend to $0$ as $\lambda\to+\infty$, then
$||\nabla v^{\lambda}||_{L^{p}(B)}^{p}$ is definitively upper bounded.
Moreover, up to subsequences, since $v^{\lambda}=f$ on $\partial\Omega$, then
[60, Chap. 11] there exists $v^{\infty}\in W^{1,p}(B)$, with $v^{\infty}=f$,
such that

$v^{\lambda}\rightharpoonup v^{\infty}\quad\text{in}\
W^{1,p}(B)\quad\text{as}\ \lambda\to+\infty.$ (5.10)

Let $\tilde{v}_{B}\in W^{1,p}(\Omega)$ be an extension of $v_{B}\in
W^{1,p}(B)$. By using $\tilde{v}_{B}$ as the test function in (5.8), and from
the rightmost inequality of assumption ( A3.ii), we have

$\mathbb{G}^{\lambda}(v^{\lambda})\leq\mathbb{G}^{\lambda}(\tilde{v}_{B})\leq\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v_{B}(x)|)dx+\frac{\overline{Q}}{E_{0}^{q}}\frac{1}{\lambda^{p-q}}\int_{A}|\nabla\tilde{v}_{B}(x)|^{q}dx+\frac{\overline{Q}}{\lambda^{p}}|A|.$
(5.11)

By using the Dominated Convergence Theorem together with (A3) and (A4) to
treat the first term on the r.h.s. of (5.11), and by taking into account that
the second and third terms vanish, we have

$\limsup_{\lambda\to+\infty}\mathbb{G}^{\lambda}(v^{\lambda})\leq\int_{B}\beta(x)|\nabla
v_{B}(x)|^{p}dx=\mathbb{B}(v_{B}).$ (5.12)

Since $v^{\lambda}$ is weakly convergent (see (5.10)), we can apply
Proposition 4.3 which gives:

$\mathbb{B}(v^{\infty})\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,|\nabla
v^{\lambda}(x)|)dx.$ (5.13)

Therefore, we have the following inequalities

$\begin{split}\mathbb{B}(v_{B})\leq\mathbb{B}(v^{\infty})&\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx\\\
&\leq\limsup_{\lambda\to+\infty}\mathbb{G}^{\lambda}(v^{\lambda})\leq\lim_{\lambda\to+\infty}\mathbb{G}^{\lambda}(\tilde{v}_{B})=\mathbb{B}(v_{B}),\end{split}$
(5.14)

where in the first inequality we have used $v^{\infty}$ as test function in
$\mathbb{B}$, the second inequality is (5.13), the third inequality is
obtained by adding the integral of the Dirichlet Energy density in $A$, the
fourth inequality is obtained by using $\tilde{v}_{B}$ as the test function in
$\mathbb{G}^{\lambda}$, and the last inequality is (5.12). This implies that
$\mathbb{B}(v_{B})=\mathbb{B}(v^{\infty})$ and $v_{B}=v^{\infty}$, because of
the uniqueness of the solution of problem (5.1). This result together with
(5.10), gives the conclusion. ∎

######  Remark 5.4.

From (5.14), we find that the fundamental inequality (5.13), also stated in
Proposition 4.3, holds as an equality:

$\mathbb{B}(v^{\infty})=\lim_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,|\nabla
v^{\lambda}(x)|)dx.$

The boundedness of $v^{\lambda}$ in $W^{1,p}(B)$ implies the boundedness of
$Tr(v^{\lambda})\in X^{p}(\partial A)$ and therefore $Tr(v^{\lambda})\in
X^{q}(\partial A)$. Then, by the Inverse Trace inequality in Besov spaces [60,
Th. 18.40], we have the boundedness of $v^{\lambda}\in W^{1,q}(A)$. Then, up
to a subsequence, there exists $v_{A}\in W^{1,q}(A)$ such that
$v^{\lambda}\rightharpoonup v_{A}$ in $W^{1,q}(A)$. Indeed, we recall from
Theorem 5.3 that $v^{\lambda}\rightharpoonup v_{B}$ in $W^{1,p}(B)$ as
$\lambda\to+\infty$; this implies that $v^{\lambda}\rightharpoonup v_{B}$ in
$W^{1,q}(B)$. Therefore, by [60, Cor. 18.4], we have $Tr(v^{\lambda})\to
Tr(v_{B})=Tr(v_{A})$ in $L^{q}(\partial A)$ as $\lambda\to+\infty$. However,
to prove the desired result, we also need the stronger convergence

$||Tr(v^{\lambda})-Tr(v_{A})||_{X^{q}(\partial A)}\to 0^{+},\quad\text{as }\
\lambda\to+\infty.$

This convergence holds when $\nabla v^{\lambda}\to\nabla v_{B}$ in $L^{p}(B)$,
by using [60, Th. 18.40]. This problem will be addressed in a forthcoming
paper. Anyway, it seems interesting to state the following conditional
results.

######  Lemma 5.5.

Let $1<q<p<+\infty$, if (A1), (A2), (A3) and (A4) hold, and
$||Tr(v^{\lambda})-Tr(v_{A})||_{X^{q}(\partial A)}\to 0^{+}$, as
$\lambda\to+\infty$, then there exist two positive constants $C$ and
$\lambda_{0}$ such that $||\nabla v^{\lambda}||_{L^{q}(A)}\leq C$ for any
$\lambda>\lambda_{0}$.

###### Proof.

For any $v\in v^{\lambda}+W^{1,q}_{0}(A)$, we have

$\begin{split}\frac{\underline{Q}}{E_{0}^{{q}}}\int_{A}|\nabla
v^{\lambda}(x)|^{q}\text{d}x{\ -\frac{\
\underline{Q}|A|}{\lambda^{q}}}&\leq\frac{1}{\lambda^{q}}\int_{A}Q_{A}(x,{\lambda}|\nabla{v^{\lambda}}(x)|)\text{d}x\\\
&\leq\frac{\overline{Q}}{E_{0}^{{q}}}\int_{A}|\nabla v(x)|^{q}\text{d}x{\
+\frac{\ \overline{Q}|A|}{\lambda^{q}}}\end{split}$ (5.15)

where in the first inequality we used assumption (A3), and in the second (A3)
and the minimality of $v^{\lambda}$. The Inverse Trace inequality in Besov
spaces [60, Th. 18.40] assures the existence of a function $g\in
v^{\lambda}+W^{1,q}_{0}(A)$ such that $||\nabla g||_{q}\leq
K(q,\Omega)||Tr(v^{\lambda})||_{X^{q}(\partial A)}$. Therefore, from (5.15)
with $v=g$ and since $Tr(g)=Tr(v^{\lambda})$ on $\partial A$, we have

$||\nabla
v^{\lambda}||^{q}_{q}\leq{\frac{\overline{Q}}{\underline{Q}}}||\nabla
g||^{q}_{q}{\ +\
\frac{(\overline{Q}+\underline{Q})E_{0}^{q}|A|}{\underline{Q}\lambda^{q}}}\leq{\frac{\overline{Q}}{\underline{Q}}K}||Tr(v^{\lambda})||^{q}_{X^{q}(\partial
A)}{\ +\
\frac{(\overline{Q}+\underline{Q})E_{0}^{q}|A|}{\underline{Q}\lambda^{q}}}.$
(5.16)

Hence, $||\nabla v^{\lambda}||^{q}_{L^{q}(A)}$ is definitively upper bounded
because of the convergence of $Tr(v^{\lambda})$. ∎

In order to identify $v_{A}$, before proving the inner convergence result, we
introduce the following functional

$\mathbb{I}^{\lambda}(v)=\frac{1}{\lambda^{q}}\int_{A}Q_{A}(x,\lambda|\nabla
v(x)|)dx,$ (5.17)

and we observe that, for any $\lambda>0$, $v^{\lambda}$ is also the minimizer
of

$\min_{\begin{subarray}{c}v\in W^{1,q}(A)\\\ v=v^{\lambda}\ \text{on }\partial
A\end{subarray}}\mathbb{I}^{\lambda}(v).$

######  Theorem 5.6.

Let $1<q<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$ and $v^{\lambda}$
be the solution of (5.8). If (A1), (A2), (A3), (A4) and (A4’) hold, and
$||Tr(v^{\lambda})-Tr(v_{A})||_{X^{q}(\partial A)}\to 0^{+}$, as
$\lambda\to+\infty$, then

  * $v^{\lambda}{\rightharpoonup}v_{A}$ in $W^{1,q}(A)$, as $\lambda\to+\infty$,

where $v_{A}\in W^{1,q}(A)$ is the unique solution of (5.2).

###### Proof.

The Inverse Trace inequality [60, Th. 18.34] assures the existence of a
function $w^{\lambda}\in Tr(v^{\lambda}-v_{A})+W^{1,q}_{0}(A)$ such that
$||\nabla w^{\lambda}||_{q}\leq
K(q,\Omega)||Tr(v^{\lambda})-Tr(v_{A})||_{X^{q}(\partial A)}$. Since, by
assumption, the last quantity is infinitesimal, then $||\nabla
w^{\lambda}||_{q}\to 0$. For any $0<t<1$, let us consider the convex
combination

$v_{A}+w^{\lambda}=t\frac{v_{A}}{t}+(1-t)\frac{w^{\lambda}}{1-t};$

then, by the convexity of $\mathbb{I}^{\lambda}$, we have

$\begin{split}\mathbb{I}^{\lambda}(v_{A}+w^{\lambda})&\leq
t\mathbb{I}^{\lambda}\left(\frac{v_{A}}{t}\right)+(1-t)\mathbb{I}^{\lambda}\left(\frac{w^{\lambda}}{1-t}\right)\\\
&\leq
t\mathbb{I}^{\lambda}\left(\frac{v_{A}}{t}\right)+(1-t)\overline{Q}\left(\int_{A}\frac{|\nabla
w^{\lambda}(x)|^{q}}{(1-t)^{q}E_{0}^{q}}dx+\frac{|A|}{\lambda^{q}}\right).\end{split}$
(5.18)

by assumption (A3.ii). By passing to the $\liminf$ in (5.18) and taking into
account that $||\nabla w^{\lambda}||_{q}\to 0$ as $\lambda\to+\infty$, we have

$\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v_{A}+w^{\lambda})\leq
t\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}\left(\frac{v_{A}}{t}\right)=t^{1-q}\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}\left(v_{A}\right).$

Therefore, the limit as $t\to 1^{-}$ provides

$\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v_{A}+w^{\lambda})\leq\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}\left(v_{A}\right).$
(5.19)

From Proposition 5.5, we know that $||\nabla v^{\lambda}||_{L^{q}(A)}$ is
upper bounded and hence there exists $v^{\infty}\in W^{1,q}(A)$ such that
$v^{\lambda}\rightharpoonup v^{\infty}$ in $W^{1,q}(A)$. Hence, we have the
following inequalities

$\begin{split}\mathbb{A}(v_{A})\leq\mathbb{A}(v^{\infty})&\leq\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v^{\lambda})\leq\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v_{A}+w^{\lambda})\\\
&\leq\liminf_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v_{A})=\lim_{\lambda\to+\infty}\mathbb{I}^{\lambda}(v_{A})=\mathbb{A}(v_{A}).\end{split}$
(5.20)

where the first inequality follows from testing $v^{\infty}$ in $\mathbb{A}$,
the second inequality follows from using the fundamental result of Proposition
4.3, the third inequality accounts the minimality of $v^{\lambda}$ for
$\mathbb{I}^{\lambda}$, the fourth inequality is (5.19), and the last
equalities follows from using the dominated convergence theorem together with
(A3) and (A4’). Proposition 4.3 was applied in the second inequality for $B$,
$p$, $\beta$ and (A4) replaced by $A$, $q$, $\alpha$ and (A4’), respectively.
Equation (5.20) implies that $\mathbb{A}(v_{A})=\mathbb{A}(v^{\infty})$ and
$v_{A}=v^{\infty}$, by the uniqueness of the solution of problem (5.1). ∎

As a corollary, we have a global convergence result.

######  Corollary 5.7.

Let $1<q<p<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$ and $v^{\lambda}$
be the solution of (5.8). If (A1), (A2), (A3), (A4) and (A4’) hold, and
$||Tr(v^{\lambda})-v_{A}||_{X^{q}(\partial A)}\to 0^{+}$, as
$\lambda\to+\infty$, then

  * $v^{\lambda}\rightharpoonup v_{\Omega}$ in $W^{1,q}(\Omega)$, as $\lambda\to+\infty$,

where $v_{\Omega}\in W^{1,q}(\Omega)$ is defined by

$v_{\Omega}=\begin{cases}v_{B}\quad\text{in }B\\\ v_{A}\quad\text{in
}A,\end{cases}$

$v_{B}\in W^{1,p}(B)$ is the unique solution of (5.1), $v_{A}\in W^{1,q}(A)$
is the unique solution of (5.2).

###  5.2. Second case: $\mathbf{p<q}$

For any fixed $f\in X_{\diamond}^{p}(\partial\Omega)$, since $p<q$, we have
$W^{1,p}(\Omega)\cup W^{1,q}(\Omega)=W^{1,p}(\Omega)$ and, hence, the
variational problem (3.5) particularizes as

$\min_{\begin{subarray}{c}v\in W^{1,p}(\Omega)\\\ v=f\ \text{on}\
\partial\Omega\end{subarray}}\mathbb{G}^{\lambda}(v).$ (5.21)

We denote by ${w}$ the solution of the limiting problem defined in (5.3).
Problem (5.3) is relevant because $v^{\lambda}$ weakly converges to $w$ for
$\lambda\to+\infty$. We observe that the condition $|\nabla v|=0$ in (5.3) is
equivalent to saying that $v$ is constant on each connected component of $A$.
In other words, region $A$ behaves as a perfect electric conductor and, in
addition, region $B$ corresponds to a $p-$Laplacian modelled material.

######  Theorem 5.8.

Let $1<p<q<+\infty$, $f\in X^{p}_{\diamond}(\partial\Omega)$ and $v^{\lambda}$
be the solution of (5.21). If (A1), (A2), (A3) and (A4) hold, then

  * $v^{\lambda}\rightharpoonup{w}$ in $W^{1,p}(\Omega)$, as $\lambda\to+\infty$,

where ${w}\in W^{1,p}(\Omega)$ is the unique solution of (5.3). Moreover, we
have

  * $v^{\lambda}\to{w}$ in $W^{1,q}(A)$, as $\lambda\to+\infty$.

###### Proof.

For the sake of simplicity, we will only treat the case when $A$ has one
connected component. Let ${w}\in W^{1,p}(\Omega)$ the solution of (5.3). By
(5.4) of Lemma 5.2, we have

$\int_{B}|\nabla
v^{\lambda}(x)|^{p}dx\leq\frac{\overline{Q}}{\underline{Q}}\left(\int_{B}|\nabla{w}(x)|^{p}dx+\frac{{E_{0}^{p}}}{\lambda^{p}}|B|\right)+\frac{{E_{0}^{p}}}{\lambda^{p}}|B|.$
(5.22)

Hence the left hand side in (5.22) is upper bounded. Moreover, we have

$\begin{split}\lambda^{q-p}\int_{A}|\nabla
v^{\lambda}(x)|^{q}dx&\leq\frac{{E_{0}^{q}}}{\lambda^{p}}|A|+\frac{E_{0}^{q}}{\underline{Q}\lambda^{p}}\int_{A}Q_{A}(x,|\lambda\nabla
v^{\lambda}(x)|)dx\\\
&\leq\frac{{E_{0}^{q}}}{\lambda^{p}}|A|+\frac{E_{0}^{q}}{\underline{Q}}\mathbb{G}^{\lambda}(v^{\lambda})\\\
&\leq\frac{{E_{0}^{q}}}{\lambda^{p}}|A|+\frac{\overline{Q}}{\underline{Q}}E_{0}^{q-p}\int_{B}|\nabla{w}(x)|^{p}dx+\frac{\overline{Q}}{\underline{Q}}\frac{{{E_{0}^{q}}}}{\lambda^{{p}}}|B|.\end{split}$
(5.23)

where in the first inequality we used the left-hand side of assumption
(A3.ii), in the second inequality we used the definition of
$\mathbb{G}^{\lambda}$ and in the third inequality we exploited (5.5) of Lemma
5.2. From (5.22) and (5.23), it is clear that $v^{\lambda}\in
W^{1,p}(\Omega)\cap W^{1,q}(A)$ and that $||v^{\lambda}||_{W^{1,p}(\Omega)}$
is upper bounded. Therefore, taking into account that $v^{\lambda}=f$ on
$\partial\Omega$ for any $\lambda>0$, there exists a function $v^{\infty}\in
W^{1,p}(\Omega)$ such that $v^{\lambda}\rightharpoonup v^{\infty}$ in
$W^{1,p}(\Omega)$, up to a subsequence, with $v^{\infty}=f$ on
$\partial\Omega$. Finally, from (5.23), we find that $\int_{A}|\nabla
v^{\lambda}(x)|^{q}dx=O(\lambda^{p-q})$. Therefore, $v^{\lambda}\to
v^{\infty}$ in $W^{1,{q}}(A)$ and $v^{\infty}$ is constant in $A$ because
$\nabla v^{\lambda}\to 0$ in $L^{q}(A)$. Since assumption (A4) holds, then we
are in position to apply Proposition 4.3 which gives

$\mathbb{B}(v^{\infty})\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx.$ (5.24)

Consequently, we have the following inequalities

$\begin{split}\mathbb{B}(v_{\Omega})\leq\mathbb{B}(v^{\infty})&\leq\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,|\lambda\nabla
v^{\lambda}(x)|)dx\\\
&\leq\liminf_{\lambda\to+\infty}\mathbb{G}^{\lambda}(v^{\lambda})\leq\lim_{\lambda\to+\infty}\mathbb{G}^{\lambda}({w})=\
\mathbb{B}({w}),\end{split}$ (5.25)

where in the first inequality we used $v^{\infty}$ as the test function in
$\mathbb{B}$, in the second inequality we used (5.24), in the third inequality
we used the definition of $\mathbb{G}^{\lambda}$, in the fourth inequality we
used $w$ as the test function in $\mathbb{G}^{\lambda}$, and in the equality
we used assumption (A4) and the dominated convergence Theorem. This implies
that $\mathbb{B}({w})=\mathbb{B}(v^{\infty})$ and hence $v^{\infty}={w}$ by
the uniqueness of the solution of problem (5.3). ∎

######  Remark 5.9.

From (5.25), we find that the fundamental inequality (5.24), also stated in
Proposition 4.3, holds as an equality:

$\mathbb{B}(v^{\infty})=\liminf_{\lambda\to+\infty}\frac{1}{\lambda^{p}}\int_{B}Q_{B}(x,\lambda|\nabla
v^{\lambda}(x)|)dx.$

##  6\. The pointwise convergence assumptions in the limiting case

The main aim of this Section is to prove that assumption (A4) for large
Dirichlet data is sharp. Specifically, we provide a counterexample where (A4)
does not hold, while (A1), (A2) and (A3) still hold. For this case, we prove
that, for the Dirichlet energy (3.5), the convergence results of Theorems 5.3
and 5.8 (and hence of Theorem 5.6) do not hold. With a similar approach, not
reported here for the sake of brevity, it is possible to prove that even
assumption (A4’) is sharp. In order to prove this result, we need to build a
Dirichlet energy function $Q_{B}$ such that the ratio $Q_{B}(x,E)/E^{p}$ does
not admit the limit for $E\to+\infty$. This is the case when the ratio
$Q_{B}(x,E)/E^{p}$ oscillates between two different values. In the following
Lemma we prove that a Dirichlet Energy density of this type exists (see Figure
8 for the geometrical interpretation).

Figure 8. The continuous line represents the function $\Phi$ for the counter
example. It gives the Dirichlet Energy density as $\Psi(E)=\Phi(E)+E^{2}$.

######  Lemma 6.1.

Let $L>1$, then there exist two sequences

$\lambda_{n}^{\prime}\uparrow+\infty\quad\text{and}\quad\lambda_{n}^{\prime\prime}\uparrow+\infty$

such that

$L\lambda^{\prime\prime}_{n}<\lambda^{\prime}_{n},\ \
L\lambda^{\prime}_{n}<\lambda^{\prime\prime}_{n+1}\quad\forall
n\in\mathbb{N},$

and a strictly convex function $\Psi:[0,+\infty[\to[0,+\infty[$ such that

$\Psi|_{[\lambda_{n}^{\prime},L\lambda_{n}^{\prime}]}(E)=2E^{2}\quad\Psi|_{[\lambda_{n}^{\prime\prime},L\lambda_{n}^{\prime\prime}]}(E)=3E^{2}.$

###### Proof.

Let us fix $\lambda_{1}^{\prime\prime}>0$. For each $n\in\mathbb{N}$ we set
the auxiliary function $\Phi$ equal to $2E^{2}$ in
$(\lambda_{n}^{\prime\prime},L\lambda_{n}^{\prime\prime})$ and equal to
$E^{2}$ in $(\lambda_{n}^{\prime},L\lambda_{n}^{\prime})$. In interval
$(L\lambda_{n}^{\prime\prime},\lambda_{n}^{\prime})$ the function $\Phi$ is
equal to the tangent line to function $2E^{2}$ evaluated at
$L\lambda_{n}^{\prime\prime}$. Point $\lambda_{n}^{\prime}$ is found at the
intersection of this tangent line with function $E^{2}$. In interval
$(L\lambda_{n}^{\prime},\lambda_{n+1}^{\prime\prime})$ the function $\Phi$ is
a straight line, continuous at $L\lambda_{n}^{\prime}$ and tangent to
$2E^{2}$. Point $\lambda_{n+1}^{\prime\prime}$ is found as the abscissa of the
tangent point between this straight line and function $2E^{2}$. This procedure
is applied iteratively from $n=1$. Function $\Phi$ turns out to be convex and
sequences $\\{\lambda_{n}^{\prime}\\}_{n\in\mathbb{N}}$ and
$\\{\lambda_{n}^{\prime\prime}\\}_{n\in\mathbb{N}}$ are monotonically
increasing to infinity. Therefore, the measure of intervals where $\Phi$ is
equal to $E^{2}$ or equal to $2E^{2}$ is nonvanishing. It is possible to prove
that the provided sequences $\\{\lambda^{\prime}_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\lambda_{n}^{\prime\prime}\\}_{n\in\mathbb{N}}$ are two geometrical
progressions. Indeed

$\lambda_{n}^{\prime}=c_{1}L\lambda_{n}^{\prime\prime},\
\lambda_{n+1}^{\prime\prime}=c_{2}L\lambda_{n}^{\prime},\
\lambda_{n+1}^{\prime}=C^{2}L^{2}\lambda_{n}^{\prime},\
\lambda_{n+1}^{\prime\prime}=C^{2}L^{2}\lambda_{n}^{\prime\prime},$

where $c_{1}=2+\sqrt{2}$, $c_{2}=1+\frac{\sqrt{2}}{2}$ and $C$ is the
geometrical average of $c_{1}$ and $c_{2}$, that is
$C^{2}=\left(3+2\sqrt{2}\right)$. Finally, we set $\Psi(E)=\Phi(E)+E^{2}$.
$\Psi$ is a strictly convex function. ∎

The Dirichlet energy density defined as $Q_{B}(x,E)=\Psi(E)$, satisfies all
the assumptions except (A4). This energy density is the basis to build a
counterexample proving that (A4) is sharp. Specifically, we consider a 2D case
($n=2$) and $p=2$ in the outer region. The asymptotic growth exponent $q$
satisfies condition $1<q<\infty$. Let $r$ be greater than or equal to 10, and
let the outer region $B$ be the annulus centred in the origin with radii $1$
and $r$. This annulus is $D_{r}\setminus\overline{D}_{1}$, where $D_{r}$ and
$D_{1}$ are the disks of radii $r$ and $1$, respectively, and centered at the
origin. The inner region is, therefore, $D_{1}$. We focus on problem (3.5),
where the Dirichlet energy density is defined as

$\begin{split}Q_{B}(x,E)&=\Psi(E)\quad\text{in}\
D_{r}\setminus\overline{D}_{1}\times[0,+\infty[,\qquad\\\
Q_{A}(x,E)&=E^{q}\qquad\text{in}\ D_{1}\times[0,+\infty[.\end{split}$

Let $\gamma$ be defined as $\gamma=7+\frac{12}{r^{2}}$. We denote
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$ and we consider the problem

$\min_{\begin{subarray}{c}v\in W^{1,q}(D_{r})\\\ v=\gamma x_{1}\ \text{on}\
\partial
D_{r}\end{subarray}}\mathbb{G}^{\lambda}(v),\quad\mathbb{G}^{\lambda}(v)=\frac{1}{\lambda^{2}}\left(\int_{D_{r}\setminus
D_{1}}\Psi(\lambda|\nabla v(x)|)dx+\int_{D_{1}}\lambda^{q}|\nabla
v(x)|^{q}dx\right).$ (6.1)

Here we prove that $\lim_{\lambda\to+\infty}\mathbb{G}^{\lambda}(v^{\lambda})$
does not exist. Specifically, the two sequences
$\\{\lambda_{n}^{\prime}\\}_{n\in\mathbb{N}}\uparrow+\infty$ and
$\\{\lambda_{n}^{\prime\prime}\\}_{n\in\mathbb{N}}\uparrow+\infty$ of Lemma
6.1 give

$\limsup_{n\to+\infty}\mathbb{G}^{\lambda_{n}^{\prime}}(v^{\lambda_{n}^{\prime}})\leq\ell_{1}<\ell_{2}\leq\liminf_{n\to+\infty}\mathbb{G}^{\lambda_{n}^{\prime\prime}}(v^{\lambda_{n}^{\prime\prime}}).$
(6.2)

As usual, $v^{\lambda}$ is the solution of (6.1). Let us consider the
following problem

$\displaystyle\min_{\begin{subarray}{c}v\in
H^{1}(D_{r}\setminus\overline{D}_{1})\\\ v=\gamma x_{1}\ \text{on}\ \partial
D_{r}\\\ v=const.\ \text{on}\ \partial
D_{1}\end{subarray}}\mathbb{C}(v),\quad\mathbb{C}(v)=\int_{D_{r}\setminus
D_{1}}|\nabla v(x)|^{2}dx.$ (6.3)

The symmetry of the domain and the zero average of the boundary data imply
that the constant appearing in (6.3) on $\partial D_{1}$ is zero. An easy
computation reveals that

$v_{D_{r}}(x)=\frac{7r^{2}+12}{r^{2}-1}\left(1-\frac{1}{x_{1}^{2}+x_{2}^{2}}\right)x_{1}\quad\text{in}\
D_{r}\setminus\overline{D}_{1}$

is the solution of (6.3), that $\Delta v_{D_{r}}=0$ in
$D_{r}\setminus\overline{D}_{1}$, and that we have

$\begin{split}\frac{7r^{2}-12}{r^{2}-1}\left(1-\frac{1}{\rho^{2}}\right)\leq|\nabla
v_{D_{r}}(x)|\leq\frac{7r^{2}+12}{r^{2}-1}\left(1+\frac{1}{\rho^{2}}\right)\quad\text{on}\
\partial D_{\rho},\ 1<\rho\leq r.\end{split}$

Consequently, when $\rho\geq 2$, we have

$1\leq\frac{3}{4}\frac{7r^{2}-12}{r^{2}-1}\leq|\nabla
v_{D_{r}}(x)|\leq\frac{5}{4}\frac{7r^{2}+12}{r^{2}-1}\leq 10\quad\text{in}\
D_{r}\setminus D_{2}.$ (6.4)

Let $L$ be greater than 10, $\lambda_{n}^{\prime}\uparrow+\infty$ and let
$\lambda_{n}^{\prime\prime}\uparrow+\infty$ be the two sequences of Lemma 6.1.
It turns out that

$\lambda_{n}^{\prime}\leq\lambda_{n}^{\prime}|\nabla v_{D_{r}}(x)|\leq
L\lambda^{\prime}_{n}\quad\text{in}\ D_{r}\setminus D_{2}.$ (6.5)

We have

$\begin{split}&\limsup_{n\to+\infty}\mathbb{G}^{\lambda^{\prime}_{n}}(v^{\lambda_{n}^{\prime}})\leq\limsup_{n\to+\infty}\mathbb{G}^{\lambda^{\prime}_{n}}(v_{D_{r}})\\\
&=\limsup_{n\to+\infty}\frac{1}{(\lambda^{\prime}_{n})^{2}}\int_{D_{r}\setminus
D_{2}}\Psi(\lambda^{\prime}_{n}|\nabla
v_{D_{r}}(x)|)dx+\limsup_{n\to+\infty}\frac{1}{(\lambda^{\prime}_{n})^{2}}\int_{D_{2}\setminus
D_{1}}\Psi(\lambda^{\prime}_{n}|\nabla v_{D_{r}}(x)|)dx\\\ &\leq
2\int_{D_{r}\setminus D_{2}}|\nabla v_{D_{r}}(x)|^{2}dx+3\int_{D_{2}\setminus
D_{1}}|\nabla v_{D_{r}}(x)|^{2}dx,\end{split}$ (6.6)

where in the first line we used the minimality of $v^{\lambda^{\prime}_{n}}$
for $\mathbb{G}^{\lambda^{\prime}_{n}}$ and that $v_{D_{r}}$ is an admissible
function for problem (6.1), in the second line we exploited the property that
the gradient of $v_{D_{r}}$ in $D_{1}$ is vanishing, and in the third line we
used (6.5). By setting $\ell_{1}$ equal to (6.6):

$\ell_{1}:=2\int_{D_{r}\setminus D_{2}}|\nabla
v_{D_{r}}(x)|^{2}dx+3\int_{D_{2}\setminus D_{1}}|\nabla v_{D_{r}}(x)|^{2}dx,$

we have the leftmost inequality in (6.2). To obtain the rightmost inequality
in (6.2), we consider the following problems

$\min_{\begin{subarray}{c}v\in H^{1}(D_{r}\setminus\overline{D}_{1})\\\
v=\gamma x_{1}\ \text{on}\ \partial
D_{r}\end{subarray}}\mathbb{H}^{\lambda}(v),\quad\mathbb{H}^{\lambda}(v)=\frac{1}{\lambda^{2}}\int_{D_{r}\setminus
D_{2}}\Psi(\lambda|\nabla v(x)|)dx+2\int_{D_{2}\setminus D_{1}}|\nabla
v(x)|^{2}dx.$ (6.7) $\min_{\begin{subarray}{c}v\in
H^{1}(D_{r}\setminus\overline{D}_{1})\\\ v=\gamma x_{1}\ \text{on}\ \partial
D_{r}\end{subarray}}\mathbb{D}(v),\quad\mathbb{D}(v)=3\int_{D_{r}\setminus
D_{2}}|\nabla v(x)|^{2}dx+2\int_{D_{2}\setminus D_{1}}|\nabla v(x)|^{2}dx.$
(6.8)

The unique solution of (6.8) is

$w_{D_{r}}(x)=\begin{cases}\left(7+\frac{12}{x_{1}^{2}+x_{2}^{2}}\right)x_{1}&\quad\text{in}\
D_{r}\setminus\overline{D}_{2}\\\
8\left(1+\frac{1}{x_{1}^{2}+x_{2}^{2}}\right)x_{1}&\quad\text{in}\
D_{2}\setminus\overline{D}_{1}.\end{cases}$

Analogously to (6.4), it can be easily proved that

$1\leq 4\leq\left(7-\frac{12}{\rho^{2}}\right)\leq|\nabla
w_{D_{r}}(x)|\leq\left(7+\frac{12}{\rho^{2}}\right)\leq 10<L\quad\text{on}\
\partial D_{\rho},\ 2\leq\rho\leq r.$

and hence we choose $L>10$ such that

$\lambda_{n}^{\prime\prime}\leq\lambda_{n}^{\prime\prime}|\nabla
w_{D_{r}}(x)|\leq L\lambda^{\prime\prime}_{n}\quad\text{in}\ D_{r}\setminus
D_{2}.$ (6.9)

Therefore, we have

$\mathbb{G}^{\lambda_{n}^{\prime\prime}}(v^{\lambda_{n}^{\prime\prime}})\geq\mathbb{H}^{\lambda^{\prime\prime}_{n}}(w_{D_{r}})=\mathbb{D}(w_{D_{r}}),$
(6.10)

where the inequality comes from the definition of $\Psi$. The equality follows
from the fact that $\mathbb{H}^{\lambda^{\prime\prime}_{n}}$ coincides with
$\mathbb{D}$ by (6.9) and the definition of $\Psi$. We highlight that
$w_{D_{r}}$ is a local minimizer in $W^{1,\infty}(D_{r})\cap W^{1,2}(D_{r})$,
since $\Psi$ does not depend on $x$ (see [68] for details). Finally,
$w_{D_{r}}$ is a global minimizer thanks to the uniqueness of (6.7). By
setting

$\ell_{2}(r):=\mathbb{D}(w_{D_{r}}).$

we have the rightmost inequality in (6.2) by passing to the limit in (6.10).
At this stage, it only remains to be proved that $\ell_{1}(r)<\ell_{2}(r)$. To
this purpose, we notice that:

$\begin{split}\ell_{1}(r)&=2\int_{D_{r}\setminus D_{2}}|\nabla
v_{D_{r}}(x)|^{2}dx+3\int_{D_{2}\setminus D_{1}}|\nabla v_{D_{r}}(x)|^{2}dx\\\
\ell_{2}(r)&=3\int_{D_{r}\setminus D_{2}}|\nabla
w_{D_{r}}(x)|^{2}dx+2\int_{D_{2}\setminus D_{1}}|\nabla
w_{D_{r}}(x)|^{2}dx.\end{split}$

Condition $\ell_{1}(r)<\ell_{2}(r)$ holds for large $r$, by observing that (i)
$v_{D_{r}}$ and $w_{D_{r}}$ solve the same associated Euler-Lagrange equation
on $D_{2}\setminus\overline{D}_{1}$, (ii) $\nabla v_{D_{r}}(x)$ and $\nabla
w_{D_{r}}(x)$ are bounded functions on the bounded domain $D_{2}\setminus
D_{1}$ by (6.5) and (6.9), respectively, and (iii) it turns out that

$\begin{split}&\lim_{r\to+\infty}\frac{\int_{D_{r}\setminus D_{2}}|\nabla
v_{D_{r}}(x)|^{2}dx}{\int_{D_{r}\setminus D_{2}}|\nabla
w_{D_{r}}(x)|^{2}dx}=1,\\\ &\lim_{r\to+\infty}\int_{D_{r}\setminus
D_{2}}|\nabla v_{D_{r}}(x)|^{2}dx=\lim_{r\to+\infty}\int_{D_{r}\setminus
D_{2}}|\nabla w_{D_{r}}(x)|^{2}dx=+\infty.\end{split}$

##  7\. Conclusions

This study is motivated by Inverse Problems in the presence of nonlinear
materials, where the treatment of nonlinear constitutive relationships is
still at an early stage of development, as clearly stated in [3]. We focus on
Electrical Resistance Tomography where the aim is to retrieve the electrical
conductivity/resistivity of a material by means of stationary (DC) currents.
Our main results prove that the original nonlinear problem can be replaced by
a proper $p-$Laplace problem, when the prescribed Dirichlet data are “large”.
Specifically, we prove that in the presence of two materials with different
asymptotic growth, the scalar potential in the outer region in contact with
the boundary where the Dirichlet data is prescribed, can be computed by (i)
replacing the interior region with either a Perfect Electric Conductor or a
Perfect Electric Insulator and (ii) replacing the original problem (material)
in the outer region with a weighted $p-$Laplace problem. In a certain sense,
the presence of the “fingerprint” of a weighted $p-$Laplace problem can be
recognized in an arbitrary nonlinear problem. From the perspective of
tomography, this is a significant result because it highlights the central
role played by the weighted $p-$Laplacian in inverse problems with nonlinear
materials. For $p=2$, i.e. when the material in the outer region is linear,
these results constitute a powerful bridge making it possible to bring all
theoretical results, imaging methods and algorithms developed for linear
materials into the arena of problems with nonlinear materials. The fundamental
tool to prove the convergence results are the inequalities appearing in
Propositions 4.3. They express the asymptotic behaviour of the Dirichlet
Energy for the outer region in terms of a factorized $p-$Laplacian form.
Moreover, we prove that our assumptions are sharp, by means of proper
counterexamples. Finally, it would be interesting to provide a numerical
example, referring to a superconducting cable, as an application of the
theoretical results proved in this paper.

## Acknowledgements

This work has been partially supported by the MiUR-Progetto Dipartimenti di
eccellenza 2018-2022 grant “Sistemi distribuiti intelligenti”of Dipartimento
di Ingegneria Elettrica e dell’Informazione “M. Scarano”, by the MiSE-FSC
2014-2020 grant “SUMMa: Smart Urban Mobility Management” and by GNAMPA of
INdAM.

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