# Monotonicity principle for the imaging of piecewise nonlinear materials

Antonio Corbo Esposito1, Luisa Faella1, Vincenzo Mottola1, 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 Scienze Economiche, Giuridiche, Informatiche e Motorie,
Università degli Studi di Napoli Parthenope, Via Guglielmo Pepe, Rione Gescal,
80035 Nola (NA), 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><EMAIL_ADDRESS>(corresponding author),
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is focused on the Monotonicity Principle (MP) for nonlinear
materials with piecewise growth exponent. This results are relevant because
enables the use of a fast imaging method based on MP, to the wide class of
problems with two or more materials, where at least one is nonlinear.

The treatment is very general and allows to model a wide variety of practical
configurations such as, for instance, Superconducting (SC) or Perfect
Electrical Conducting (PEC) or Perfect Electrical Insulating (PEI) materials.

A key role is played by the average Dirichlet-to-Neumann operator, introduced
in [16], where the MP for a single type of nonlinearity was treated.

Realistic numerical examples confirm the theoretical findings.

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 Scienze Economiche, Giuridiche, Informatiche e Motorie,
Università degli Studi di Napoli Parthenope, Via Guglielmo Pepe, Rione Gescal,
80035 Nola (NA), 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><EMAIL_ADDRESS>(corresponding author),
<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

This contribution is in the framework of a nonlinear generalization of the
Calderon problem [4, 11, 12]. Specifically, it consists in retrieving the
electrical conductivity $\sigma=\sigma(x,|\nabla u(x)|)$ appearing in the
nonlinear Laplace type equation:

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

where $f$ is the boundary data and $\Omega$ is a region filled by an
electrical conducting material. The nonlinear electrical conductivity $\sigma$
in $\Omega$ is retrieved tomographically from the knowledge of boundary data
evaluated on $\partial\Omega$. Specifically, the electrical conductivity is
retrieved from the knowledge of the Dirichlet-to-Neumann operator

$\Lambda_{\sigma}:f(x)\ \mapsto\ \sigma(x,|\nabla
u^{f}(x)|)\partial_{n}u^{f}(x),$

that is the operator mapping the applied boundary potential $f$ on
$\partial\Omega$, into the electrical current density entering the domain
$\Omega$ through its boundary $\partial\Omega$. As it is well known, this is a
strongly ill-posed inverse problem that, moreover, presents nonlinear
constitutive relationships. In the case of linear materials, i.e.
$\sigma=\sigma(x)$, the Dirichlet-to-Neumann (DtN) operator $\Lambda_{\sigma}$
is of paramount importance for retrieving $\sigma$ from boundary data. In this
framework, a relevant Monotonicity Property was found by Gisser, Isaacson and
Newell in [28]:

$\sigma_{1}(x)\leq\sigma_{2}(x)\text{ for a.e.
}x\in\Omega\quad\Longrightarrow\quad\left\langle{\Lambda}_{\sigma_{1}}f,f\right\rangle\leq\left\langle{\Lambda}_{\sigma_{2}}f,f\right\rangle\quad\forall
f.$ (1.2)

Then, Tamburrino and Rubinacci [58, 59] exploited the negation of (1.2):

$\exists f\ :\
\left\langle{\Lambda}_{\sigma_{1}}f,f\right\rangle\not\leq\left\langle{\Lambda}_{\sigma_{2}}f,f\right\rangle\quad\Longrightarrow\quad\sigma_{1}(x)\not\leq\sigma_{2}(x)\text{
for a.e. }x\in\Omega$

to develop a real-time imaging method, the so-called Monotonicity Principle
Method (MPM), to solve the inverse obstacle problems, where one aims to
reconstruct the shape of the support of anomalies in a given background, from
boundary data. The four features rendering the MPM highly suitable for solving
inverse problems are: (i) the computational cost is negligible making the
method suitable for online and real-time applications, (ii) the processing can
be carried out in parallel, (iii) under proper assumptions, the MPM provides
rigorous upper and lower bounds to the regions occupied by the unknown
anomalies, even in the presence of noise and a finite number of measurements
(see [64], based on [32, 58]), and (iv) MPM gives an exact reconstruction of
the outer boundary of the unknown anomalous region, in the ideal setting of
complete knowledge of the boundary measurement operator and noise free data
(see [2, 19, 29, 31]). The MP and the related imaging method MPM are general
physical/mathematical features which appear in many physical problems governed
by different PDEs, from stationary PDEs to evolutive PDEs and hyperbolic
evolutive PDEs. MPM was originally developed for stationary PDEs arising from
static problems (such as, Electrical Resistance Tomography, Electrical
Capacitance Tomography, Eddy Current Tomography and Linear Elastostatics
Tomography) [13, 24, 31, 54, 58, 60], and for stationary PDEs arising as the
limit of quasi-static problems, such as Eddy Current Tomography for either
large or small skin depth operations [54, 59, 65]. Other monotonicity-based
reconstruction methods can be found in [23] for reconstructing piecewise
constant layered conductivities and in [6] for the Helmholtz equation in a
closed cylindrical waveguide with penetrable scattering objects. MP was
discovered and applied to tomography for problems governed by parabolic
evolutive PDEs, (see [52, 53, 57, 61, 62, 63]), to phenomena governed by
Helmotz equations arising from wave propagation problems (hyperbolic evolutive
PDEs) in time-harmonic operations (see [2, 3, 19, 29, 55]), and to the
quasilinear generalizations of the classical biharmonic operator (see [36]).
MP for the Helmoltz equation arising from (steady-state) optical diffuse
tomography was introduced in [42]. Regularization for MPM is introduced in
[25, 46]. The MPM has been applied to many different engineering problems. The
first experimental validation of the MPM for Eddy Current Tomography is shown
in [56]. Other results of the MPM applied to Tomography and Nondestructive
Testing, can be found in [51, 66]. MPM is applied to locate breast cancer via
Electrical Impedance Tomography in [22]. MPM is applied to the tomography of
two-phase materials by means of either Electrical Capacitance Tomography,
Electrical Resistance Tomography, or Magneto Inductance Tomography in [50].
Finally, MPM has also been applied to the homogenization of materials [41] and
to the inspection of concrete rebars via Eddy Current Testing [20, 47]. It is
worth noting that regardless the underlying governing equation and/or the
specific measurement technique, at the foundation of the MPM imaging method,
there is some form of the Monotonicity Principle. The subject of inverse
problems with nonlinear materials is a new area of research, as clearly stated
in [37]: “… the mathematical analysis for inverse problems governed by
nonlinear Maxwell’s equations is still in the early stages of development.”.
In this framework, the specific case of $p$-Laplacian is treated in [9, 14,
30, 48]. The most comprehensive contribution in the area of inverse problems
with quasilinear constitutive relationships (1.1) is [16], where the
Monotonicity Principle for the (quasilinear) elliptic case is proved.
Specifically, the authors introduced the new operator
$\overline{\Lambda}_{\sigma}$, termed as the average DtN, defined as:

$\overline{\Lambda}_{\sigma}:f\in
X^{p}_{\diamond}(\partial\Omega)\mapsto\int_{0}^{1}\Lambda_{\sigma}\left(\alpha
f\right)\text{d}\alpha\in X^{p}_{\diamond}(\partial\Omega)^{\prime},$

where $X^{p}_{\diamond}(\partial\Omega)$ is a proper functional space
introduced in Section 2. The average DtN is of paramount importance because it
satisfies a MP (see [16, Th. 4.3]), unlike the “classical” DtN which does not
satisfy the MP, apart from the very special case of the $p-$Laplacian. In [16]
it was treated the case of a quasilinear constitutive relationship where the
electrical conductivity has a growth exponent∗222∗We say that a function
$Q(E)$ has a $p-$growth if $0<c_{0}\leq|\sigma(E)/E^{p}|\leq c_{1}<+\infty$,
for large $E>0$ and for some proper constants $c_{0}$ and $c_{1}$. $2\leq
p<+\infty$ which is constant over the domain $\Omega$. The original content of
the present contribution is twofold and consists in the treatment of a much
larger class of materials. Specifically, we consider materials made by
different phases where, in each phase, the material has either a different
growth exponent, or is a Perfect Electric Conductor (PEC), or is a Perfect
Electric Insulator (PEI). Also, growth exponents $1<p<2$ are accounted for.
This latter case is relevant for including in the treatment type II
supeconductors [45]. This mathematical generalization is relevant for real-
world applications, where one can found either problems where the growth
exponent is smaller than 2, and/or it is non constant over $\Omega$, and/or
some regions are well approximated by either a PEC or a PEI. For instance,
type II superconductors [1, 8, 5, 27] can be modeled by a growth exponent
$1<p<2$. From the mathematical standpoint, this means that the electrical
conductivity $\sigma$ is bounded by monotonically decreasing functions (growth
exponent $p-2$), rather than monotonically increasing functions, as for $2\leq
p<+\infty$. Regarding the non constancy of the growth exponent, this is a case
appearing in many practical configurations involving two or more different
materials. For instance, one may have a nonlinear material embedded in a
linear material. In all these cases the non constant growth exponent $p$
results to be piecewise constant. Similarly, PEC (infinite electrical
conductivity) and PEI (vanishing electrical conductivity) are excellent
approximations of real-world applications, because the electrical conductivity
varies over a large range of orders of magnitude [33]. PEC and PEI appear
naturally in nonlinear problems where the boundary data is either large or
small enough (see [15, 17]). The paper is organized as follows: in Section 2
we present the problem; in Section 3 we give a fundamental convergence result
for the solutions; in Section 4 we study the connection between the Dirichlet
Energy and the Dirichlet-to-Neumann operator; in Section 5 we prove the
Monotonicity Principles for this very general setting. In Section 6 we provide
numerical validations of the previous results and in Section 7, we give the
conclusions.

##  2\. A unified treatment for Nonlinear Inverse Problems

This section presents a general and unified framework for treating the steady
ohmic conduction, i.e. problem (1.1), in the presence of materials made by
different phases. For the sake of simplicity we refer to materials made by two
different phases, however the treatment is general and it can be extended to
materials consisting in multiple phases. We first introduce the basic
notations, than the formal statement of the problem, the required assumptions
and, finally, we prove existence and uniqueness of the solution.

###  2.1. Notations

Throughout this paper, $\Omega$ is the region occupied by the conducting
material. We assume $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, to be an open
bounded domain (i.e. open and connected set) with Lipschitz continuous
boundary. Hereafter we assume that the first material occupies the open
bounded set $A\subset\subset\Omega$, with a Lipschitz continuous boundary and
made by a finite number of connected components $A_{i}$, for $i=1,...,M$. The
second material occupies $B:=\Omega\setminus\overline{A}$, which is still a
domain (see Figure 1). We denote by $\mathbf{\hat{n}}$ the outer unit normal
defined on $\partial\Omega$, by $\langle\cdot,\cdot\rangle$ the duality
product on $L^{2}(\partial\Omega)$ and 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, for any bounded Lipschitz
open set, is a Besov space (refer to [35, 38]), 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 $|u|_{B^{1-\frac{1}{p},p}(\partial\Omega)}$ is the Slobodeckij seminorm:

$|u|_{B^{1-\frac{1}{p},p}(\partial\Omega)}=\left(\int_{\partial\Omega}\int_{\partial\Omega}\frac{|u(x)-u(y)|^{p}}{||x-y||^{N-1+(1-\frac{1}{p})p}}dS(y)dS(x)\right)^{\frac{1}{p}},$

see Definition 18.32, Definition 18.36 and Exercise 18.37 in [38]. This
guarantees the existence of a function in $W^{1,p}(\Omega)$ whose trace is $f$
[38, Th. 18.40]. For the sake of brevity, we denote this space by
$X^{p}(\partial\Omega)$ and 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 $f-g\in W^{1,p}_{0}(\Omega)$,
see [38, Th. 18.7]. Finally, we denote by $X^{p}_{\diamond}(\partial\Omega)$
the set of elements in $X^{p}(\partial\Omega)$ with vanishing integral mean on
$\partial\Omega$ with respect to the measure $dS$.

###  2.2. Statement of the Problem

Our aim is to prove a Monotonicity Principle for two-phase materials operating
in steady ohmic conduction. In this regime, the electric field $\bf E$ is
expressed in terms of the electric scalar potential as ${\bf E}(x)=-\nabla
u(x)$. For a quasilinear material the Ohm’s law ${\bf J}(x)=\sigma(x,E(x)){\bf
E}(x)$ is expressed as

${\bf J}(x)=-\sigma(x,|\nabla u(x)|)\nabla u(x)\quad\text{in }\Omega,$ (2.1)

where $\sigma$ is the electrical conductivity, and ${\bf J}$ is the electric
current density. A quasilinear material reduces to a linear material when
$\sigma$ does not depend on the electric field, i.e. $\sigma=\sigma(x)$. For a
PEC we have $\bf E=\bf 0$, i.e. $\nabla u=\bf 0$, whereas for a PEI we have
$\bf J=\bf 0$, i.e. $\sigma(x,|\nabla u(x)|)\nabla u(x)=\bf 0$. It is worth
noting that a PEC corresponds to $\sigma=+\infty$, whereas a PEI corresponds
to $\sigma=0$. We assume that the material in $B$ has a $p-$growth
($1<p<+\infty$) and that the material in $A$ has either a $q-$growth, or is a
PEC, or is a PEI.

Figure 1. The outer region admits $p$ as growth exponent, the inner region
admits $q$ as growth exponent.

This setting is quite general from the mathematical point of view and it is
very versatile from the physical/engineering point of view, as showed in the
Table 1, which summarizes the cases accounted in this contribution.
Specifically, we highlight that: (i) a growth exponent smaller than 2
corresponds to an electrical conductivity monotonically decreasing with the
magnitude of the electric field $E$, (ii) a growth exponent equal to 2
corresponds to an electrical conductivity independent upon $E$, i.e. a linear
material, and (iii) a growth exponent larger than 2 corresponds to an
electrical conductivity increasing with $E$ (see Section 2.3). The setting
also include cases where the electrical conductivity is increasing or constant
with the electric field in one phase and is decreasing in the other phase, for
instance $p\geq 2$ and $1<q<2$. Other cases included in this treatment refer
to materials where the order relation between the electrical conductivities in
regions $A$ and $B$ changes with the value of the magnitude of the electric
field $E$. Also, we notice that $p=q$ corresponds to $A=\emptyset$. In
addition, for $p=q=2$ this case corresponds to [28, 58], whereas for $p=q>2$
it corresponds to [16].

$B$ $A$ | $1<q<2$ | $q=2$ | $2<q<+\infty$ | PEC | PEI
---|---|---|---|---|---
$1<p<2$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$p=2$ | $\bullet$ | [28, 58] | $\bullet$ | $\bullet$ | $\bullet$
$2<p<+\infty$ | $\bullet$ | $\bullet$ |  $\bullet$, ([16] for $p=q$) | $\bullet$ | $\bullet$
Table 1. Scheme of all cases accounted with this general framework. The symbol
$\bullet$ stays for “yes”, i.e. the specific combination of materials in
phases $A$ and $B$ can be treated within this setting. References are reported
for those specific cases treated in past works.

###  2.3. The assumptions

The well-posedness of the forward problem in the sense of Hadamard is the
minimal requirement to formulate the associated inverse problem. This is
guaranteed by the following assumptions on
$\sigma_{B}:B\times[0,+\infty)\to\mathbb{R}$ and
$\sigma_{A}:A\times[0,+\infty)\to\mathbb{R}$. Firstly, we recall the
definition of the Carathéodory function.

######  Definition 2.1.

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

  * •

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

  * •

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

  * (A1)

$\sigma_{B}$ and $\sigma_{A}$ are Carathéodory function.

  * (A2)

$E\in\ [0,+\infty)\mapsto\sigma_{B}(x,E)E$ is strictly increasing for a.e.
$x\in B$,
$E\in\ [0,+\infty)\mapsto\sigma_{A}(x,E)E$ is strictly increasing for a.e.
$x\in A$.

  * (A3)

For fixed $1<p<+\infty$, there exists three positive constants
$\underline{\sigma}\leq\overline{\sigma}$ and $E_{0}$ such that:

$\begin{split}&\underline{\sigma}\left(\frac{E}{E_{0}}\right)^{p-2}\leq\sigma_{B}(x,E)\leq\overline{\sigma}\left[1+\left(\frac{E}{E_{0}}\right)^{p-2}\right]\qquad\text{if}\
p\geq 2,\\\
&\underline{\sigma}\left(\frac{E}{E_{0}}\right)^{p-2}\leq\sigma_{B}(x,E)\leq\overline{\sigma}\left(\frac{E}{E_{0}}\right)^{p-2}\qquad\qquad\quad\
\text{if}\ 1<p<2,\\\ \end{split}$

$\text{for a.e.}\ x\in{\overline{B}}\ \text{and}\ \forall E\geq 0$.

  * (A4)

For fixed $1<p<+\infty$, there exists $\kappa>0$ such that:

$\begin{split}(\sigma_{B}(x,E_{2})&{\bf E}_{2}-\sigma_{B}(x,E_{1}){\bf
E}_{1})\cdot({\bf E}_{2}-{\bf E}_{1})\\\ &\geq\begin{cases}\kappa|{\bf
E}_{2}-{\bf E}_{1}|^{p}\ &\text{if}\ p\geq 2\\\ \kappa(1+|{\bf
E}_{2}|^{2}+|{\bf E}_{1}|^{2})^{\frac{p-2}{2}}|{\bf E}_{2}-{\bf E}_{1}|^{2}\
&\text{if}\ 1<p<2\end{cases}\\\ \end{split}$

$\text{for a.e.}\ x\in B$, and for any ${\bf E}_{1},{\bf
E}_{2}\in\mathbb{R}^{n}$.

  * (B1)

For fixed $1<q<+\infty$, there exists three positive constants
$\underline{\sigma}\leq\overline{\sigma}$ and $E_{0}$ such that:

$\begin{split}&\underline{\sigma}\left(\frac{E}{E_{0}}\right)^{q-2}\leq\sigma_{A}(x,E)\leq\overline{\sigma}\left[1+\left(\frac{E}{E_{0}}\right)^{q-2}\right]\qquad\text{if}\
q\geq 2,\\\
&\underline{\sigma}\left(\frac{E}{E_{0}}\right)^{q-2}\leq\sigma_{A}(x,E)\leq\overline{\sigma}\left(\frac{E}{E_{0}}\right)^{q-2}\qquad\qquad\quad\
\text{if}\ 1<q<2,\end{split}$

$\text{for a.e.}\ x\in A\ \text{and}\ \forall E>0$.

  * (B2)

For fixed $1<q<+\infty$, there exists a positive constant $\kappa$ such that:

$\begin{split}(\sigma_{A}(x,E_{2})&{\bf E}_{2}-\sigma_{A}(x,E_{1}){\bf
E}_{1})\cdot({\bf E}_{2}-{\bf E}_{1})\\\ &\geq\begin{cases}\kappa|{\bf
E}_{2}-{\bf E}_{1}|^{q}\ &\text{if}\ q\geq 2\\\ \kappa(1+|{\bf
E}_{2}|^{2}+|{\bf E}_{1}|^{2})^{\frac{q-2}{2}}|{\bf E}_{2}-{\bf E}_{1}|^{2}\
&\text{if}\ 1<q<2\end{cases}\\\ \end{split}$

$\ \text{for a.e.}\ x\in A,$ for any ${\bf E}_{1},{\bf
E}_{2}\in\mathbb{R}^{n}$.

  * (C)

$\sigma_{A}(x,E)=0$, $\text{for a.e.}\ x\in A$.

  * (D)

$\sigma_{A}(x,E)=+\infty$, $\text{for a.e.}\ x\in A$.

Assumptions (BX), (C) and (D) are alternative. Through the paper, we will
treat all the cases arising from these assumptions. Moreover, let us remark
that $A$ could be $\emptyset$; in this case the assumptions (BX), (C) and (D)
are not considered. Figure 2 represents the geometrical interpretations of
(A3). Similar arguments are valid for (B1).

Figure 2. The impact of constraints in (A3) on the electrical conductivity,
for $1<p<2$ (left) and $p\geq 2$ (right). Dashed lines represent the upper and
lower bounds, whereas the solid line represents the actual electrical
conductivity.

######  Remark 2.2.

We notice that assumptions (A3) and (B1) imply that

$\underline{\sigma}\left(\frac{E}{E_{0}}\right)^{r-2}\leq\sigma(x,E)\leq\overline{\sigma}\left[1+\left(\frac{E}{E_{0}}\right)^{s-2}\right]$
(2.2)

for a.e. $x\in\Omega$ and for all $E>0$, being $r=\min\\{p,q\\}$ and
$s=\max\\{p,q\\}$.

######  Remark 2.3.

The first inequality in hypothesis (A4) is nothing but a generalization of the
standard inequality (see [40, Sec.12, eq (I)] and [10, Lem. 2.2])

$(E_{2}^{k}\ {\bf E}_{2}-E_{1}^{k}\ {\bf E}_{1})\cdot({\bf E}_{2}-{\bf
E}_{1})\geq\frac{1}{2^{k+1}}|{\bf E}_{2}-{\bf E}_{1}|^{k+2}\quad\forall k\geq
0.$

In addition, the first inequality in (A4) is also used in [37, eq. (3.4)] for
the particular case of $p=2$ and $A=\emptyset$.

######  Remark 2.4.

The second inequality in (A4) is nothing but a generalization of the standard
inequality (see [40, Sec.12, eq (VII)] and [10, Lem. 2.2]):

$(E_{2}^{k}\ {\bf E}_{2}-E_{1}^{k}\ {\bf E}_{1})\cdot({\bf E}_{2}-{\bf
E}_{1})\geq(k+1)(1+|{\bf E}_{2}|^{2}+|{\bf E}_{1}|^{2})^{\frac{k}{k+2}}|{\bf
E}_{2}-{\bf E}_{1}|^{2}\quad\forall 0>k>-1,$

where $k=p-2$.

######  Remark 2.5.

Dealing with (C) and (D) means that we are treating a problem in a holed
domain, as it is easly seen by the associated variational formulations of
problems (2.6)-(2.8). We highlight that optimization problems on domains with
holes have received much interest in recent years, see e.g. [21, 26, 43, 44]
and reference therein.

###  2.4. The mathematical model

The electric scalar potential $u$ solves the steady current problem (1.1),
where $f\in X_{\diamond}^{p}(\partial\Omega)$. Problem (1.1) is meant in the
weak sense, that is

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

and $u$ restricted to $B$ belongs to $W^{1,p}(B)$, whereas $u$ restricted to
$A$ belongs to $W^{1,q}(A)$. Moreover, the solution $u$, as a whole, is an
element of the larger functional space $W^{1,p}(\Omega)\cup W^{1,q}(\Omega)$.
In the sequel we will exploit that (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
$u|_{\partial\Omega}=f$. The solution $u$ is variationally characterized as

$\min_{\begin{subarray}{c}u\in W^{1,p}(\Omega)\cup W^{1,q}(\Omega)\\\ u=f\
\text{on}\ \partial\Omega\end{subarray}}\mathbb{E}_{\sigma}\left(u\right).$
(2.3)

In (2.3), the functional $\mathbb{E}_{\sigma}\left(u\right)$ is the Dirichlet
Energy

$\mathbb{E}_{\sigma}\left(u\right)=\int_{\Omega}Q_{\sigma}(x,|\nabla u(x)|)\
\text{d}x=\int_{B}Q_{B}(x,|\nabla u(x)|)\ \text{d}x+\int_{A}Q_{A}(x,|\nabla
u(x)|)\ \text{d}x$ (2.4)

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 $A$. At this stage let us stress that when
$A=\emptyset$, then we fall in the case treated in [16] for $p\geq 2$.
Moreover, if $\sigma_{A}(x,E)=0$, we have a PEI in $A$. The corresponding
problem is

$\begin{cases}\operatorname{div}(\sigma_{B}(x,|\nabla v|)\nabla v)=0&\text{in
}B\\\ \sigma_{B}(x,|\nabla v|)\partial_{\nu}v=0&\text{on }\partial A\\\ v\in
W^{1,p}(B),\ v=f&\text{on }\partial\Omega.\end{cases}$ (2.5)

The solution $u$ is variationally defined as the minimum of the following
Dirichlet Energy

$\min_{\begin{subarray}{c}u\in W^{1,p}(B)\\\ u=f\text{on
}\partial\Omega\end{subarray}}\mathbb{E}_{\sigma}(u),$ (2.6)

where

$\mathbb{E}_{\sigma}(u)=\int_{B}Q_{B}(x,|\nabla u(x)|)dx.$

Similarly, if $\sigma_{A}(x,E)=+\infty$, we have a PEC in $A$. The
corresponding problem is

$\begin{cases}\operatorname{div}(\sigma_{B}(x,|\nabla v|)\nabla v)=0&\text{in
}B\\\ |\nabla v|=0&\text{a.e. in }A\\\ \int_{\partial A_{i}}\sigma(x,|\nabla
v(x)|)\partial_{\nu}v(x)dS=0&i=1,...,M\\\ v\in W^{1,p}(\Omega),\ v=f&\text{on
}\partial\Omega.\end{cases}$ (2.7)

The solution $u$ is variationally defined as the minimum of the following
Dirichlet Energy

$\min_{\begin{subarray}{c}u\in W^{1,p}(\Omega)\\\ u=f\text{ on
}\partial\Omega\\\ |\nabla u|=0\text{ in
}A\end{subarray}}\mathbb{E}_{\sigma}(u),$ (2.8)

where

$\mathbb{E}_{\sigma}(u)=\int_{B}Q_{B}(x,|\nabla u(x)|)dx.$ (2.9)

Let us stress that, since the solution in (2.8) assumes a constant value in
$A$, then we can equivalently consider $\mathbb{E}_{\sigma}$ equal to (2.9)
instead of (2.8). Throughout the paper, we will use the following notation to
denote the solutions of the previously described minimum problems.

######  Definition 2.6.

We denote by $u^{g}$ the unique solution arising from problems (2.3), (2.6)
and (2.8), corresponding to the boundary data $g\in
X_{\diamond}^{p}(\partial\Omega)$.

The proof of existence and uniqueness of the solution for (2.3), (2.6) and
(2.8) in its variational form, relies on standard methods of the Calculus of
Variations, when the Dirichlet Energy Density presents the same growth
exponent in any point of the domain $\Omega$. The case treated in this work is
nonstandard, because the Dirichlet Energy Density presents different growth
exponents in $B$ and $A$ and, hence, we provide a proof in the following.

######  Theorem 2.7.

Let $1<p,q<+\infty$, $f\in X_{\diamond}^{p}(\partial\Omega)$ and $\sigma$
satisfies (A1), (A2) and (A3).

  * •

If $A=\emptyset$, then there exists a unique solution of problem (2.3);

  * •

If (B1) hold, then there exists a unique solution of problem (2.3);

  * •

If (C) hold, then there exists a unique solution of problem (2.6);

  * •

If (D) hold, then there exists a unique solution of problem (2.8).

###### Proof.

When (B1) holds, (2.2) implies that there exist $\underline{Q}>0$ such that

$Q_{\sigma}(x,E)\geq\underline{Q}\left(\frac{E}{E_{0}}\right)^{r}\quad\text{for
a.e. }x\in\Omega\text{ and }\forall E>0.$

Hence, the function $Q_{\sigma}(x,\cdot)$ is strictly convex and coercive for
a.e. $x\in\Omega$. Similarly, when (C) or (D) holds, the convexity of $Q_{B}$
follows from assumption (A2) whereas the coercitivity follows from the the
left hand side of assumption (A3). In all three cases, standard direct methods
of the calculus of variations [18, Th. 3.30] provide the existence and
uniqueness of the solution. ∎

######  Remark 2.8.

The 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 $\mathbb{E}_{\sigma}(f)<+\infty$ (see [18,
Th. 3.30] for details).

###  2.5. The DtN operator.

The Dirichlet-to-Neumann (DtN) operator maps the Dirichlet data into the
corresponding Neumann data:

$\Lambda_{\sigma}:f\in X_{\diamond}(\partial\Omega)\mapsto\sigma(x,|\nabla
u^{f}|)\ \partial_{n}u^{f}|_{\partial\Omega}\in
X_{\diamond}(\partial\Omega)^{\prime},$

where $X_{\diamond}(\partial\Omega)^{\prime}$ is the dual space of
$X(\partial\Omega)$ and $u^{f}$ is the solution of (1.1). From the physical
standpoint, the DtN operator maps the boundary electric scalar potential into
the normal component of the electrical current density entering
$\partial\Omega$ (see 2.1). In weak form, the DtN operator is

$\langle\Lambda_{\sigma}\left(f\right),\varphi\rangle=\int_{\partial\Omega}\varphi(x){\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\partial_{{{n}}}u^{f}(x)\text{d}S\quad\forall\varphi\in
X_{\diamond}(\partial\Omega).$ (2.10)

Furthermore, by testing the DtN operator (2.10) with the solution $u^{f}$ of
(1.1) and using a divergence Theorem, we obtain the ohmic power dissipated by
the conducting material:

$\langle\Lambda_{\sigma}\left(f\right),f\rangle=\int_{\Omega}{\sigma}(x,\nabla
u^{f}(x)){|\nabla u^{f}}(x)|^{2}\ \text{d}x.$

If $\varphi\neq f$ in (2.10), we have the so-called _virtual power_ product
that plays an important role since it is equal to the Gâteaux derivative of
the Dirichlet Energy, when evaluated at the solution $u^{f}$, as showed in
Section 4. The injectivity of the DtN operator is guaranteed by the assumption
of zero average of $f$.

##  3\. Convergence of the Scalar Potential w.r.t. the Boundary Data

The aim of this section is to prove a fundamental convergence result of the
solutions with respect to the boundary data. Specifically, it is possible to
prove that the operator $f\mapsto\nabla u^{f}$ is continuous with respect to
the boundary data $f$, along any arbitrary direction (see Proposition 3.2).
This generalizes the result of [16, Lem. 3.1], where it was proved a
convergence result for $p\geq 2$ and $q=p$. Here we generalize these previous
results to arbitrary $p$ and $q$ with $1<p,q<+\infty$. Firstly, we prove the
following Lemma that provides an upper bound to the norm of the solutions of
problems (2.3), (2.6), (2.8), through the norm of the boundary data. The
proof, based on well-established methods, is given explicitly because it
refers to an atypical configuration.

######  Lemma 3.1.

Let $1<p,q<+\infty$, $f\in X_{\diamond}^{p}(\partial\Omega)$, ${\sigma}$
satisfying (A1), (A2), (A3), (A4). Then, there exists $C=C(p,q,\Omega)$ such
that

  * (i)

If $A=\emptyset$, it results

$\begin{split}||\nabla u^{f}||_{L^{p}(\Omega)}\leq
C(1+||f||_{X^{p}(\partial\Omega)}),\end{split}$ (3.1)

where $u^{f}\in W^{1,p}(\Omega)$ is the minimizer of (2.3).

  * (ii)

If ${\sigma}$ satisfies (BX), it results

$\begin{split}||\nabla u^{f}||_{L^{p}(B)},||\nabla
u^{f}||^{\frac{p}{q}}_{L^{q}(A)}&\leq\left(||\nabla
u^{f}||^{p}_{L^{p}(B)}+||\nabla u^{f}||^{q}_{L^{q}(A)}\right)^{\frac{1}{p}}\\\
&\leq C(1+||f||_{X^{p}(\partial\Omega)}),\end{split}$ (3.2)

where $u^{f}\in W^{1,p}(\Omega)$ is the minimizer of (2.3).

  * (iii)

If ${\sigma}$ satisfies (C), it results

$||\nabla u^{f}||_{L^{p}(B)}\leq C(1+||f||_{X^{p}(\partial\Omega)}),$ (3.3)

where $u^{f}\in W^{1,p}(\Omega)$ is the minimizer of (2.6).

  * (iv)

If ${\sigma}$ satisfies (D), it results

$||\nabla u^{f}||_{L^{p}(B)}\leq C(1+||f||_{X^{p}(\partial\Omega)}),$ (3.4)

where $u^{f}\in W^{1,p}(\Omega)$ is the minimizer of (2.8).

###### Proof.

We give the proof for each individual case. $\bullet$ _Claim (i)_. Let us
assume $p\geq 2$. Since $u^{f}$ realizes the minimum in (2.3) with
$A=\emptyset$, by using (A3), we have

$\underline{c}\int_{\Omega}|\nabla
u^{f}(x)|^{p}\text{d}x\leq\int_{\Omega}Q_{\sigma}(x,|\nabla{u^{f}}(x)|)\text{d}x\leq\overline{c}\int_{\Omega}\left(|\nabla
u(x)|^{2}+|\nabla u(x)|^{p}\right)\text{d}x$ (3.5)

for any $u\in f+W^{1,p}_{0}(\Omega)$, and where the two constants
$\overline{c}$ and $\underline{c}$ are defined as

$\overline{c}=\frac{\overline{\sigma}}{pE_{0}^{p-2}},\
\underline{c}=\frac{\underline{\sigma}}{pE_{0}^{p-2}}.$ (3.6)

The Inverse Trace inequality in Besov spaces [38, Th. 18.34] assures the
existence of a function $w\in f+W^{1,p}_{0}(\Omega)$ such that $||\nabla
w||_{p}\leq K(p,\Omega)||f||_{X^{p}(\partial\Omega)}$. Therefore, we have

$\begin{split}||\nabla u^{f}||^{p}_{p}\leq C_{1}(||\nabla w||^{2}_{2}+||\nabla
w||^{p}_{p})\leq C_{1}(C_{2}||\nabla w||^{2}_{p}+||\nabla w||^{p}_{p})\\\ \leq
C_{1}(C_{2}+(C_{2}+1)||\nabla w||^{p}_{p})\leq
C_{1}(C_{2}+(C_{2}+1)K^{p}_{p}||f||^{p}_{X^{p}(\partial\Omega)}).\end{split}$

where in the first inequality we have used (3.5) with $u=w$ and
$C_{1}=\overline{c}/\underline{c}$; in the second inequality that, for $p\geq
2$ there exists a constant $C$ such that $||\nabla w||_{2}\leq C_{2}||\nabla
w||_{p}$ by the continuous embedding of the Sobolev spaces; in the third
inequality that $||\nabla w||_{p}^{2}\leq 1+||\nabla w||_{p}^{p}$ and in the
fourth inequality the inverse trace inequality. Therefore, (3.1) follows by
setting $C=C_{1}\max\\{C_{2},(C_{2}+1)K_{p}^{p}\\}$. The inequality for
$1<p<2$ analogously follows in easier way using the second line of the
assumption (A3). $\bullet$ _Claim (ii)_. We have that

$\begin{split}\underline{c}\left(\int_{B}|\nabla
u^{f}(x)|^{p}dx+\int_{A}|\nabla
u^{f}(x)|^{q}dx\right)&\leq\mathbb{E}_{\sigma}(u^{f})\\\
&\leq\mathbb{E}_{\sigma}(u_{\infty}^{f})=\int_{B}Q_{B}(x,|\nabla{u^{f}_{\infty}}(x)|)\text{d}x\leq\overline{c}C\left(1+||f||^{p}_{X^{p}(\partial\Omega)}\right),\end{split}$

where $\overline{c}$ and $\underline{c}$ are the constants defined in (3.6)
and $u_{\infty}^{f}$ is the solution of the PEC problem (2.8), the first
inequality follows from assumption (A3)-left and (B1)-left, the second
inequality follows from using $u_{\infty}^{f}$ as test function, and the third
inequality follows by using the argument of the previous case. $\bullet$
_Claim (iii)_. Regarding the case when (C) holds, that is the PEI case, let us
consider the solution $\hat{u}^{f}$ of the problem
$\min_{\begin{subarray}{c}u\in W^{1,p}(\Omega)\\\ u=f\ \text{on}\
\partial\Omega\end{subarray}}\int_{\Omega}\hat{Q}(x,|\nabla{u}(x)|)\text{d}x$,
where

$\hat{Q}(x,E)=\begin{cases}Q_{B}(x,E)\ &\text{on}\ B,\\\ E^{p}\ &\text{on}\
A.\end{cases}$

It is easy to see that there exists two positive constants $\underline{c}$ and
$\overline{c}$ such that

$\begin{split}\underline{c}\int_{B}|\nabla
u^{f}(x)|^{p}dx&\leq\mathbb{E}_{\sigma}(u^{f})\leq\mathbb{E}_{\sigma}(\hat{u}^{f})\leq\overline{c}\left(\int_{B}|\nabla\hat{u}^{f}(x)|^{p}\text{d}x+\int_{A}|\nabla\hat{u}^{f}(x)|^{p}\text{d}x\right)\\\
&\leq C||f||_{X^{p}(\partial\Omega)}.\end{split}$

where in the first inequality we have used the l.h.s. of (A3), in the second
inequality we have $\hat{u}^{f}$ as admissible test function, in the third
inequality we have used the r.h.s of assumption (A3) and the last inequality
follows as in Claim (i). $\bullet$ _Claim (iv)_. The conclusion for the PEC
case follows arguing as in Claim (i). ∎

Lemma 3.1 is at the foundation of the following fundamental convergence
result.

######  Proposition 3.2.

Let $1<p,q<+\infty$, $f\in X_{\diamond}^{p}(\partial\Omega)$ and ${\sigma}$
satisfying (A1), (A2), (A3), (A4).

  * (i)

If $A=\emptyset$, then

$\begin{split}&\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}\ \text{in}\
{L^{p}(\Omega)}\qquad\text{as}\ {\varepsilon\to 0^{+}},\end{split}$ (3.7)

for any $\varphi\in W^{1,p}(\Omega)$, where $u^{f+\varepsilon\varphi}\in
W^{1,p}(\Omega)$ (resp. $u^{f}\in W^{1,p}(\Omega)$) is the minimizer of (2.3)
with boundary data $f+\varepsilon\varphi$ (resp. $f$).

  * (ii)

If ${\sigma}$ satisfies (BX), then

$\begin{split}&\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}\ \text{in}\
{L^{p}(B)}\qquad\text{as}\ {\varepsilon\to 0^{+}},\\\ &\nabla
u^{f+\varepsilon\varphi}\to\nabla u^{f}\ \text{in}\ {L^{q}(A)}\qquad\text{as}\
{\varepsilon\to 0^{+}},\\\ &\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}\
\text{in}\ {L^{r}(\Omega)}\qquad\text{as}\ {\varepsilon\to 0^{+}},\end{split}$
(3.8)

for any $\varphi\in(W^{1,p}(\Omega)\cup W^{1,q}(\Omega))\cap W^{1,p}(B)\cap
W^{1,q}(A)$, where $r=\min\\{p,q\\}$ and $u^{f+\varepsilon\varphi}\in
W^{1,p}(\Omega)$ (resp. $u^{f}\in W^{1,p}(\Omega)$) is the minimizer of (2.3)
with boundary data $f+\varepsilon\varphi$ (resp. $f$).

  * (iii)

If ${\sigma}$ satisfies (C), then

$\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}\ \text{in}\
{L^{p}(B)}\qquad\text{as}\ {\varepsilon\to 0^{+}},$ (3.9)

where $u^{f+\varepsilon\varphi}\in W^{1,p}(\Omega)$ (resp. $u^{f}\in
W^{1,p}(\Omega)$) is the minimizer of (2.6) with boundary data
$f+\varepsilon\varphi$ (resp. $f$).

  * (iv)

If ${\sigma}$ satisfies (D), then

$\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}\ \text{in}\
{L^{p}(B)}\qquad\text{as}\ {\varepsilon\to 0^{+}},$ (3.10)

where $u^{f+\varepsilon\varphi}\in W^{1,p}(\Omega)$ (resp. $u^{f}\in
W^{1,p}(\Omega)$) is the minimizer of (2.8) with boundary data
$f+\varepsilon\varphi$ (resp. $f$).

###### Proof.

For any fixed $0<\varepsilon<1$, we denote by $I$ and $II$ the following
quantities:

$\displaystyle I:$ $\displaystyle=\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right){\nabla
u^{f+\varepsilon\varphi}(x)}\cdot(\nabla u^{f+\varepsilon\varphi}(x)-\nabla
u^{f}(x))\text{d}x,$ (3.11) $\displaystyle II:$
$\displaystyle=\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right){\nabla u^{f}(x)}\cdot(\nabla
u^{f+\varepsilon\varphi}(x)-\nabla u^{f}(x))\text{d}x.$ (3.12)

$\bullet$ Claim (i). We distinguish the proof according the value of the
exponent $p$. For $p\geq 2$, the divergence Theorem gives

$\displaystyle I$
$\displaystyle=\varepsilon\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\nabla
u^{f+\varepsilon\varphi}(x)\cdot\nabla\varphi(x)\text{d}x$ (3.13)
$\displaystyle II$
$\displaystyle=\varepsilon\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\text{d}x.$ (3.14)

By subtracting (3.14) from (3.13), we have

$\begin{split}I-II&=\varepsilon\int_{\Omega}\nabla\varphi(x)\left[{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\nabla
u^{f+\varepsilon\varphi}(x)-{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\right]\ \text{d}x\\\ &\leq\varepsilon
D_{1}\int_{\Omega}|\nabla\varphi(x)|\left[\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|+\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|^{p-1}+\left|\nabla
u^{f}(x)\right|+\left|\nabla u^{f}(x)\right|^{p-1}\right]\ \text{d}x,\\\
\end{split}$

where in the second line we have used (A3)-right with
$D_{1}=\overline{\sigma}\max\left\\{1,{E_{0}^{2-p}}\right\\}$. Then, we have

$\begin{split}I-II&\leq\varepsilon
D_{1}\left[\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{2}(\Omega)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{p}(\Omega)}^{p-1}\right.\\\
&\qquad\qquad\qquad\qquad\quad\left.+\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}\left|\left|\nabla
u^{f}\right|\right|_{L^{2}(\Omega)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}\left|\left|\nabla
u^{f}\right|\right|_{L^{p}(\Omega)}^{p-1}\right]\\\ &\leq\varepsilon
CD_{1}\left[\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}(1+\left|\left|f+\varepsilon\varphi\right|\right|_{X^{2}(\partial\Omega)})+\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}(1+||f+\varepsilon\varphi||_{X^{p}(\partial\Omega)})^{p-1}\right.\\\
&\qquad\qquad\qquad\qquad\quad\left.+\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}(1+\left|\left|f\right|\right|_{X^{2}(\partial\Omega)})+\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}(1+||f||_{X^{p}(\partial\Omega)})^{p-1}\right]\\\
&\leq\varepsilon
CD_{1}\bigg{[}\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}\left(1+\left|\left|f\right|\right|_{X^{2}(\partial\Omega)}+\left|\left|\varphi\right|\right|_{X^{2}(\partial\Omega)}\right)\\\
&\qquad\qquad\qquad\qquad\quad+\left.\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}\left(1+||f||_{X^{p}(\partial\Omega)}+||\varphi||_{X^{p}(\partial\Omega)}\right)^{p-1}\right],\end{split}$
(3.15)

where in the first inequality we used the Hölder inequality, in the second
inequality we used (3.1) of Lemma 3.1, in the third inequality we used the
elementary properties of the norm and that $\varepsilon<1$. The quantity in
the squared bracket of the last term in (3.15) is finite because $f,\varphi\in
X^{p}_{\diamond}(\partial\Omega)$. Hence, $I-II$ is upper bounded:

$I-II\leq\varepsilon F,$ (3.16)

where $F$ is the product of $C$, $D_{1}$ and the quantity in the squared
brackets at the r.h.s. of (3.15). On the other hand, by using (A4), we have:

$||\nabla u^{f+\varepsilon g}-\nabla
u^{f}||^{p}_{L^{p}(\Omega)}\leq\frac{1}{\kappa}(I-II).$ (3.17)

By joining (3.16) and (3.17), we have

$||\nabla u^{f+\varepsilon g}-\nabla
u^{f}||^{p}_{L^{p}(\Omega)}\leq\frac{1}{\kappa}(I-II)\leq\frac{F}{\kappa}\varepsilon.$

This gives (3.7) by passing the to limit as $\varepsilon\to 0^{+}$. If
$1<p<2$, the inequality (3.16) analogously follows, but with slightly
different constants. Regarding the lower bound, we have

$\begin{split}&||\nabla u^{f+\varepsilon\varphi}-\nabla u^{f}||^{p}_{p}\\\
&\leq\frac{1}{\kappa}\int_{\Omega}(1+|\nabla u^{f+\varepsilon
g}(x)|^{2}+|\nabla u^{f}(x)|^{2})^{\frac{2-p}{4}}\\\
&\qquad\qquad((\sigma(x,|\nabla u^{f+\varepsilon\varphi}(x)|)-\sigma(x,|\nabla
u^{f}(x)|))(\nabla u^{f+\varepsilon\varphi}(x)-\nabla
u^{f}(x)))^{\frac{p}{2}}\text{d}x\\\
&\leq\frac{1}{\kappa}\left(\int_{\Omega}(1+|\nabla
u^{f+\varepsilon\varphi}(x)|^{2}+|\nabla
u^{f}(x)|^{2})^{\frac{1}{2}}\text{d}x\right)^{\frac{2-p}{2}}\\\
&\qquad\qquad\left(\int_{\Omega}(\sigma(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)-\sigma(x,|\nabla u^{f}(x)|))(\nabla
u^{f+\varepsilon\varphi}(x)-\nabla u^{f}(x))\text{d}x\right)^{\frac{p}{2}}\\\
&=\frac{G}{\kappa}(I-II)^{\frac{p}{2}},\end{split}$ (3.18)

where in the first inequality we used the assumption (A4) raised to the power
$p/2$, in the second inequality we used the Hölder inequality with exponents
$2/p$ and $2/(2-p)$, and $G=\left(\int_{\Omega}(1+|\nabla
u^{f+\varepsilon\varphi}(x)|^{2}+|\nabla
u^{f}(x)|^{2})^{\frac{1}{2}}\text{d}x\right)^{\frac{2-p}{2}}$ is a finite
quantity. By taking into account (3.16) and (3.18), we have

$||\nabla u^{f+\varepsilon g}-\nabla
u^{f}||^{p}_{L^{p}(\Omega)}\leq\frac{G}{\kappa}(I-II)^{\frac{p}{2}}\leq\frac{GF^{\frac{p}{2}}}{\kappa}\varepsilon^{\frac{p}{2}},$

that gives (3.7) by passing the to limit as $\varepsilon\to 0^{+}$. $\bullet$
Claim (ii). First, we notice that the divergence Theorem gives (3.13) and
(3.14), also in this case. For $p,q\geq 2$, we have

$\begin{split}I-II&=\varepsilon\int_{\Omega}\nabla\varphi(x)\left[{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\nabla
u^{f+\varepsilon\varphi}(x)-{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\right]\ \text{d}x\\\ &\leq\varepsilon
D_{1}\int_{B}|\nabla\varphi(x)|\left[\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|+\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|^{p-1}+\left|\nabla
u^{f}(x)\right|+\left|\nabla u^{f}(x)\right|^{p-1}\right]\ \text{d}x\\\ &\ \
+\varepsilon D_{1}\int_{A}|\nabla\varphi(x)|\left[\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|+\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|^{q-1}+\left|\nabla
u^{f}(x)\right|+\left|\nabla u^{f}(x)\right|^{q-1}\right]\
\text{d}x,\end{split}$

where in the first inequality we used (A3) and (B1). Then, we have

$\begin{split}I-II&\leq\varepsilon
D_{1}\left[\left|\left|\nabla\varphi\right|\right|_{L^{2}(B)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{2}(B)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(B)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{p}(B)}^{p-1}\right.\\\
&\qquad\qquad+\left|\left|\nabla\varphi\right|\right|_{L^{2}(B)}\left|\left|\nabla
u^{f}\right|\right|_{L^{2}(B)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(B)}\left|\left|\nabla
u^{f}\right|\right|_{L^{p}(B)}^{p-1}\\\
&\qquad\qquad\qquad+\left|\left|\nabla\varphi\right|\right|_{L^{2}(A)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{2}(A)}+\left|\left|\nabla\varphi\right|\right|_{L^{q}(A)}\left|\left|\nabla
u^{f+\varepsilon\varphi}\right|\right|_{L^{q}(A)}^{q-1}\\\
&\qquad\qquad\qquad\qquad\
+\left.\left|\left|\nabla\varphi\right|\right|_{L^{2}(A)}\left|\left|\nabla
u^{f}\right|\right|_{L^{2}(A)}+\left|\left|\nabla\varphi\right|\right|_{L^{q}(A)}\left|\left|\nabla
u^{f}\right|\right|_{L^{q}(A)}^{q-1}\right]\\\ &\leq\varepsilon
CD_{1}\left[\left|\left|\nabla\varphi\right|\right|_{L^{2}(B)}\left|\left|f+\varepsilon\varphi\right|\right|_{X^{2}(\partial\Omega)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(B)}||f+\varepsilon\varphi||^{p-1}_{X^{p}(\partial\Omega)}\right.\\\
&\qquad\qquad+\left|\left|\nabla\varphi\right|\right|_{L^{2}(B)}\left|\left|f\right|\right|_{X^{2}(\partial\Omega)}+\left|\left|\nabla\varphi\right|\right|_{L^{p}(B)}||f||^{p-1}_{X^{p}(\partial\Omega)}\\\
&\qquad\qquad+\left|\left|\nabla\varphi\right|\right|_{L^{2}(A)}\left|\left|f+\varepsilon\varphi\right|\right|_{X^{2}(\partial\Omega)}^{\frac{p}{2}}+\left|\left|\nabla\varphi\right|\right|_{L^{q}(A)}||f+\varepsilon\varphi||^{\frac{p(q-1)}{q}}_{X^{p}(\partial\Omega)}\\\
&\qquad\qquad\left.+\left|\left|\nabla\varphi\right|\right|_{L^{2}(A)}\left|\left|f\right|\right|_{X^{2}(\partial\Omega)}^{\frac{p}{2}}+\left|\left|\nabla\varphi\right|\right|_{L^{q}(A)}||f||^{p-1}_{X^{p}(\partial\Omega)}+||f||^{\frac{p(q-1)}{q}}_{X^{p}(\partial\Omega)}\right]\\\
&\leq\varepsilon
CD_{1}\bigg{[}\left|\left|\nabla\varphi\right|\right|_{L^{2}(\Omega)}\left(\left|\left|f\right|\right|_{X^{2}(\partial\Omega)}+\left|\left|\varphi\right|\right|_{X^{2}(\partial\Omega)}\right)\\\
&\qquad\qquad+\left.\left|\left|\nabla\varphi\right|\right|_{L^{p}(\Omega)}\left(||f||^{p-1}_{X^{p}(\partial\Omega)}+||\varphi||^{p-1}_{X^{p}(\partial\Omega)}+||f||^{\frac{p(q-1)}{q}}_{X^{p}(\partial\Omega)}+||\varphi||^{\frac{p(q-1)}{q}}_{X^{p}(\partial\Omega)}\right)\right],\end{split}$
(3.19)

where in the first inequality we used the Hölder inequality, in the second
inequality we used (3.2) of Lemma 3.1, in the third inequality we used the
elementary properties of the norm and the fact that $\varepsilon<1$. The
quantity in the squared bracket of the last term in (3.19) is finite because
$f,\varphi\in X^{p}_{\diamond}(\partial\Omega)$. Hence, the quantity $I-II$ is
upper bounded:

$I-II\leq F\varepsilon,$ (3.20)

where $F$ is the product of $C_{1}$, $D$ and the last term in the squared
parenthesis of (3.19). In the remaining cases ($1<p<2$ and/or $1<q<2$), the
inequality (3.20) follows analogously, but with slightly different constants.
The lower bound for $I-II$ depends on the values of $p$ and $q$. Specifically,
for $p\geq 2$, by using (A4), it results that

$||\nabla u^{f+\varepsilon g}-\nabla
u^{f}||^{p}_{L^{p}(B)}\leq\frac{1}{\kappa}(I-II),$ (3.21)

whereas for $1<p<2$, it results that

$\begin{split}&||\nabla u^{f+\varepsilon\varphi}-\nabla
u^{f}||^{p}_{L^{p}(B)}\\\ &\leq\frac{1}{\kappa}\int_{B}[1+|\nabla
u^{f+\varepsilon g}(x)|^{2}+|\nabla u^{f}(x)|^{2}]^{\frac{2-p}{4}}\\\
&\qquad\qquad((\sigma(x,|\nabla u^{f+\varepsilon\varphi}(x)|)-\sigma(x,|\nabla
u^{f}(x)|))(\nabla u^{f+\varepsilon\varphi}(x)-\nabla
u^{f}(x)))^{\frac{p}{2}}\text{d}x\\\
&\leq\frac{1}{\kappa}\left(\int_{B}(1+|\nabla
u^{f+\varepsilon\varphi}(x)|^{2}+|\nabla
u^{f}(x)|^{2})^{\frac{1}{2}}\text{d}x\right)^{\frac{2-p}{2}}\\\
&\qquad\qquad\left(\int_{B}(\sigma(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)-\sigma(x,|\nabla u^{f}(x)|))(\nabla
u^{f+\varepsilon\varphi}(x)-\nabla u^{f}(x))\text{d}x\right)^{\frac{p}{2}}\\\
&\leq\frac{G}{\kappa}(I-II)^{\frac{p}{2}},\end{split}$ (3.22)

where in the first inequality we used the assumption (A4) raised to the power
$p/2$, in the second inequality we used the Hölder inequality with exponents
$2/p$ and $2/(2-p)$. Regarding the inner convergence result, for $q\geq 2$ and
by using (B2), it results that

$||\nabla u^{f+\varepsilon g}-\nabla
u^{f}||^{q}_{L^{q}(A)}\leq\frac{1}{\kappa}(I-II),$ (3.23)

whereas for $1<q<2$ it results that

$||\nabla u^{f+\varepsilon\varphi}-\nabla
u^{f}||^{q}_{L^{q}(A)}\leq\frac{G}{\kappa}(I-II)^{\frac{q}{2}},$ (3.24)

as follows by using (B2) instead of (A4) and by substituting $q$ with $p$ and
$A$ with $B$ in (3.22). By combining (3.20) with either (3.21) or (3.22), and
by letting $\varepsilon\to 0^{+}$, we have the outer convergence result of
(3.8). Similarly, by combining (3.20) with either (3.23) or (3.24), and by
letting $\varepsilon\to 0^{+}$, we have the inner convergence result of (3.8).
Finally, the convergence result on the whole domain $\Omega$ (see (3.8))
follows by observing that $L^{r}(\Omega)=L^{p}(\Omega)\cup L^{q}(\Omega)$.
$\bullet$ Claim (iii). By the divergence theorem, and taking into account that
in the PEI case the solutions have vanishing normal derivative on $\partial A$
(see (2.5)), we have

$\displaystyle I$
$\displaystyle=\varepsilon\int_{B}{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\nabla
u^{f+\varepsilon\varphi}(x)\cdot\nabla\varphi(x)\text{d}x$ $\displaystyle II$
$\displaystyle=\varepsilon\int_{B}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\text{d}x.$

The conclusion (3.9) follows by arguing as in the previous case, but using
(3.3) of Lemma 3.1. $\bullet$ Claim (iv). First, it results that

$\begin{split}I&=\varepsilon\int_{B}{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\nabla
u^{f+\varepsilon\varphi}(x)\cdot\nabla\varphi(x)\text{d}x\\\
II&=\varepsilon\int_{B}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla
u^{f}(x)\cdot\nabla\varphi(x)\text{d}x.\end{split}$

Indeed, from the definition of $I$ appearing in (3.11), it follows that

$\begin{split}I&=\int_{B}\operatorname{div}(\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}|)\nabla u^{f+\varepsilon\varphi}(x)\
(u^{f+\varepsilon\varphi}(x)-u^{f}(x)))dx\\\
&=\varepsilon\int_{\partial\Omega}\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)\partial_{\nu}u^{f+\varepsilon\varphi}(x)\
\varphi(x)|_{\Omega}dS\\\
&\qquad+\sum_{i=1}^{M}(c^{i}_{\varepsilon}-c^{i}_{0})\int_{\partial
A}\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)\partial_{\nu}u^{f+\varepsilon\varphi}(x)\ dS\\\
&=\varepsilon\int_{B}\operatorname{div}(\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)\nabla u^{f+\varepsilon\varphi}(x)\
\varphi(x))dx\\\ &=\varepsilon\int_{B}\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}(x)|)\nabla
u^{f+\varepsilon\varphi}(x)\cdot\nabla\varphi(x)dx,\end{split}$

where in the first equality we exploited that
$\operatorname{div}(\sigma_{B}(x,|\nabla u^{f+\varepsilon\varphi}|)\nabla
u^{f+\varepsilon\varphi})=0$, since $u^{f+\varepsilon\varphi}$ is a solution
of (2.7); in the second equality we applied the divergence Theorem, being
$c^{i}_{\varepsilon}$ (resp. $c^{i}_{0}$) the constant value of
$u^{f+\varepsilon\varphi}$ (resp. $u^{f}$) on $\partial A_{i}$, $i=1,..,M$; in
the third equality we applied the divergence Theorem, together with the third
constraint of (2.7) but written for $u^{f+\varepsilon\varphi}$ and, in the
fourth equality, we exploited that $\operatorname{div}(\sigma_{B}(x,|\nabla
u^{f+\varepsilon\varphi}|)\nabla u^{f+\varepsilon\varphi}(x)=0$. Similar
considerations can be applied to term $II$. The, the conclusion (3.10) follows
by arguing as for Claim (i), but using (3.4) of Lemma 3.1. ∎

##  4\. The connection between the Dirichlet Energy and the DtN

The main aim of this Section is to prove that, in each of the four cases of
nonlinearity we are interested in, there is a connection between the Dirichlet
Energy and the DtN operator. The DtN operator is of paramount relevance in
inverse problems, because it can be measured from boundary data, at difference
of the Dirichlet Energy that is an integral quantity that requires the
knowledge of the solution inside the domain $\Omega$. On the other hand, the
Dirichlet Energy is a relevant quantity because it satisfies a Monotonicity
Principle, as shown in Section 5.1. The intimate connection between the
Dirichlet Energy and the DtN operator is given by the the Gâteaux derivative.
Specifically, it turns out that the Gâteaux derivative of the mapping
$f\mapsto\mathbb{E}_{\sigma}\circ\mathbb{U}_{\sigma}$ is equal to
$\Lambda_{\sigma}$, where $\mathbb{U}_{\sigma}$ is the operator mapping the
boundary data $f$ into the corresponding solution $u^{f}$:

$\mathbb{U}_{\sigma}:f\in X^{p}_{\diamond}(\partial\Omega)\mapsto u^{f}\in W,$

where $W$ is a proper functional space (see Proposition 4.1). In line with
[16], it is possible to prove the following Proposition.

######  Proposition 4.1.

Let $1<p,q<+\infty$, $f\in X_{\diamond}^{p}(\partial\Omega)$ and ${\sigma}$
satisfying (A1), (A2), (A3), (A4). Then

$\text{\emph{d}}(\mathbb{E}_{\sigma}\circ\mathbb{U_{\sigma}})(f)=\Lambda_{\sigma}(f).$

###### Proof.

In the following we prove that

$\text{\emph{d}}(\mathbb{E}_{\sigma}\circ\mathbb{U_{\sigma}})(f;\varphi)=\langle\Lambda_{\sigma}(f),\varphi\rangle\quad\forall\varphi\in
W$ (4.1)

where

  * (i)

if $A=\emptyset$, then $W=W^{1,p}(\Omega)$;

  * (ii)

if ${\sigma}$ satisfies (BX), then $W=W^{1,r}(\Omega)$ where
$r=\min\\{p,q\\}$;

  * (iii)

if ${\sigma}$ satisfies (C), then $W=W^{1,p}(B)$;

  * (iv)

if ${\sigma}$ satisfies (D), then $W=W^{1,p}(B)$.

Since the $\sigma(x,E)E$ product is strictly increasing w.r.t. $E>0$ and
$\partial_{E}Q_{\sigma}(x,E)=\sigma(x,E)E$, it results that

$\begin{split}0\leq\sigma(x,E_{1})E_{1}(E_{2}-E_{1})&\leq
Q_{\sigma}(x,E_{2})-Q_{\sigma}(x,E_{1})\leq\sigma(x,E_{2})E_{2}(E_{2}-E_{1}),\end{split}$
(4.2)

for a.e. $x\in\overline{\Omega}$ and for any $0<E_{1}\leq E_{2}$. $\bullet$
Case (i). Let $u^{f+\varepsilon\varphi}\in W^{1,p}(\Omega)$ (resp. $u^{f}\in
W^{1,p}(\Omega)$) be the minimizer of (2.3) corresponding to
$f+\varepsilon\varphi$ (resp. $f$). For sake of simplicity, we set

$\mathbb{G}_{\sigma}(f):=(\mathbb{E}_{\sigma}\circ\mathbb{U}_{\sigma})(f)=\mathbb{E}_{\sigma}\left(u^{f}\right).$
(4.3)

For any $\varepsilon>0$, we consider the following finite quantities

$\displaystyle\mathbb{G}_{\sigma}(f)$
$\displaystyle=\int_{\Omega}Q_{\sigma}(x,|\nabla
u^{f}(x)|)\text{d}x\leq\int_{\Omega}Q_{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)\right|\right)\text{d}x,$
(4.4) $\displaystyle\mathbb{G}_{\sigma}(f+\varepsilon\varphi)$
$\displaystyle=\int_{\Omega}Q_{\sigma}\left(x,\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|\right)\text{d}x\leq\int_{\Omega}Q_{\sigma}\left(x,\left|\nabla
u^{f}(x)+\varepsilon\nabla\varphi(x)\right|\right)\text{d}x,$ (4.5)

where we have used the minimality of the Dirichlet Energy on the solutions.
Let us consider the incremental ratio
$\left[\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)\right]/\varepsilon$.
By (4.5), we have

$\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\leq\frac{1}{\varepsilon}\int_{\Omega}Q_{\sigma}\left(x,|\nabla
u^{f}+\varepsilon\nabla\varphi(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f}(x)|\right)\text{d}x.$ (4.6)

The magnitude of the integrand in (4.6) can be bounded easily from above.
Indeed, by (4.2), we have

$\begin{split}&\left|\frac{Q_{\sigma}\left(x,|\nabla{u}(x)+\varepsilon\nabla\varphi(x)|\right)-Q_{\sigma}\left(x,|\nabla
u(x)|\right)}{\varepsilon}\right|\\\
&\qquad\leq\frac{1}{|\varepsilon|}\sigma\left(x,\left|\nabla{u}(x)|+|\varepsilon\nabla\varphi(x)\right|\right)(\left|\nabla
u(x)|+|\varepsilon\nabla\varphi(x)\right|)|\varepsilon\nabla\varphi(x)|\\\
&\qquad\leq\sigma\left(x,\left|\nabla
u(x)|+|\nabla\varphi(x)\right|\right)(\left|\nabla
u(x)|+|\nabla\varphi(x)\right|)|\nabla\varphi(x)|,\end{split}$ (4.7)

for any $\varepsilon<1$. By the assumption (A3), the last term in (4.7) is a
$L^{1}$ function and hence, by Lebesgue Dominate Convergence Theorem, we can
pass to the limit in (4.6):

$\begin{split}&\limsup_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\\\
&\leq\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon}\int_{\Omega}Q_{\sigma}\left(x,|\nabla
u^{f}(x)+\varepsilon\nabla\varphi(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f}(x)|\right)\text{d}x\\\ &=\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\
\text{\emph{d}}x.\end{split}$ (4.8)

To prove that the limit of the incremental ratio at the l.h.s. of (4.6)
exists, for $\varepsilon\to 0^{+}$, it is sufficient to prove that the
$\liminf$ is greater or equal to the r.h.s. of (4.8), that is

$\liminf_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\geq\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\ \text{\emph{d}}x.$

Specifically, let $\\{\varepsilon_{j}\\}_{j\in\mathbb{N}}$ be a sequence such
that $\varepsilon_{j}\to 0^{+}$ and

$\liminf_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}=\lim_{j\to+\infty}\frac{\mathbb{G}_{\sigma}(f+\varepsilon_{j}\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon_{j}}.$

By using (4.4), we have

$\begin{split}\frac{\mathbb{G}_{\sigma}(f+\varepsilon_{j}\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon_{j}}&\geq\frac{1}{\varepsilon_{j}}\int_{\Omega}Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j}\varphi}(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j}\varphi}(x)-\varepsilon_{j}\nabla\varphi(x)|\right)\text{d}x.\end{split}$
(4.9)

To show that it is possible to pass to the limit, as $j\to+\infty$, in the
r.h.s of (4.9), we prove that (i) the integrand is convergent and (ii) there
exists a dominating summable function for the integrand. Then, the limit
exists thanks to the Dominated Convergence Theorem. In order to prove the
pointwise convergence of the argument of r.h.s. of (4.9), we observe that
$Q_{\sigma}(x,\cdot)$ is in $C^{1}([0,+\infty[)$ for a.e. $x\in\Omega$ and,
therefore, the integrand of the r.h.s. of (4.9) can be written as

$\begin{split}&\frac{Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon\varphi}(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)|\right)}{\varepsilon}\\\
&\qquad\qquad=\sigma\left(x,\xi_{\varepsilon}^{\prime}\right)\xi_{\varepsilon}^{\prime}\frac{\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|-\left|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)\right|}{\varepsilon}\\\
\end{split}$ (4.10)

with

$\begin{split}\xi_{\varepsilon}^{\prime}\in&[\min\\{\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|,\left|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)\right|\\},\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\max\\{\left|\nabla
u^{f+\varepsilon\varphi}(x)\right|,\left|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)\right|\\}].\end{split}$

The r.h.s. in (4.10) is equal to

$\left\\{\begin{array}[]{@{}l@{}}\sigma\left(x,\xi_{\varepsilon}^{\prime}\right)\xi_{\varepsilon}^{\prime}\dfrac{2\nabla
u^{f+\varepsilon\varphi}(x)\cdot\nabla\varphi(x)-\varepsilon\left|\nabla\varphi(x)\right|^{2}}{|\nabla
u^{f+\varepsilon\varphi}(x)|+|\nabla
u^{f+\varepsilon\varphi}(x)-\varepsilon\nabla\varphi(x)|}\quad\text{if}\
|\nabla u^{f+\varepsilon\varphi}(x)|\neq 0,\vspace{0.2cm}\\\
\sigma\left(x,\xi_{\varepsilon}^{\prime}\right)\xi_{\varepsilon}^{\prime}\operatorname{sign}(-\varepsilon)|\nabla\varphi(x)|\qquad\qquad\qquad\qquad\qquad\text{otherwise}.\end{array}\right.$
(4.11)

Exploiting that $\xi_{\varepsilon}^{\prime}\to|\nabla u^{f}(x)|$ as
$\varepsilon\to 0^{+}$ for a.e. $x\in\Omega$ thanks to (3.7), we have that
both expression appearing in (4.11) tend to ${\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)$, as
$\varepsilon\to 0^{+}$. To prove the existence of a summable dominating
function for the integrand of (4.9), we remind from Proposition 3.2 (see the
convergence result (3.7)), that $\nabla u^{f+\varepsilon\varphi}\to\nabla
u^{f}$ strongly in $L^{p}(\Omega)$, when $\varepsilon\to 0^{+}$. By standard
arguments, we can say that there exists a subsequence
$\\{\varepsilon_{j_{h}}\\}_{h\in\mathbb{N}}$ such that $\nabla
u^{f+\varepsilon_{j_{h}}\varphi}\to\nabla u^{f}$ a.e. in $\Omega$, and there
exists a measurable real function $\psi$, defined on $\Omega$, such that

$|\nabla u^{f+\varepsilon_{j_{h}}\varphi}|\leq\psi\ \text{in}\
\Omega,\quad\int_{\Omega}\psi(x)^{p}dx<+\infty.$ (4.12)

When $p\geq 2$, for any $\varepsilon_{j_{h}}<1$, we have

$\begin{split}&\left|\frac{Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)-\varepsilon_{j_{h}}\nabla\varphi(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|\right)}{\varepsilon_{j_{h}}}\right|\\\
&\leq\sigma(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|)(|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|)|\nabla\varphi(x)|\\\
&\leq\overline{\sigma}\left[1+\left(\frac{|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|}{E_{0}}\right)^{p-2}\right]\left(|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|\right)|\nabla\varphi(x)|\\\
&\leq\overline{\sigma}\left[1+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{p-2}\right](\psi(x)+|\nabla\varphi(x)|)^{2}\\\
&\leq\overline{\sigma}\left[\left({\psi(x)+|\nabla\varphi(x)|}\right)^{2}+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{p}\right],\end{split}$
(4.13)

where in the first inequality we have used (4.2) and the monotonicity (A2); in
the second inequality we have used (A3); in the third inequality we have used
(4.12); the fourth inequality is straightforward. It is worth noting that
$\overline{\sigma}$ is independent of $\varepsilon_{j_{h}}$. The inequalities
chain (4.13) for $1<p<2$ can be similarly proven by using the second line of
the assumption (A3). Summing up, by applying the Dominate Convergence Theorem
and (4.8), it turns out that

$\lim_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\\\
=\int_{\Omega}{\sigma}\left(x,|\nabla u^{f}(x)|\right)\nabla
u^{f}(x)\cdot\nabla\varphi(x)\text{d}x.$

The conclusion follows by observing that this equation is nothing but (4.1).
$\bullet$ Case (ii). Let $u^{f+\varepsilon\varphi}\in W^{1,r}(\Omega)$ (resp.
$u^{f}\in W^{1,r}(\Omega)$) be the minimizer of (2.3) corresponding to
$f+\varepsilon\varphi$ (resp. $f$). As in the case (i), we consider the
function $\mathbb{G}_{\sigma}$ defined in (4.3). Analogously to case (i), we
can upper bound the lim sup and lower bound the lim inf of the incremetal
ratio
$\left[\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)\right]/\varepsilon$
with the same quantity. Regarding the lim sup, arguing as in case (i), we
obtain (4.8), that is

$\begin{split}\limsup_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\leq\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\
\text{\emph{d}}x.\end{split}$ (4.14)

Regarding the lim inf, we have to show that is possible to pass to the limit
in the r.h.s. of (4.9) by using the Dominated Convergence Theorem. Following
line by line the analogous part of the proof of the case (i), one can show
that the integrand is convergent to ${\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)$, as
$\varepsilon\to 0^{+}$. Therefore, it remains to prove that there exists a
dominating summable function for the integrand appearing at the r.h.s of
(4.9). To prove this, we notice that in Proposition 3.2 (see the convergence
result (3.8)), we have showed that $\nabla u^{f+\varepsilon\varphi}\to\nabla
u^{f}$ in $L^{p}(B)$ and $\nabla u^{f+\varepsilon\varphi}\to\nabla u^{f}$ in
$L^{q}(A)$, when $\varepsilon\to 0^{+}$. Then by standard arguments, we can
say that there exists a subsequence
$\\{\varepsilon_{j_{h}}\\}_{h\in\mathbb{N}}$ such that $\nabla
u^{f+\varepsilon_{j_{h}}\varphi}\to\nabla u^{f}$ a.e. in $\Omega$, and there
exists a measurable real function $\psi$, defined on $\Omega$, such that

$|\nabla u^{f+\varepsilon_{j_{h}}\varphi}|\leq\psi\ \text{in}\
\Omega,\quad\int_{B}\psi(x)^{p}dx+\int_{A}\psi(x)^{q}dx<+\infty.$ (4.15)

For any $p,q\geq 2$ and for any $\varepsilon_{j_{h}}$, we have

$\begin{split}&\int_{\Omega}\left|\frac{Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)-\varepsilon_{j_{h}}\nabla\varphi(x)|\right)-Q_{\sigma}\left(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|\right)}{\varepsilon_{j_{h}}}\right|dx\\\
&\leq\displaystyle\int_{\Omega}\sigma(x,|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|)(|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|)|\nabla\varphi(x)|dx\\\
&\leq\overline{\sigma}\displaystyle\int_{B}\left[1+\left(\frac{|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|}{E_{0}}\right)^{p-2}\right]\left(|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|\right)|\nabla\varphi(x)|dx\\\
&\ \ +\overline{\sigma}\displaystyle\int_{A}\left[1+\left(\frac{|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|}{E_{0}}\right)^{q-2}\right]\left(|\nabla
u^{f+\varepsilon_{j_{h}}\varphi}(x)|+|\varepsilon_{j_{h}}\nabla\varphi(x)|\right)|\nabla\varphi(x)|dx\\\
&\leq\overline{\sigma}\displaystyle\int_{B}\left[1+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{p-2}\right](\psi(x)+|\nabla\varphi(x)|)^{2}dx\\\
&\ \
+\overline{\sigma}\displaystyle\int_{A}\left[1+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{q-2}\right](\psi(x)+|\nabla\varphi(x)|)^{2}dx\\\
&\leq\overline{\sigma}\displaystyle\int_{B}\left(\psi(x)+|\nabla\varphi(x)|\right)^{2}+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{p}dx\\\
&\ \
+\overline{\sigma}\displaystyle\int_{A}\left(\psi(x)+|\nabla\varphi(x)|\right)^{2}+\left(\frac{\psi(x)+|\nabla\varphi(x)|}{E_{0}}\right)^{q}dx,\end{split}$
(4.16)

where in the first inequality we used (4.2) and the monotonicity (A2); in the
second inequality we used (A3); in the third inequality we used (4.15); the
fourth inequality is straightforward. The inequalities chain (4.16) for
$1<p<2$ can be similarly proven by using the second line of the assumption
(A3). By applying the Dominate Convergence Theorem, the limit as
$\varepsilon_{j_{h}}\to 0$ in (4.9) provides

$\begin{split}\liminf_{\varepsilon\to
0^{+}}\frac{\mathbb{G}_{\sigma}(f+\varepsilon\varphi)-\mathbb{G}_{\sigma}(f)}{\varepsilon}\geq\int_{\Omega}{\sigma}\left(x,\left|\nabla
u^{f}(x)\right|\right)\nabla u^{f}(x)\cdot\nabla\varphi(x)\
\text{\emph{d}}x.\end{split}$ (4.17)

Finally, the conclusion follows by combining (4.14) and (4.17). $\bullet$ Case
(iii). The proof is obtained as in the case (ii) but replacing the role of
$\Omega$ with $B$ and exploiting the strong convergence result of Proposition
3.2, case (iii). $\bullet$ Claim (iv). The proof is obtained as in the case
(ii) but replacing the role of $\Omega$ with $B$ and exploiting the strong
convergence result of Proposition 3.2, case (iv). ∎

As straightforward consequence of Proposition 4.1, we have the following
expressions for the derivative of $\mathbb{E}_{\sigma}$, with respect to $u$
and $u^{f}$, respectively.

######  Proposition 4.2.

Let $f\in X_{\diamond}(\partial\Omega)$, ${\sigma}$ satisfying (A1), (A2),
(A3) and (A4), and $u,\varphi\in W$, then

$\displaystyle\text{\emph{d}}\mathbb{F}_{\sigma}(u;\varphi)=\int_{\Omega}{\sigma}\left(x,\left|\nabla
u(x)\right|\right)\nabla u(x)\cdot\nabla\varphi(x)\ \text{\emph{d}}x,$
$\displaystyle\text{\emph{d}}\mathbb{F}_{\sigma}(u^{f};\varphi)=\langle\Lambda_{\sigma}(f),\varphi\rangle,$

where

  * (i)

if $A=\emptyset$, then $W=W^{1,p}(\Omega)$ and $u^{f}$ is the minimizer of
(2.3);

  * (ii)

if ${\sigma}$ satisfies (BX), then $W=W^{1,r}(\Omega)$ where $r=\min\\{p,q\\}$
and $u^{f}$ is the minimizer of (2.3);

  * (iii)

if ${\sigma}$ satisfies (C), then $W=W^{1,p}(B)$ and $u^{f}$ is the minimizer
of (2.6);

  * (iv)

if ${\sigma}$ satisfies (D), then $W=W^{1,p}(B)$ and $u^{f}$ is the minimizer
of (2.8).

##  5\. The Monotonicity Principle

Here we prove the Monotonicity Principle of the average-DtN, with respect to
the electrical conductivity. Specifically, first we demonstrate a Monotonicity
Principle for the Dirichlet Energy (Theorem 5.1)

$\sigma_{1}\leq\sigma_{2}\quad\Longrightarrow\quad\mathbb{E}_{\sigma_{1}}(u_{1}^{f})\leq\mathbb{E}_{\sigma_{2}}(u_{2}^{f})\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega),$

where $u_{i}^{f}$ is the minimizer of the Dirichlet Energy with
$\sigma=\sigma_{i}$, for $i=1,2$ and $f$ is the applied boundary voltage and,
then, we “translate” the Dirichlet Energy in terms of boundary data via the
following fundamental relationship (see Theorem 5.2)

$\mathbb{E}_{\sigma}\left(u^{f}\right)=\left\langle\overline{\Lambda}_{\sigma}\left(f\right),f\right\rangle\quad\forall
f\in X^{p}_{\diamond}(\partial\Omega).$ (5.1)

The operator $\overline{\Lambda}_{\sigma}$ appearing in (5.1) is called
Average DtN and is defined as

$\overline{\Lambda}_{\sigma}:f\in
X^{p}_{\diamond}(\partial\Omega)\mapsto\int_{0}^{1}\Lambda_{\sigma}\left(\alpha
f\right)\text{d}\alpha\in X^{p}_{\diamond}(\partial\Omega)^{\prime},$

where

$\langle\overline{\Lambda}_{\sigma}(f),\varphi\rangle=\int_{0}^{1}\left\langle\Lambda_{\sigma}\left(\alpha
f\right),\varphi\right\rangle\text{d}\alpha\quad\forall\varphi\in
X^{p}_{\diamond}(\partial\Omega).$

Operator $\overline{\Lambda}_{\sigma}$ gives the average flow of the
electrical current density through $\partial\Omega$ for an applied boundary
potential of the type $\alpha f$, for $\alpha\in\left[0,1\right]$. It is the
key “tool” for transferring the Monotonicity Principle for the Dirichlet
Energy to the boundary data and it replaces the usual DtN operator
$\Lambda_{\sigma}$ when treating nonlinear problems (see [16] for its very
first introduction). Finally, as a byproduct of Theorems 5.1 and 5.2, we get
the Monotonicity Principle for the operator $\overline{\Lambda}_{\sigma}$ (see
Theorem 5.3), i.e.

$\sigma_{1}\leq\sigma_{2}\quad\Longrightarrow\quad\left\langle\overline{\Lambda}_{\sigma_{1}}\left(f\right),f\right\rangle\leq\left\langle\overline{\Lambda}_{\sigma_{2}}\left(f\right),f\right\rangle\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega).$ (5.2)

In (5.2) $\sigma_{1}\leq\sigma_{2}$ means that

$\sigma_{1}(x,E)\leq\sigma_{2}(x,E)\quad\text{for a.e.}\
x\in\overline{\Omega}\ \text{and}\ \forall\ E>0.$ (5.3)

Furthermore, let us also observe that there exist four possibly different
subsets $B_{i},A_{i}$ of $\Omega$, $i=1,2$, such that $\sigma_{i}$ is defined
on $\Omega=B_{i}\cup A_{i}$, $i=1,2$, with the same growth exponent $p$ on
$B_{i}$ and possibly different growth exponents $q_{i}$ on $A_{i}$, $i=1,2$.
Therefore, no inclusion relation is imposed between $A_{1}$ and $A_{2}$ but
only the validity of the relation (5.3). The Monotonicity Principle for the
Dirichlet Energy in the nonlinear case can be easily proved as follows.

######  Theorem 5.1.

Let $1<p,q<+\infty$, $\sigma_{1}$, $\sigma_{2}$ satisfying (A1), (A2), (A3)
and (A4), then

$\sigma_{1}\leq\sigma_{2}\quad\Longrightarrow\quad\mathbb{E}_{\sigma_{1}}(u_{1}^{f})\leq\mathbb{E}_{\sigma_{2}}(u_{2}^{f})\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega),$

where $\sigma_{1}\leq\sigma_{2}$ is meant in the sense (5.3) and

  * (i)

if $A=\emptyset$, then $u_{1}^{f}$ (resp. $u_{2}^{f}$) is the minimizer of
(2.3) corresponding to $\sigma_{1}$ (resp. $\sigma_{2}$);

  * (ii)

if $\sigma_{1},\sigma_{2}$ satisfy (BX), then $u_{1}^{f}$ (resp. $u_{2}^{f}$)
is the minimizer of (2.3) corresponding to $\sigma_{1}$ (resp. $\sigma_{2}$);

  * (iii)

if $\sigma_{1},\sigma_{2}$ satisfy (C), then $u_{1}^{f}$ (resp. $u_{2}^{f}$)
is the minimizer of (2.6) corresponding to $\sigma_{1}$ (resp. $\sigma_{2}$)

  * (iv)

if $\sigma_{1},\sigma_{2}$ satisfy (D), then then $u_{1}^{f}$ (resp.
$u_{2}^{f}$) is the minimizer of (2.8) corresponding to $\sigma_{1}$ (resp.
$\sigma_{2}$).

###### Proof.

Since $u_{2}^{f}$ is an admissible function for problem (2.4) with
$\sigma=\sigma_{1}$, we have

$\begin{split}\mathbb{E}_{\sigma_{1}}(u_{1}^{f})&\leq\mathbb{E}_{\sigma_{1}}(u_{2}^{f})=\int_{\Omega}Q_{\sigma_{1}}(x,|\nabla
u_{2}^{f}(x)|)\ \text{d}x\\\ &=\int_{\Omega}\int_{0}^{|\nabla
u_{2}^{f}(x)|}\sigma_{1}\left(x,\xi\right)\xi\ \text{d}\xi\
\text{d}x\leq\int_{\Omega}\int_{0}^{|\nabla
u_{2}^{f}(x)|}\sigma_{2}\left(x,\xi\right)\xi\ \text{d}\xi\
\text{d}x=\mathbb{E}_{\sigma_{2}}(u_{2}^{f}),\end{split}$

where the second inequality follows from the assumption
$\sigma_{1}\leq\sigma_{2}$. ∎

The Dirichlet Energy can be transferred to a boundary measurement involving
$\overline{\Lambda}_{\sigma}$ as follows.

######  Theorem 5.2.

Let $1<p,q<+\infty$ and $\sigma$ satisfying (A1), (A2), (A3) and (A4), then

$\mathbb{E}_{\sigma}\left(u^{f}\right)=\left\langle\overline{\Lambda}_{\sigma}\left(f\right),f\right\rangle\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega),$

where

  * (i)

if $A=\emptyset$, then $u^{f}$ is the minimizer of (2.3);

  * (ii)

if ${\sigma}$ satisfies (BX), then $u^{f}$ is the minimizer of (2.3);

  * (iii)

if ${\sigma}$ satisfies (C), then then $u^{f}$ is the minimizer of (2.6);

  * (iv)

if ${\sigma}$ satisfies (D), then then $u^{f}$ is the minimizer of (2.8).

###### Proof.

For $\alpha\in[0,1]$, we set

$g(\alpha):=(\mathbb{F}_{\sigma}\circ\mathbb{U_{\sigma}})(\alpha f).$

By Proposition 4.1, since $\mathbb{F}_{\sigma}\circ\mathbb{U_{\sigma}}$ is
Gâteaux-differentiable, $g$ is differentiable. By replacing $f$ and $\varphi$
with $\alpha f$ and $f$, we have

$g^{\prime}(\alpha)=\text{d}(\mathbb{F}_{\sigma}\circ\mathbb{U_{\sigma}})(\alpha
f;f)=\langle\Lambda_{\sigma}(\alpha f),f\rangle.$

Eventually, by integrating over the interval $\left[0,1\right]$, we obtain

$\mathbb{F}_{\sigma}(u^{f})=(\mathbb{F}_{\sigma}\circ\mathbb{U_{\sigma}})(f)=g(1)=\int_{0}^{1}g^{\prime}(\alpha)\text{d}\alpha=\int_{0}^{1}\langle\Lambda_{\sigma}(\alpha
f),f\rangle\
\text{d}\alpha=\left\langle\overline{\Lambda}_{\sigma}\left(f\right),f\right\rangle.$

∎

Finally, we have the Monotonicity Principle for the average DtN
$\overline{\Lambda}_{\sigma}$.

######  Theorem 5.3.

Let $1<p,q<+\infty$ and $\sigma_{1}$, $\sigma_{2}$ satisfying (A1), (A2), (A3)
and (A4), then

$\sigma_{1}\leq\sigma_{2}\quad\Longrightarrow\quad\left\langle\overline{\Lambda}_{\sigma_{1}}\left(f\right),f\right\rangle\leq\left\langle\overline{\Lambda}_{\sigma_{2}}\left(f\right),f\right\rangle\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega),$ (5.4)

where $\sigma_{1}\leq\sigma_{2}$ is meant in the sense (5.3), and one of the
following holds

  1. (i)

$A=\emptyset$;

  2. (ii)

${\sigma_{1}},{\sigma_{2}}$ satisfy (BX);

  3. (iii)

${\sigma_{1}},{\sigma_{2}}$ satisfy (C);

  4. (iv)

${\sigma_{1}},{\sigma_{2}}$ satisfy (D).

###### Proof.

We consider $u_{i}^{f}$ be the unique solution with $\sigma=\sigma_{i}$,
$i=1,2$, and boundary value $f$. By Theorems 5.1 and 5.2, we have

$\begin{split}\left\langle\overline{\Lambda}_{\sigma_{1}}\left(f\right),f\right\rangle=\mathbb{E}_{\sigma_{1}}(u^{f}_{1})\leq\mathbb{E}_{\sigma_{2}}(u^{f}_{2})=\left\langle\overline{\Lambda}_{\sigma_{2}}\left(f\right),f\right\rangle\end{split}$

that proves (5.4). ∎

If the constitutive relationship $J_{\sigma}=J_{\sigma}\left(x,E\right)$ is
strictly decreasing with respect to $E$, then the Monotonicity is reversed,
i.e.

$\sigma_{1}\leq\sigma_{2}\quad\Longrightarrow\quad\left\langle\overline{\Lambda}_{\sigma_{1}}\left(f\right),f\right\rangle\geq\left\langle\overline{\Lambda}_{\sigma_{2}}\left(f\right),f\right\rangle\quad\forall
f\in X_{\diamond}^{p}(\partial\Omega).$

##  6\. Numerical Examples

In this Section we provide a numerical evidence of the Monotonicity Principle
(5.4), proved in Theorem 5.3. The numerical examples are inspired by a real-
world application: nonlinear conductive materials in superconductive wires. In
the following, we refer to a standard Bi-2212 commercial round wire. The
geometry of the cable is shown in Figure 3.

Figure 3. Typical geometry for the cross section of superconducting cables.
The cable is made of several petals embedded in a linear material (36 petals
for (a), and 18 petals for (b) and (c)). Each petal is constituted by many
thin superconductive filaments (19 wires for (a), 37 wires for (b) and 61 for
(c)). This figure is reproduced with the kind permission of the publisher MDPI
(see [49, Fig. 4]).

It is made by several superconducting “petals”embedded in a linear matrix. The
matrix is made by an AgMg alloy (Ag 0.2%, Mg 99.8%, electrical conductivity of
$55.5\,\text{MS/m}$ [39]). The nonlinear electrical conductivity of petals,
can be derived by the so-called _E-J Power Law_ [45]

$\sigma(E)=\frac{J_{c}}{E_{0}}\left(\frac{E}{E_{0}}\right)^{\frac{1-n}{n}}.$

For the case of interest, we assume

$J_{c}=8000\,\text{A/mm${}^{2}$},\quad n=27,$

that are realistic values for such type of materials [7]. The parameter
$E_{0}=0.1\,\text{mV/m}$ is a reference value independent from the particular
material [67]. The external radius $R_{e}$ is equal to 0.6 mm [34]. We
highlight that region $A$ corresponds to the petals (growth exponent $1<q<2$),
while region $B$ corresponds to the linear AgMg matrix (growth exponents
$p=2$). For the sake of simplicity, the numerical simulations are carried out
by considering each petalmade by a single superconductive wire, instead of
many thin filaments (see Figure 4 (left)). As first example, we consider a
damaged wire with a crack in the AgMg matrix. Specifically, we assume the
crack to be a perfect electric insulator (PEI). The geometry of the crack is
showed in Figure 4.

Figure 4. Left: the geometry of the healthy wire. In white the linear matrix
and in gray the “petals” made by the superconductive material. Right: the
geometry of the damaged wire. The black region represents the crack.

Let $f$ be the applied boundary data,
$\mathbb{E}_{\sigma_{0}}(f)=\langle\overline{\Lambda}_{\sigma_{0}}(f),f\rangle$
the Dirichlet Energy related to the healthy wire and
$\mathbb{E}_{\sigma_{1}}(f)=\langle\overline{\Lambda}_{\sigma_{1}}(f),f\rangle$
the Dirichlet Energy related to the damaged wire. The presence of the crack
reduces the electrical conductivity a region of the domain and, therefore, we
expect a decrease of the Dirichlet Energy decreases, i.e.
$\mathbb{E}_{\sigma_{1}}(f)\geq\mathbb{E}_{\sigma_{0}}(f)$, $\forall f$. The
numerical results in Table 2 confirm the validity of the Monotonicity
Principle.

$\begin{array}[]{cccc}\hline\cr\hline\cr
f\,\text{(V)}&\mathbb{E}_{\sigma_{0}}(f)\,\text{(W)}&\mathbb{E}_{\sigma_{1}}(f)\,\text{(W)}&\mathbb{E}_{\sigma_{0}}(f)\,-\mathbb{E}_{\sigma_{1}}(f)\,\text{(W)}\\\
\hline\cr 100x&6.6610\times 10^{5}&6.4155\times 10^{5}&2.4548\times 10^{4}\\\
300x&3.4927\times 10^{6}&3.4194\times 10^{6}&7.3253\times 10^{4}\\\
500x&7.4529\times 10^{6}&7.3145\times 10^{5}&1.3846\times 10^{5}\\\
0.1\sin(\theta)&1.4723\times 10^{6}&1.4501\times 10^{6}&2.2170\times 10^{4}\\\
0.3\sin(\theta)&7.4530\times 10^{6}&7.3499\times 10^{6}&1.0313\times 10^{5}\\\
0.5\sin(\theta)&1.6446\times 10^{7}&1.6202\times 10^{7}&2.4423\times 10^{5}\\\
0.1\sin(\theta)-0.2\cos(2\theta)&7.7056\times 10^{6}&7.5517\times
10^{6}&1.5391\times 10^{5}\\\ 100\exp{(x^{2}+2y)}&1.9288\times
10^{6}&1.9009\times 10^{6}&2.7932\times 10^{4}\\\
300\exp{(x^{2}+2y)}&9.8394\times 10^{6}&9.7004\times 10^{6}&1.3895\times
10^{5}\\\ 500\exp{(x^{2}+2y)}&2.2051\times 10^{7}&2.1712\times
10^{7}&3.3846\times 10^{5}\\\ \hline\cr\hline\cr\end{array}$ Table 2.
Numerical evaluation of $\mathbb{E}_{\sigma_{0}}(f)$ and
$\mathbb{E}_{\sigma_{1}}(f)$ for the case of damaged matrix. The difference
$\mathbb{E}_{\sigma_{0}}(f)-\mathbb{E}_{\sigma_{1}}(f)$ is greater than $0$.

Another configuration of interested is that of a superconductive wire with a
damaged petal, that is assumed to be a perfect insulator. The geometry is
depicted in Figure 5.

Figure 5. Left: the geometry for the healthy cable. Right: the cable with the
damaged petal (PEI) in black.

As in the previous case, since the damage causes a loss of conductive
material, we expect that the Dirichlet Energy
$\mathbb{E}_{\sigma_{1}}(f)=\langle\overline{\Lambda}_{\sigma_{1}}(f),f\rangle$
for the damaged wire is always less then the Dirichlet Energy
$\mathbb{E}_{\sigma_{0}}(f)=\langle\overline{\Lambda}_{\sigma_{0}}(f),f\rangle$
evaluated in presence of the healthy wire. In Table 3 we report the results of
the numerical simulations which agree with Theorem 5.3.

$\begin{array}[]{cccc}\hline\cr\hline\cr
f\,\text{(V)}&\mathbb{E}_{\sigma_{0}}(f)\,\text{(W)}&\mathbb{E}_{\sigma_{1}}(f)\,\text{(W)}&\mathbb{E}_{\sigma_{0}}(f)-\mathbb{E}_{\sigma_{1}}(f)\,\text{(W)}\\\
\hline\cr 100x&6.6610\times 10^{5}&6.2682\times 10^{5}&3.9284\times 10^{4}\\\
300x&3.4927\times 10^{6}&3.3531\times 10^{6}&1.3951\times 10^{5}\\\
500x&7.4529\times 10^{6}&7.2170\times 10^{5}&2.3591\times 10^{5}\\\
0.1\sin(\theta)&1.4723\times 10^{6}&1.4082\times 10^{6}&6.4086\times 10^{4}\\\
0.3\sin(\theta)&7.4530\times 10^{6}&7.2645\times 10^{6}&1.8850\times 10^{5}\\\
0.5\sin(\theta)&1.6446\times 10^{7}&1.6139\times 10^{7}&3.0696\times 10^{5}\\\
0.1\sin(\theta)-0.2\cos(2\theta)&7.7056\times 10^{6}&7.4816\times
10^{6}&2.2401\times 10^{5}\\\ 100\exp{(x^{2}+2y)}&1.9288\times
10^{6}&1.8513\times 10^{6}&7.7496\times 10^{4}\\\
300\exp{(x^{2}+2y)}&9.8394\times 10^{6}&9.6152\times 10^{6}&2.2418\times
10^{5}\\\ 500\exp{(x^{2}+2y)}&2.2051\times 10^{7}&2.1685\times
10^{7}&3.6611\times 10^{5}\\\ \hline\cr\hline\cr\end{array}$ Table 3.
Numerical evaluation of $\mathbb{E}_{\sigma_{0}}(f)$ and
$\mathbb{E}_{\sigma_{1}}(f)$ for the case of the damaged petal. The difference
$\mathbb{E}_{\sigma_{0}}(f)-\mathbb{E}_{\sigma_{1}}(f)$ is greater than $0$.

##  7\. Conclusions

The contribution of this paper is focused on the Monotonicity Principle for
nonlinear materials. Specifically, it covers the cases of (i) nonlinear
materials with piecewise growth exponents and (ii) nonlinear materials with
growth exponent in $(1,2)$. These results enables to extend a fast imaging
method based on the Monotonicity Principle, to the wide class of problems with
two or more materials, where at least one is nonlinear. The treatment is very
general and allows to model a wide variety of practical configurations, such
as Superconductors (SC), or Perfect Electrical Conductors(PEC), or Perfect
Electrical Insulators (PEI). The key results for proving the Monotonicity
Principle in this setting are the convergence of the scalar potential with
respect to the boundary data $f$ (Proposition 3.1), and the Proposition 4.1,
which gives the connection between the Dirichlet Energy and the measured
boundary data. Numerical results related to a superconducting cable confirm
the validity of the Monotonicity Principle.

## Acknowledgements

This work has been partially supported by GNAMPA of INdAM.

## Data availability statement

No new data were created or analysed in this study.

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