# Existence results for a nonlinear nonautonomus transmission problem via
domain perturbation

Matteo Dalla Riva  Dipartimento di Ingegneria, Università degli Studi di
Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy.
<EMAIL_ADDRESS>,  Riccardo Molinarolo  Dipartimento di Matematica e
Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico
II, Via Cintia, Monte S. Angelo, 80216 Napoli, Italy.
<EMAIL_ADDRESS>and  Paolo Musolino  Dipartimento di Scienze
Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, via Torino 155,
30170 Venezia Mestre, Italy<EMAIL_ADDRESS>
Abstract: In this paper we study the existence and the analytic dependence
upon domain perturbation of the solutions of a nonlinear nonautonomous
transmission problem for the Laplace equation. The problem is defined in a
pair of sets consisting of a perforated domain and an inclusion whose shape is
determined by a suitable diffeomorphism $\phi$. First we analyse the case in
which the inclusion is a fixed domain. Then we will perturb the inclusion and
study the arising boundary value problem and the dependence of a specific
family of solutions upon the perturbation parameter $\phi$.

Keywords: nonlinear nonautonomous transmission problem, domain perturbation,
Laplace equation, real analyticity, special nonlinear operators

2020 Mathematics Subject Classification: 35J25; 35B20; 31B10; 35J65; 47H30

##  1\. Introduction

We begin by introducing the geometric framework of our problem. We fix once
and for all a natural number

$n\in\mathbb{N}\setminus\\{0,1\\}$

that will be the dimension of the Euclidean space $\mathbb{R}^{n}$ we are
going to work in. We also fix a parameter

$\alpha\in]0,1[\,,$

which we use to define the regularity of our sets and functions. In order to
introduce the domains where our problem is defined, we take two sets
$\Omega^{o}$ and $\Omega^{i}$ that satisfy the following conditions:

(1) $\begin{split}&\mbox{$\Omega^{o}$, $\Omega^{i}$ are bounded open connected
subsets of $\mathbb{R}^{n}$ of class $C^{1,\alpha}$,}\\\ &\mbox{with exteriors
$\mathbb{R}^{n}\setminus\overline{\Omega^{o}}$ and
$\mathbb{R}^{n}\setminus\overline{\Omega^{i}}$ connected and
$\overline{\Omega^{i}}\subset\Omega^{o}$}\end{split}$

Figure 1. The domains $\Omega^{o}$ and $\Omega^{i}$ ($n=2$)

(see Figure 1). Here the superscript “$o$” stands for “outer domain” whereas
the superscript “$i$” stands for “inner domain.” We first want to introduce a
transmission problem in the pair of domains consisting of
$\Omega^{o}\setminus\overline{\Omega^{i}}$ and $\Omega^{i}$. Therefore, to
define the boundary conditions, we fix three functions

(2) $F_{1}\in C^{0}(\partial\Omega^{i}\times\mathbb{R}\times\mathbb{R}),\qquad
F_{2}\in C^{0}(\partial\Omega^{i}\times\mathbb{R}\times\mathbb{R}),\quad
f^{o}\in C^{0,\alpha}(\partial\Omega^{o}).$

The functions $F_{1}$ and $F_{2}$ determine the transmission conditions on the
inner boundary $\partial\Omega^{i}$. Instead, $f^{o}$ plays the role of the
Neumann datum on the outer boundary $\partial\Omega^{o}$. We consider the
following nonlinear transmission boundary value problem for a pair
$(u^{o},u^{i})\in C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i})\times
C^{1,\alpha}(\overline{\Omega^{i}})$:

(3) $\begin{cases}\Delta u^{o}=0&\mbox{in
}\Omega^{o}\setminus\overline{\Omega^{i}},\\\ \Delta u^{i}=0&\mbox{in
}\Omega^{i},\\\ \nu_{\Omega^{o}}(x)\cdot\nabla u^{o}(x)=f^{o}(x)&\forall
x\in\partial\Omega^{o},\\\ \nu_{\Omega^{i}}(x)\cdot\nabla
u^{o}(x)=F_{1}(x,u^{o}(x),u^{i}(x))&\forall x\in\partial\Omega^{i},\\\
\nu_{\Omega^{i}}(x)\cdot\nabla u^{i}(x)=F_{2}(x,u^{o}(x),u^{i}(x))&\forall
x\in\partial\Omega^{i},\end{cases}$

where $\nu_{\Omega^{o}}$ and $\nu_{\Omega^{i}}$ denote the outward unit normal
vector field to $\partial\Omega^{o}$ and to $\partial\Omega^{i}$,
respectively. We note that, _a priori_ , it is not clear why problem
$(\ref{princeq})$ should admit a classical solution. As a first result, we
prove that under suitable conditions on $F_{1}$ and $F_{2}$, problem (3) has
at least one solution $(u^{o},u^{i})\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i})\times
C^{1,\alpha}(\overline{\Omega^{i}})$. We notice that a problem similar to (3)
has been studied in Dalla Riva and Mishuris [7]. More precisely, in [7] the
authors consider a nonlinear transmission problem with a Dirichlet boundary
condition on $\partial\Omega^{o}$ and a jump type condition for the normal
derivative across the interface $\partial\Omega^{i}$. Then we study the
existence and the analytic dependence of the solutions of the transmission
problem (3) upon domain perturbation of the inclusion, i.e. of the inner set
$\Omega^{i}$. Hence, we introduce a “perturbed” version of problem (3): we fix
the external domain $\Omega^{o}$ and we assume that the boundary of the
internal domain is of the form $\phi(\partial\Omega^{i})$, where $\phi$ is a
diffeomorphism of $\partial\Omega^{i}$ into a subset of $\mathbb{R}^{n}$ that
belongs to the class

(4) $\begin{split}\mathcal{A}_{\partial\Omega^{i}}\equiv\Big{\\{}\phi&\in
C^{1,\alpha}(\partial\Omega^{i},\mathbb{R}^{n}):\,\phi\text{ is injective
and}\\\ &\text{the differential}\,d\phi(y)\text{ is injective for all
}y\in\partial\Omega^{i}\Big{\\}}.\end{split}$

Clearly, the identity function of $\partial\Omega^{i}$ belongs to the class
$\mathcal{A}_{\partial\Omega^{i}}$, and, for convenience, we set

(5) $\phi_{0}\equiv\text{id}_{\partial\Omega^{i}}.$

Then by the Jordan Leray Separation Theorem (cf., e.g., Deimling [9, Theorem
5.2] and [6, §A.4]), $\mathbb{R}^{n}\setminus\phi(\partial\Omega^{i})$ has
exactly two open connected components for all
$\phi\in\mathcal{A}_{\partial\Omega^{i}}$, and we define $\Omega^{i}[\phi]$ to
be the unique bounded open connected component of
$\mathbb{R}^{n}\setminus\phi(\partial\Omega^{i})$. We set

$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\equiv\left\\{\phi\in\mathcal{A}_{\partial\Omega^{i}}:\overline{\Omega^{i}[\phi]}\subset\Omega^{o}\right\\}\,.$

By assumption (1), $\phi_{0}\in\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}$.
Now let $\phi\in\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}$. We wish to
consider the following nonlinear transmission boundary value problem for a
pair of functions $(u^{o},u^{i})\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i}[\phi])\times
C^{1,\alpha}(\overline{\Omega^{i}[\phi]})$:

(6) $\begin{cases}\Delta u^{o}=0&\mbox{in
}\Omega^{o}\setminus\overline{\Omega^{i}[\phi]},\\\ \Delta u^{i}=0&\mbox{in
}\Omega^{i}[\phi],\\\ \nu_{\Omega^{o}}(x)\cdot\nabla u^{o}(x)=f^{o}(x)&\forall
x\in\partial\Omega^{o},\\\ \nu_{\Omega^{i}[\phi]}(x)\cdot\nabla
u^{o}(x)=F_{1}(\phi^{(-1)}(x),u^{o}(x),u^{i}(x))&\forall
x\in\phi(\partial\Omega^{i}),\\\ \nu_{\Omega^{i}[\phi]}(x)\cdot\nabla
u^{i}(x)=F_{2}(\phi^{(-1)}(x),u^{o}(x),u^{i}(x))&\forall
x\in\phi(\partial\Omega^{i}),\end{cases}$

where $\nu_{\Omega^{i}[\phi]}$ denotes the outward unit normal vector field to
$\Omega^{i}[\phi]$. We prove that, under suitable conditions, problem (6)
admits a family of solutions $\\{(u^{o}_{\phi},u^{i}_{\phi})\\}_{\phi\in
Q_{0}}$, where $Q_{0}$ is a neighbourhood of $\phi_{0}$ in
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}$ and
$(u^{o}_{\phi},u^{i}_{\phi})\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i}[\phi])\times
C^{1,\alpha}(\overline{\Omega^{i}[\phi]})$ for every $\phi\in Q_{0}$. In
literature, the existence of solutions of nonlinear boundary value problems
has been largely investigated by means of variational techniques (see, e.g.,
the monographs of Nečas [26] and of Roubíček [28] and the references therein).
Moreover, potential theoretic techniques have been widely exploited to study
nonlinear boundary value problems with transmission conditions by Berger,
Warnecke, and Wendland [3], by Costabel and Stephan [5], by Gatica and Hsiao
[11], and by Barrenechea and Gatica [2]. Boundary integral methods have been
applied also by Mityushev and Rogosin for the analysis of transmission
problems in the plane (cf. [24, Chap. 5]). Several authors have investigated
the dependence upon domain perturbation of the solutions to boundary value
problems and it is impossible to provide a complete list of contributions.
Here we mention, for example, Henrot and Pierre [13], Henry [14], Keldysh
[15], Novotny and Sokołowski [27], and Sokołowski and Zolésio [31]. Most of
the contributions on this topic deal with first or second order shape
derivability of functionals associated to the solutions of linear boundary
value problems. In the present paper, instead, we are interested into higher
order regularity properties (namely real analiticity) of the solutions of a
nonlinear problem. To do so, we choose to adopt the Functional Analytic
Approach, which has revealed to be a powerful tool to analyse perturbed linear
and nonlinear boundary value problems. This method has been first applied to
investigate regular and singular domain perturbation problems for elliptic
equations and systems with the aim of proving real analytic dependence upon
the perturbation parameter (cf. Lanza de Cristoforis [16, 18, 19]). An
application to the study of the behaviour of the effective conductivity of a
periodic two-phase composite upon perturbations of the inclusion can be found
in Luzzini and Musolino [22]. The key point of the strategy of the method is
the transformation of the perturbed boundary value problem into an equivalent
functional equation that can be studied by the Implicit Function Theorem.
Typically, such a transformation is achieved by exploiting classical results
of potential theory, for example integral representation of harmonic functions
in terms of layer potentials. Nonlinear transmission problems in perturbed
domains have been studied by Lanza de Cristoforis in [19] and by the authors
of the present paper in [8, 25], where they have investigated the behavior of
the solution of a nonlinear transmission problem for the Laplace equation in a
domain with a small inclusion shrinking to a point.

The paper is organised as follows. In Section 2 we define some of the symbols
used later on. In Section 3 we introduce some classical results of potential
theory that we need. Section 4 is devoted to the study of problem (3). We
first prove a representation result for harmonic functions in
$\overline{\Omega^{o}}\setminus\Omega$ and $\overline{\Omega}$ (where $\Omega$
is an open bounded connected subset of class $C^{1,\alpha}$ contained in
$\Omega^{o}$) in terms of single layer potentials with appropriate densities
and constant functions (cf. Lemma 4.1). Then we prove an uniqueness result in
$C^{1,\alpha}(\Omega^{o}\setminus\overline{\Omega^{i}})\times
C^{1,\alpha}(\overline{\Omega^{i}})$ for an homogeneous linear transmission
problem in the pair of domains $\Omega^{o}\setminus\overline{\Omega^{i}}$ and
$\Omega^{i}$ and we analyse an auxiliary boundary operator arising from the
integral formulation of that problem (cf. Lemma 4.2 and Proposition 4.3). In
Proposition 4.4 we provide a formulation of problem (3) in terms of integral
equations. The obtained integral system is solved by means of a fixed-point
theorem, namely the Leray-Schauder Theorem (cf. Proposition 4.5 and
Proposition 4.7). Finally, under suitable conditions on the functions $F_{1}$
and $F_{2}$, we obtain an existence results in
$C^{1,\alpha}(\Omega^{o}\setminus\overline{\Omega^{i}})\times
C^{1,\alpha}(\overline{\Omega^{i}})$ for problem (3) (cf. Proposition 4.8).
Section 5 is devoted to the study of problem (6). We provide a formulation of
problem (6) in terms of integral equations depending on the diffeomorphism
$\phi$ which we rewrite into an equation of the type $M[\phi,\mu]=0$ for an
auxiliary map $M:\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times X\to Y$
(with $X$ and $Y$ suitable Banach spaces), where the variable $\mu$ is related
to the densities of the integral representation of the solution (cf.
Proposition 5.1). Then, by analiticity results for the dependence of single
and double layer potentials upon the perturbation of the support, we prove
that $M$ is real analytic (cf. Proposition 5.2) and the differential of $M$
with respect to the variable $\mu\in X$ is an isomorphism (cf. Proposition
5.3). Hence, by the Implicit Function Theorem, we show the existence of a
family of solutions $\\{(u^{o}_{\phi},u^{i}_{\phi})\\}_{\phi\in Q_{0}}$ of (6)
(cf. Theorem 5.6) and we prove that it can be represented in terms of real
analytic functions (cf. Theorem 5.7).

##  2\. Notation

We denote by $\mathbb{N}$ the set of natural numbers including $0$. We denote
the norm of a real normed space $X$ by $\|\cdot\|_{X}$. We denote by $I_{X}$
the identity operator from $X$ to itself and we omit the subscript $X$ where
no ambiguity can occur. If $X$ and $Y$ are normed spaces we consider on the
product space $X\times Y$ the norm defined by $\|(x,y)\|_{X\times
Y}\equiv\|x\|_{X}+\|y\|_{Y}$ for all $(x,y)\in X\times Y$, while we use the
Euclidean norm for $\mathbb{R}^{d}$, $d\in\mathbb{N}\setminus\\{0,1\\}$. If
$U$ is an open subset of $X$, and $F:U\to Y$ is a Fréchet-differentiable map
in $U$, we denote the differential of $F$ by $dF$. The inverse function of an
invertible function $f$ is denoted by $f^{(-1)}$, while the reciprocal of a
non-zero scalar function $g$ or the inverse of an invertible matrix $A$ are
denoted by $g^{-1}$ and $A^{-1}$ respectively. Let
$\Omega\subseteq\mathbb{R}^{n}$. Then $\overline{\Omega}$ denotes the closure
of $\Omega$ in $\mathbb{R}^{n}$, $\partial\Omega$ denotes the boundary of
$\Omega$, and $\nu_{\Omega}$ denotes the outward unit normal to
$\partial\Omega$. For $x\in\mathbb{R}^{d}$, $x_{j}$ denotes the $j$-th
coordinate of $x$, $|x|$ denotes the Euclidean modulus of $x$ in
$\mathbb{R}^{d}$. If $x\in\mathbb{R}^{d}$ and $r>0$, we denote by $B_{d}(x,r)$
the open ball of center $x$ and radius $r$. Let $\Omega$ be an open subset of
$\mathbb{R}^{n}$ and $m\in\mathbb{N}\setminus\\{0\\}$. The space of $m$ times
continuously differentiable real-valued function on $\Omega$ is denoted by
$C^{m}(\Omega,\mathbb{R})$ or more simply by $C^{m}(\Omega)$. Let
$r\in\mathbb{N}\setminus\\{0\\}$, $f\in(C^{m}(\Omega))^{r}$. The $s$-th
component of $f$ is denoted by $f_{s}$ and the gradient of $f_{s}$ is denoted
by $\nabla f_{s}$. Let $\eta=(\eta_{1},\dots,\eta_{n})\in\mathbb{N}^{n}$ and
$|\eta|=\eta_{1}+\dots+\eta_{n}$. Then
$D^{\eta}f\equiv\frac{\partial^{|\eta|}f}{\partial
x^{\eta_{1}}_{1},\dots,\partial x^{\eta_{n}}_{n}}$. We retain the standard
notation for the space $C^{\infty}(\Omega)$ and its subspace
$C^{\infty}_{c}(\Omega)$ of functions with compact support. The subspace of
$C^{m}(\Omega)$ of those functions $f$ such that $f$ and its derivatives
$D^{\eta}f$ of order $|\eta|\leq m$ can be extended with continuity to
$\overline{\Omega}$ is denoted $C^{m}(\overline{\Omega})$. We denote by
$C^{m}_{b}(\overline{\Omega})$ the space of functions of
$C^{m}(\overline{\Omega})$ such that $D^{\eta}f$ is bounded for $|\eta|\leq
m$. Then the space $C^{m}_{b}(\overline{\Omega})$ equipped with the usual norm
$\|f\|_{C^{m}_{b}(\overline{\Omega})}\equiv\sum_{|\eta|\leq
m}\sup_{\overline{\Omega}}|D^{\eta}f|$ is well known to be a Banach space. Let
$f\in C^{0}(\overline{\Omega})$. Then we define its Hölder constant as

$|f:\Omega|_{\alpha}\equiv\mbox{sup}\left\\{\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}:x,y\in\overline{\Omega},x\neq
y\right\\}.$

We define the subspace of $C^{0}(\overline{\Omega})$ of Hölder continuous
functions with exponent $\alpha\in]0,1[$ by
$C^{0,\alpha}(\overline{\Omega})\equiv\\{f\in
C^{0}(\overline{\Omega}):\,|f:\Omega|_{\alpha}<\infty\\}$. Similarly, the
subspace of $C^{m}(\overline{\Omega})$ whose functions have $m$-th order
derivatives that are Hölder continuous with exponent $\alpha\in]0,1[$ is
denoted $C^{m,\alpha}(\overline{\Omega})$. Then the space
$C^{m,\alpha}_{b}(\overline{\Omega})\equiv C^{m,\alpha}(\overline{\Omega})\cap
C^{m}_{b}(\overline{\Omega})\,,$ equipped with its usual norm
$\|f\|_{C^{m,\alpha}_{b}(\overline{\Omega})}\equiv\|f\|_{C^{m}_{b}(\overline{\Omega})}+\sum_{|\eta|=m}{|D^{\eta}f:\Omega|_{\alpha}}$,
is a Banach space. If $\Omega$ is bounded, then
$C^{m,\alpha}_{b}(\overline{\Omega})=C^{m,\alpha}(\overline{\Omega})$, and we
omit the subscript $b$. We denote by
$C^{m,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus\Omega)$ the space of
functions on $\mathbb{R}^{n}\setminus\Omega$ whose restriction to
$\overline{U}$ belongs to $C^{m,\alpha}(\overline{U})$ for all open bounded
subsets $U$ of $\mathbb{R}^{n}\setminus\Omega$. On
$C^{m,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus\Omega)$ we consider the
natural structure of Fréchet space. Finally if $\Omega$ is bounded, we set

$C^{m,\alpha}_{\mathrm{h}}(\overline{\Omega})\equiv\\{u\in
C^{m,\alpha}(\overline{\Omega})\cap C^{2}(\Omega):\Delta u=0\text{ in
}\Omega\\},$
$\begin{split}C^{m,\alpha}_{\mathrm{h}}(\mathbb{R}^{n}\setminus\Omega)\equiv\\{u\in
C^{m,\alpha}(\mathbb{R}^{n}\setminus\Omega)&\cap
C^{2}(\mathbb{R}^{n}\setminus\overline{\Omega}):\Delta u=0\text{ in
}\mathbb{R}^{n}\setminus\overline{\Omega},\\\ &|u(x)|=O(|x|^{2-n})\text{ as
}x\to+\infty\\}.\end{split}$

The condition $|u(x)|=O(|x|^{2-n})$ as $x\to+\infty$ in the above definition
is equivalent for an harmonic function to the so-called harmonicity at
infinity (see Folland [10, Prop. (2.74), p. 112]). We say that a bounded open
subset of $\mathbb{R}^{n}$ is of class $C^{m,\alpha}$ if it is a manifold with
boundary imbedded in $\mathbb{R}^{n}$ of class $C^{m,\alpha}$. In particular
if $\Omega$ is a $C^{1,\alpha}$ subset of $\mathbb{R}^{n}$, then
$\partial\Omega$ is a $C^{1,\alpha}$ sub-manifold of $\mathbb{R}^{n}$ of co-
dimension $1$. If $M$ is a $C^{m,\alpha}$ sub-manifold of $\mathbb{R}^{n}$ of
dimension $d\geq 1$, we define the space $C^{m,\alpha}(M)$ by exploiting a
finite local parametrization. We retain the standard definition of the
Lebesgue spaces $L^{p}$, $p\geq 1$. If $\Omega$ is of class $C^{1,\alpha}$, we
denote by $d\sigma$ the area element on $\partial\Omega$. If $Z$ is a subspace
of $L^{1}(\partial\Omega)$, we set

$Z_{0}\equiv\left\\{f\in Z:\int_{\partial\Omega}f\,d\sigma=0\right\\}.$

Then we introduce a notation for superposition operators: if $H$ is a function
from $\partial\Omega^{i}\times\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$,
then we denote by $\mathcal{N}_{H}$ the nonlinear nonautonomous superposition
operator that take a pair $(h^{1},h^{2})$ of functions from
$\partial\Omega^{i}$ to $\mathbb{R}$ to the function
$\mathcal{N}_{H}(h^{1},h^{2})$ defined by

$\mathcal{N}_{H}(h^{1},h^{2})(x)\equiv H(x,h^{1}(x),h^{2}(x))\quad\forall
x\in\partial\Omega^{i}.$

Here the letter “$\mathcal{N}$” stands for “Nemytskii operator.” Finally, we
have the following by Lanza de Cristoforis and Rossi [21, Lemma 3.3, Prop.
3.13].

######  Lemma 2.1.

Let $\Omega^{i}$ be as in (1) and let $\mathcal{A}_{\partial\Omega^{i}}$ be as
in (4). Let $\phi\in\mathcal{A}_{\partial\Omega^{i}}$. Then there exists a
unique function $\tilde{\sigma}_{n}[\phi]\in C^{0,\alpha}(\partial\Omega^{i})$
such that

$\int_{\phi(\partial\Omega^{i})}f(y)\,d\sigma_{y}=\int_{\partial\Omega^{i}}f(\phi(s))\,\tilde{\sigma}_{n}[\phi](s)\,d\sigma_{s}\quad\forall
f\in L^{1}(\phi(\partial\Omega^{i})).$

Moreover the map from $\mathcal{A}_{\partial\Omega^{i}}$ to
$C^{0,\alpha}(\partial\Omega^{i})$ that takes $\phi$ to
$\tilde{\sigma}_{n}[\phi]$ and the map from $\mathcal{A}_{\partial\Omega^{i}}$
to $C^{0,\alpha}(\partial\Omega^{i})$ that takes $\phi$ to
$\nu_{\Omega^{i}[\phi]}(\phi(\cdot))$ are real analytic.

##  3\. Some preliminaries of potential theory

As we have mentioned, a key point of the Functional Analytic Approach is the
reformulation of the boundary value problem in terms of an equivalent integral
equation. To this aim, we exploit representation formulas for harmonic
functions in terms of layer potentials. In this section, we collect some
classical results of potential theory. We do not present proofs that can be
found, for example, in Folland [10, Chap. 3], in Gilbarg and Trudinger [12,
Sec. 2].

######  Definition 3.1.

We denote by $S_{n}$ the function from $\mathbb{R}^{n}\setminus\\{0\\}$ to
$\mathbb{R}$ defined by

$S_{n}(x)\equiv\begin{cases}\frac{1}{s_{n}}\log|x|&\forall
x\in\mathbb{R}^{n}\setminus\\{0\\}\qquad\mbox{if }n=2\\\
\frac{1}{(2-n)s_{n}}|x|^{2-n}&\forall
x\in\mathbb{R}^{n}\setminus\\{0\\}\qquad\mbox{if }n>2\end{cases}$

where $s_{n}$ denotes the $(n-1)$-dimensional measure of $\partial
B_{n}(0,1)$.

$S_{n}$ is well known to be a fundamental solution of the Laplace operator
$\Delta=\sum_{j=1}^{n}\partial^{2}_{x_{j}}$. We now assume that

$\text{$\Omega$ is an open bounded subset of $\mathbb{R}^{n}$ of class
$C^{1,\alpha}$}.$

In the following definition we introduce the single layer potential, which we
use to transform our problems into integral equations.

######  Definition 3.2.

We denote by $v_{\Omega}[\mu]$ the single layer potential with density $\mu$,
i.e. the function defined by

$v_{\Omega}[\mu](x)\equiv\int_{\partial\Omega}{S_{n}(x-y)\mu(y)\,d\sigma_{y}}\qquad\forall
x\in\mathbb{R}^{n}\,,\forall\mu\in L^{2}(\partial\Omega)\,.$

It is well known that if $\mu\in C^{0,\alpha}(\partial\Omega)$, then
$v_{\Omega}[\mu]\in C^{0}(\mathbb{R}^{n})$. We set

$v^{+}_{\Omega}[\mu]\equiv v_{\Omega}[\mu]_{|\overline{\Omega}},\qquad
v^{-}_{\Omega}[\mu]\equiv v_{\Omega}[\mu]_{|\mathbb{R}^{n}\setminus\Omega}.$

Then we define the boundary integral operators associated to the trace of the
single layer potential and its normal derivative.

######  Definition 3.3.

We denote by $V_{\partial\Omega}$ the operator from $L^{2}(\partial\Omega)$ to
itself that takes $\mu$ to the function $V_{\partial\Omega}[\mu]$ defined in
the trace sense by

$V_{\partial\Omega}[\mu]\equiv v_{\Omega}[\mu]_{|\partial\Omega}.$

We denote by $W_{\partial\Omega}$ the integral operator from
$L^{2}(\partial\Omega)$ to itself defined by

$W_{\partial\Omega}[\mu](x)\equiv-\int_{\partial\Omega}\\!\\!{\nu_{\Omega}(y)\cdot\nabla
S_{n}(x-y)\mu(y)\,d\sigma_{y}}\ \ \text{for a.e.
}x\in\partial\Omega,\forall\mu\in L^{2}(\partial\Omega).$

We denote by $W^{\ast}_{\partial\Omega}$ the integral operator from
$L^{2}(\partial\Omega)$ to itself which is the transpose of
$W_{\partial\Omega}$ and that is defined by

$W^{\ast}_{\partial\Omega}[\mu](x)\equiv\int_{\partial\Omega}\\!\\!{\nu_{\Omega}(x)\cdot\nabla
S_{n}(x-y)\mu(y)\,d\sigma_{y}}\ \ \mbox{for a.e.
}x\in\partial\Omega,\forall\mu\in L^{2}(\partial\Omega).$

As it is well known, since $\Omega$ is of class $C^{1,\alpha}$,
$W_{\partial\Omega}$ and $W^{\ast}_{\partial\Omega}$ are compact operators
from $L^{2}(\partial\Omega)$ to itself (both display a weak singularity). In
particular $\left(\pm\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)$ are
Fredholm operators of index $0$ from $L^{2}(\partial\Omega)$ to itself.
Moreover, one verifies that
$W_{\partial\Omega}:C^{1,\alpha}(\partial\Omega)\to
C^{1,\alpha}(\partial\Omega)$ and
$W^{\ast}_{\partial\Omega}:C^{0,\alpha}(\partial\Omega)\to
C^{0,\alpha}(\partial\Omega)$ are transpose to one another with respect to the
duality of $C^{1,\alpha}(\partial\Omega)\times C^{0,\alpha}(\partial\Omega)$
induced by the inner product of $L^{2}(\partial\Omega)$. We collect some well
known properties of the single layer potential in the theorem below. In
particular, we note that the operator of statement (iv) is an isomorphism both
in the case of dimension $n=2$ and $n\geq 3$ (see, e.g., [6, Theorem 6.47]).

######  Theorem 3.4 (Properties of the single layer potential).

The following statements hold.

  1. (i)

For all $\mu\in L^{2}(\partial\Omega)$, the function $v_{\Omega}[\mu]$ is
harmonic in $\mathbb{R}^{n}\setminus\partial\Omega$. If $n\geq 3$ or if $n=2$
and $\int_{\partial\Omega}\mu\,d\sigma=0$ then $v_{\Omega}[\mu]$ is also
harmonic at infinity.

  2. (ii)

If $\mu\in C^{0,\alpha}(\partial\Omega)$, then $v^{+}_{\Omega}[\mu]\in
C^{1,\alpha}(\overline{\Omega})$ and the map from
$C^{0,\alpha}(\partial\Omega)$ to $C^{1,\alpha}(\overline{\Omega})$ that takes
$\mu$ to $v^{+}_{\Omega}[\mu]$ is linear and continuous. Moreover,
$v^{-}_{\Omega}[\mu]\in
C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus\Omega)$ and the map from
$C^{0,\alpha}(\partial\Omega)$ to
$C^{1,\alpha}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus\Omega)$ that takes $\mu$
to $v^{-}_{\Omega}[\mu]$ is linear and continuous.

  3. (iii)

If $\mu\in C^{0,\alpha}(\partial\Omega)$, then we have following jump
relations

$\nu_{\Omega}(x)\cdot\nabla
v^{\pm}_{\Omega}[\mu](x)=\left(\mp\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu](x)\qquad\forall
x\in\partial\Omega.$

  4. (iv)

The map from $C^{0,\alpha}(\partial\Omega)_{0}\times\mathbb{R}$ to
$C^{0,\alpha}(\partial\Omega)$ that takes a pair $(\mu,\rho)$ to
$V_{\partial\Omega}[\mu]+\rho$ is an isomorphism.

Since $\Omega$ is of class $C^{1,\alpha}$, the following classical compactness
result holds (cf. Schauder [29, 30]).

######  Theorem 3.5.

The map that takes $\mu$ to $W^{\ast}_{\partial\Omega}[\mu]$ is compact from
$C^{0,\alpha}(\partial\Omega)$ to itself.

Theorem 3.5 implies that
$\left(\pm\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)$ are Fredholm
operators of index $0$ from $C^{0,\alpha}(\partial\Omega)$ into itself. We now
collect some regularity results for integral operators. We first introduce the
following (see Folland [10, Chap. 3 §B]).

######  Definition 3.6.

Let $K$ be a measurable function from $\partial\Omega\times\partial\Omega$ to
$\mathbb{R}$ and let $0\leq\beta<n-1$. We say that $K$ is a continuous kernel
of order $\beta$ if

$K(x,y)=k(x,y)|x-y|^{-\beta}\quad\forall(x,y)\in\partial\Omega\times\partial\Omega,$

for some continuous function $k$ on $\partial\Omega\times\partial\Omega$.
If $K$ is a continuous kernel of order $\beta$, we denote by $\mathcal{K}_{K}$
the integral operator from $L^{2}(\partial\Omega)$ to itself defined by

$\mathcal{K}_{K}[\mu](x)\equiv\int_{\partial\Omega}K(x,y)\mu(y)\,d\sigma_{y}\qquad\text{for
a.e. }x\in\partial\Omega\,,\forall\mu\in L^{2}(\partial\Omega)\,.$

We observe that the functions $K_{1}(x,y)\equiv S_{n}(x-y)$ and
$K_{2}(x,y)\equiv\nu_{\Omega}(y)\cdot\nabla S_{n}(x-y)$ of
$(x,y)\in\partial\Omega\times\partial\Omega$, $x\neq y$, are continuous
kernels of order $n-2$ (cf. Folland [10, Prop. 3.17]). Clearly, we can extend
the notion of integral operator with a continuous kernel to the vectorial case
just applying the definition above component-wise. Then we present a vectorial
version of a classical regularity result (see, for example, Folland [10, Prop.
3.13]).

######  Theorem 3.7.

Let $0\leq\beta<n-1$. Let $K_{i}^{j}$ with $i,j\in\\{1,2\\}$ be continuous
kernels of order $\beta$. Let $\mathcal{K}=(\mathcal{K}_{1},\mathcal{K}_{2})$
be the operator from $(L^{2}(\partial\Omega))^{2}$ to itself defined by

$\mathcal{K}_{1}[\mu_{1},\mu_{2}]=\mathcal{K}_{K_{1}^{1}}[\mu_{1}]+\mathcal{K}_{K_{2}^{1}}[\mu_{2}],\qquad\mathcal{K}_{2}[\mu_{1},\mu_{2}]=\mathcal{K}_{K_{1}^{2}}[\mu_{1}]+\mathcal{K}_{K_{2}^{2}}[\mu_{2}],$

for all $(\mu_{1},\mu_{2})\in(L^{2}(\partial\Omega))^{2}$. If
$(I+\mathcal{K})[\mu_{1},\mu_{2}]\in(C^{0}(\partial\Omega))^{2}$, then
$(\mu_{1},\mu_{2})\in(C^{0}(\partial\Omega))^{2}$.

Finally, we present in Theorem 3.8 a regularity result that will be widely
used in what follows. The proof exploits a standard argument on iterated
kernels and can be found, e.g., in Dalla Riva and Mishuris [7, Lem. 3.3].

######  Theorem 3.8.

Let $\mu\in L^{2}(\partial\Omega)$. Let $\beta\in[0,\alpha]$. If
$\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu]$ or
$\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu]$ belongs to
$C^{0,\beta}(\partial\Omega)$, then $\mu\in C^{0,\beta}(\partial\Omega)$.

##  4\. Existence result for problem (3)

The aim of this section is to prove an existence result for problem (3). We
start with the following representation result for harmonic functions in
$\Omega^{o}\setminus\overline{\Omega}$ and in $\Omega$ in terms of single
layer potentials plus constant functions. The set $\Omega$ in the Lemma 4.1
will be later replaced by the set $\Omega^{i}$ and by the perturbed set
$\Omega^{i}[\phi]$.

######  Lemma 4.1.

Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^{n}$ of class
$C^{1,\alpha}$, such that $\mathbb{R}^{n}\setminus\overline{\Omega}$ is
connected and $\overline{\Omega}\subset\Omega^{o}$. Then the map from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times C^{0,\alpha}(\partial\Omega)\times
C^{0,\alpha}(\partial\Omega)_{0}\times\mathbb{R}^{2}$ to
$C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega^{o}}\setminus\Omega)\times
C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega})$ that takes a quintuple
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the pair of functions
$(U^{o}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])$
defined by

(7) $\displaystyle
U^{o}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]\equiv(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]+\rho^{o})_{|\overline{\Omega^{o}}\setminus\Omega}$
$\displaystyle
U^{i}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]\equiv
v^{+}_{\Omega}[\eta^{i}]+\rho^{i}$

is bijective.

###### Proof.

The map is well defined. Indeed, by the harmonicity and regularity properties
of single layer potentials (cf. Theorem 3.4 (i)-(ii)), we know that

$\displaystyle\Delta
U^{o}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]=0\quad\text{on
}\Omega^{o}\setminus\overline{\Omega},$ $\displaystyle\Delta
U^{i}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]=0\quad\text{on
}\Omega,$
$\displaystyle(U^{o}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega)\times
C^{1,\alpha}(\overline{\Omega}),$

for all $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times C^{0,\alpha}(\partial\Omega)\times
C^{0,\alpha}(\partial\Omega)_{0}\times\mathbb{R}^{2}$. We now show that it is
bijective. So, we take a pair of functions $(h^{o},h^{i})\in
C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega^{o}}\setminus\Omega)\times
C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega})$ and we prove that there exists a
unique quintuple $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times C^{0,\alpha}(\partial\Omega)\times
C^{0,\alpha}(\partial\Omega)_{0}\times\mathbb{R}^{2}$ such that

(8)
$(U^{o}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])=(h^{o},h^{i}).$

By the uniqueness of the classical solution of the Dirichlet boundary value
problem, the second equation in (8) is equivalent to

(9) $V_{\partial\Omega}[\eta^{i}]+\rho^{i}=h^{i}_{|\partial\Omega}$

(notice that, since $h^{i}$ is an element of
$C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega})$, we have
$h^{i}_{|\partial\Omega}\in C^{1,\alpha}(\partial\Omega)\subseteq
C^{0,\alpha}(\partial\Omega)$). By Theorem 3.4 (iv), there exists a unique
pair $(\eta^{i},\rho^{i})\in C^{0,\alpha}(\partial\Omega)_{0}\times\mathbb{R}$
such that (9) holds. Then it remains to show that there exists a unique triple
$(\mu^{o},\mu^{i},\rho^{o})\in C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega)\times\mathbb{R}$ such that

(10)
$(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega^{i}}[\mu^{i}]+\rho^{o})_{|\overline{\Omega^{o}}\setminus\Omega}=h^{o}.$

By the jump relations for the single layer potential (cf. Theorem 3.4 (iii))
and by the uniqueness of the classical solution of the Neumann-Dirichlet mixed
boundary value problem, equation (10) is equivalent to the following system of
integral equations:

(11) $\displaystyle
V_{\partial\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]_{|\partial\Omega^{o}}+\rho^{o}=h^{o}_{|\partial\Omega^{o}},$
$\displaystyle\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu^{i}]+\nu_{\Omega}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega}=\nu_{\Omega}\cdot\nabla
h^{o}_{|\partial\Omega}$

(notice that, by $h^{o}\in C^{1,\alpha}_{\mathrm{h}}(\overline{\Omega})$, we
get $h^{o}_{|\partial\Omega}\in C^{1,\alpha}(\partial\Omega)\subseteq
C^{0,\alpha}(\partial\Omega)$ and $\nu_{\Omega}\cdot\nabla
h^{o}_{|\partial\Omega}\in C^{0,\alpha}(\partial\Omega)$). Then we observe
that by Theorem 3.4 (iv), the map from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega)\times\mathbb{R}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times C^{0,\alpha}(\partial\Omega)$ that
takes a triple $(\mu^{o},\mu^{i},\rho^{o})$ to the pair of functions
$\left(V_{\partial\Omega^{o}}[\mu^{o}]+\rho^{o},\frac{1}{2}\mu^{i}\right)$ is
an isomorphism. Moreover, by the properties of integral operators with real
analytic kernel and no singularities (cf. Lanza de Cristoforis and Musolino
[20, Prop. 4.1]) and by Theorem 3.5, the map from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega)\times\mathbb{R}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times C^{0,\alpha}(\partial\Omega)$ that
takes a triple $(\mu^{o},\mu^{i},\rho^{o})$ to the pair of functions
$(v^{-}_{\Omega}[\mu^{i}]_{|\partial\Omega^{o}},W^{\ast}_{\partial\Omega}[\mu^{i}]+\nu_{\Omega}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega})$ is compact. Hence, the map
from $C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega)\times\mathbb{R}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times C^{0,\alpha}(\partial\Omega)$ that
takes a triple $(\mu^{o},\mu^{i},\rho^{o})$ to the pair of functions

$\left(V_{\partial\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]_{|\partial\Omega^{o}}+\rho^{o},\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu^{i}]+\nu_{\Omega}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega}\right)$

is a compact perturbation of an isomorphism and therefore it is a Fredholm
operator of index 0. Thus, to complete the proof, it suffices to show that
(11) with $(h^{o}_{|\partial\Omega^{o}},\nu_{\Omega}\cdot\nabla
h^{o}_{|\partial\Omega})=(0,0)$ implies $(\mu^{o},\mu^{i},\rho^{o})=(0,0,0)$.
If

(12)
$\left(V_{\partial\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]_{|\partial\Omega^{o}}+\rho^{o},\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega}\right)[\mu^{i}]+\nu_{\Omega}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega}\right)=(0,0),$

then by the jump relations for the single layer potential (cf. Theorem 3.4
(iii)) and by the uniqueness of the classical solution of Neumann-Dirichlet
mixed boundary value problem, one deduces that
$(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]+\rho^{o})_{|\overline{\Omega^{o}}\setminus\Omega}=0$.
Moreover, by the continuity of $v_{\Omega}[\mu^{i}]$ in $\mathbb{R}^{n}$, we
have that
$(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]+\rho^{o})_{|\partial\Omega}=(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{+}_{\Omega}[\mu^{i}]+\rho^{o})_{|\partial\Omega}=0$.
Then by the uniqueness of the classical solution of Dirichlet boundary value
problem in $\Omega$ we deduce that

(13)
$(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{+}_{\Omega}[\mu^{i}]+\rho^{o})_{|\overline{\Omega}}=0.$

By the jump relations for the single layer potential (cf. Theorem 3.4 (iii)),
adding and subtracting the term
$\nu_{\Omega}\cdot\nabla(v^{+}_{\Omega^{o}}[\mu^{o}]+\rho^{o})_{|\partial\Omega}$
and taking into account (13), we get

$\begin{split}\mu^{i}&=\nu_{\Omega}\cdot\nabla
v^{-}_{\Omega}[\mu^{i}]_{|\partial\Omega}-\nu_{\Omega}\cdot\nabla
v^{+}_{\Omega}[\mu^{i}]_{|\partial\Omega}\\\
&=\nu_{\Omega}\cdot\nabla(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega}[\mu^{i}]+\rho^{o})_{|\partial\Omega}-\nu_{\Omega}\cdot\nabla(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{+}_{\Omega}[\mu^{i}]+\rho^{o})_{|\partial\Omega}=0.\end{split}$

Thus, by (12), we obtain $V_{\Omega^{o}}[\mu^{o}]+\rho^{o}=0$ on
$\partial\Omega^{o}$, which implies $(\mu^{o},\rho^{o})=(0,0)$ (cf. Theorem
3.4 (iv)). Hence $(\mu^{o},\mu^{i},\rho^{o})=(0,0,0)$ and the proof is
complete. ∎

To represent the boundary conditions of a linearised version of problem (3),
we find convenient to introduce a matrix function

$A(\cdot)=\begin{pmatrix}A_{11}(\cdot)&A_{12}(\cdot)\\\
A_{21}(\cdot)&A_{22}(\cdot)\end{pmatrix}:\partial\Omega^{i}\to
M_{2}(\mathbb{R}).$

Here above, the symbol $M_{2}(\mathbb{R})$ denotes the set of $2\times 2$
matrices with real entries. We set

$\tilde{A}(\cdot)\equiv\begin{pmatrix}A_{11}(\cdot)&A_{12}(\cdot)\\\
-A_{21}(\cdot)&-A_{22}(\cdot)\end{pmatrix}\,.$

We will assume the following conditions on the matrix $A$:

(14) $\begin{split}&\bullet\,A_{j,k}\in C^{0,\alpha}(\partial\Omega^{i})\mbox{
for all }j,k\in\\{1,2\\};\\\ &\bullet\,\mbox{For every
}(\xi_{1},\xi_{2})\in\mathbb{R}^{2},(\xi_{1},\xi_{2})\tilde{A}(\xi_{1},\xi_{2})^{T}\geq
0\mbox{ on }\partial\Omega^{i};\\\ &\bullet\,\mbox{If
}(c_{1},c_{2})\in\mathbb{R}^{2}\mbox{ and }A(x)(c_{1},c_{2})^{T}=0\mbox{ for
all }x\in\partial\Omega^{i},\mbox{ then }(c_{1},c_{2})=(0,0).\end{split}$

We remark that in literature the third condition in (14) is often replaced by
a condition on the invertibility of the matrix $A$, namely

(15) $\bullet\mbox{There exists a point }x\in\partial\Omega^{i}\mbox{ such
that }A(x)\mbox{ is invertible}.$

We point out that, for instance, the matrix
$A(x)=\begin{pmatrix}x_{1}^{2}&x_{1}\\\ -x_{1}&-1\end{pmatrix}$ with
$x=(x_{1},\dots,x_{n})\in\partial\Omega^{i}$ satisfies the third condition in
(14) but not condition (15). Then by a standard energy argument we deduce the
following result on the uniqueness of the solution of a transmission problem.

######  Lemma 4.2.

Let $A$ be as in (14). Then the unique solution in
$C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i})\times
C^{1,\alpha}(\overline{\Omega^{i}})$ of problem

(16) $\begin{cases}\Delta u^{o}=0&\quad\mbox{in
}\Omega^{o}\setminus\overline{\Omega^{i}},\\\ \Delta u^{i}=0&\quad\mbox{in
}\Omega^{i},\\\ \nu_{\Omega^{o}}(x)\cdot\nabla u^{o}(x)=0&\quad\forall
x\in\partial\Omega^{o},\\\ \nu_{\Omega^{i}}(x)\cdot\nabla
u^{o}(x)-A_{11}(x)u^{o}(x)-A_{12}(x)u^{i}(x)=0&\quad\forall
x\in\partial\Omega^{i},\\\ \nu_{\Omega^{i}}(x)\cdot\nabla
u^{i}(x)-A_{21}(x)u^{o}(x)-A_{22}(x)u^{i}(x)=0&\quad\forall
x\in\partial\Omega^{i},\end{cases}$

is $(u^{o},u^{i})=(0,0)$.

In the following proposition, we investigate the properties of an auxiliary
boundary operator, $J_{A}$, which we will exploit in the integral formulation
of our problem, in order to recast a fixed point equation. More precisely, we
prove that $J_{A}$ is an isomorphism in $L^{2}$, in $C^{0}$, and in
$C^{0,\alpha}$. All those three frameworks will be important: the first
setting is suitable to use Fredholm theory and to directly prove the
isomorphic property of $J_{A}$, the second setting will be used in order to
apply Leray-Shauder Theorem to the aforementioned obtained fixed point
equation (see Propositions 4.5 and 4.7 below) and the third setting will be
central to deduce that the solution of problem (3) we built is actually a
classical solution, in particular of class $C^{1,\alpha}$ (cf. Propositions
4.4 and 4.8)

######  Proposition 4.3.

Let $A$ be as in (14). Let $J_{A}$ be the map from
$L^{2}(\partial\Omega^{o})_{0}\times L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}$ that takes a
quintuple $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the triple
$J_{A}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$ defined by

(17) $\displaystyle J_{A,1}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{o}}\right)[\mu^{o}]+\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\qquad\mbox{on
}\partial\Omega^{o},$ $\displaystyle
J_{A,2}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\mu^{i}]+\nu_{\Omega^{i}}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}$
$\displaystyle\quad-(A_{11},A_{12})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\qquad\mbox{on
}\partial\Omega^{i},$ $\displaystyle
J_{A,3}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\eta^{i}]$
$\displaystyle\quad-(A_{21},A_{22})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\qquad\mbox{on
}\partial\Omega^{i}.$

Then the following statements hold.

  1. (i)

$J_{A}$ is a linear isomorphism from $L^{2}(\partial\Omega^{o})_{0}\times
L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}$.

  2. (ii)

$J_{A}$ is a linear isomorphism from $C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$.

  3. (iii)

$J_{A}$ is a linear isomorphism from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$.

###### Proof.

We first prove (i). We write $J_{A}$ in the form
$J_{A}=\tilde{J}^{+}_{A}\circ\tilde{J}_{A}\circ\tilde{J}^{-}_{A}$, where
$\tilde{J}^{-}_{A}$ is the inclusion of $L^{2}(\partial\Omega^{o})_{0}\times
L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ into
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}$,
$\tilde{J}_{A}$ is the map from
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}$
into itself that takes $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the
quintuple $\tilde{J}_{A}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$ defined
by

$\displaystyle\tilde{J}_{A,1}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{o}}\right)[\mu^{o}]+\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\qquad\mbox{on
}\partial\Omega^{o},$
$\displaystyle\tilde{J}_{A,2}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\mu^{i}]+\nu_{\Omega^{i}}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}$
$\displaystyle-(A_{11},A_{12})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{i}])\qquad\mbox{on
}\partial\Omega^{i},$
$\displaystyle\tilde{J}_{A,3}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\eta^{i}]$
$\displaystyle-(A_{21},A_{22})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{i}])\qquad\mbox{on
}\partial\Omega^{i},$
$\displaystyle\tilde{J}_{A,4}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\rho^{o},$
$\displaystyle\tilde{J}_{A,5}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle\equiv\rho^{i},$

and $\tilde{J}^{+}_{A}$ is the map from
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}$
into $L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}$ that
takes a quintuple $(f,g_{1},g_{2},c_{1},c_{2})$ to the triple
$\tilde{J}^{+}_{A}[f,g_{1},g_{2},c_{1},c_{2}]$ defined by

$\tilde{J}^{+}_{A}[f,g_{1},g_{2},c_{1},c_{2}]\equiv(f,g_{1}-(A_{11},A_{12})\cdot(c_{1},c_{2}),g_{2}-(A_{21},A_{22})\cdot(c_{1},c_{2})).$

Then we observe that $\tilde{J}^{+}_{A}$ is a Fredholm operator of index $2$,
because $\mathrm{Coker}\,\tilde{J}^{+}_{A}=\\{0\\}$ and
$\mathrm{Ker}\,\tilde{J}^{+}_{A}=\mathrm{Span}\,\\{(0,A_{11},A_{21},1,0),(0,A_{12},A_{22},0,1)\\}$,
and that $\tilde{J}^{-}_{A}$ is Fredholm of index $-2$, because
$\mathrm{Ker}\,\tilde{J}^{-}_{A}=\\{0\\}$ and
$\mathrm{Coker}\,\tilde{J}^{-}_{A}=\mathrm{Span}\,\\{(1,0,0,0,0),(0,0,1,0,0)\\}$.
Next, we observe that the map from
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}$
into itself that takes a quintuple
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the quintuple
$(-\frac{1}{2}\mu^{o},\frac{1}{2}\mu^{i},-\frac{1}{2}\eta^{i},\rho^{o},\rho^{i})$
is a linear isomorphism. Moreover, by the mapping properties of the integral
operators with real analytic kernel and no singularity (cf. Lanza de
Cristoforis and Musolino [20, Prop. 4.1]), by the compactness of the operators
$W^{\ast}_{\partial\Omega^{o}}$ and $W^{\ast}_{\partial\Omega^{i}}$ from
$L^{2}(\partial\Omega^{o})$ to itself and from $L^{2}(\partial\Omega^{i})$ to
itself, respectively (see comments below Definition 3.3), by the compactness
of the operator $V_{\partial\Omega^{i}}$ from $L^{2}(\partial\Omega^{i})$ into
itself (see Costabel [4, Thm. 1]), and by the bilinearity and continuity of
the product from $C^{0,\alpha}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})$ to $L^{2}(\partial\Omega^{i})$, we deduce that the
map from
$L^{2}(\partial\Omega^{o})\times(L^{2}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}$
into itself that takes a quintuple
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the quintuple
$\tilde{J}^{C}_{A}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$ defined by

$\displaystyle\tilde{J}^{C}_{A,1}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle=W^{\ast}_{\partial\Omega^{o}}[\mu^{o}]+\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\qquad\mbox{on
}\partial\Omega^{o},$
$\displaystyle\tilde{J}^{C}_{A,2}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle=W^{\ast}_{\partial\Omega^{i}}[\mu^{i}]+\nu_{\Omega^{i}}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}$
$\displaystyle-(A_{11},A_{12})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{i}])\qquad\mbox{on
}\partial\Omega^{i},$
$\displaystyle\tilde{J}^{C}_{A,3}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle=W^{\ast}_{\partial\Omega^{i}}[\eta^{i}]$
$\displaystyle-(A_{21},A_{22})$
$\displaystyle\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}],V_{\partial\Omega^{i}}[\eta^{i}])\qquad\mbox{on
}\partial\Omega^{i},$
$\displaystyle\tilde{J}^{C}_{A,4}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle=0,$
$\displaystyle\tilde{J}^{C}_{A,5}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]$
$\displaystyle=0,$

is compact. Hence, we conclude that $\tilde{J}_{A}$ is a compact perturbation
of an isomorphism and therefore it is Fredholm of index 0. Since the index of
a composition of Fredholm operators is the sum of the indexes of the
components, we deduce that $J_{A}$ is a Fredholm operator of index $0$.
Therefore, in order to complete the proof of point $(i)$, it suffices to prove
that $J_{A}$ is injective. Thus, we now assume that
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
L^{2}(\partial\Omega^{o})_{0}\times L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ and that

(18) $J_{A}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]=(0,0,0).$

We first verify that $(\mu^{o},\mu^{i},\eta^{i})$ is actually in
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}$. In fact, by the definition of $J_{A,1}$ in
(17), by the fact that $\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\in
C^{0}(\partial\Omega^{o})$ (cf. Lanza de Cristoforis and Musolino [20, Prop.
4.1]), and by Theorem 3.8, we deduce that $\mu^{o}\in
C^{0}(\partial\Omega^{o})$. Let
$\mathcal{K}\equiv(\mathcal{K}_{1},\mathcal{K}_{2})$ be the map from
$L^{2}(\partial\Omega^{i})\times L^{2}(\partial\Omega^{i})_{0}$ to itself that
takes a pair $(\mu^{i},\eta^{i})\in L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}$ to

$\displaystyle\mathcal{K}_{1}[\mu^{i},\eta^{i}]$ $\displaystyle\equiv
2W^{\ast}_{\partial\Omega^{i}}[\mu^{i}]-2A_{11}V_{\partial\Omega^{i}}[\mu^{i}]-2A_{12}V_{\partial\Omega^{i}}[\eta^{i}]$
$\displaystyle\mbox{on }\partial\Omega^{i},$
$\displaystyle\mathcal{K}_{2}[\mu^{i},\eta^{i}]$
$\displaystyle\equiv-2W^{\ast}_{\partial\Omega^{i}}[\eta^{i}]+2A_{21}V_{\partial\Omega^{i}}[\mu^{i}]+2A_{22}V_{\partial\Omega^{i}}[\eta^{i}]$
$\displaystyle\mbox{on }\partial\Omega^{i}.$

Notice that each component of $\mathcal{K}$ is a linear combinations of
integral operators with a continuous kernel of order $n-2$ (see Definition 3.6
and comments below). By the fact that $\mu^{o}\in C^{0}(\partial\Omega^{i})$
and by the first condition in (14), we know that
$2(A_{11},A_{12})\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+\rho^{o},\rho^{i}),-2(A_{21},A_{22})\cdot(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+\rho^{o},\rho^{i})$
belong to $C^{0,\alpha}(\partial\Omega^{i})\subseteq
C^{0}(\partial\Omega^{i})$. Then (18) and the definition of the operator
$\mathcal{K}$ imply that
$(I+\mathcal{K})[\mu^{i},\eta^{i}]\in(C^{0}(\partial\Omega^{i}))^{2}$. Hence,
by Theorem 3.7 we conclude that $(\mu^{i},\eta^{i})\in
C^{0}(\partial\Omega^{i})\times C^{0}(\partial\Omega^{i})_{0}$. Then by
mapping properties of integral operators with real analytic kernel and no
singularity (cf. Lanza de Cristoforis and Musolino [20, Prop. 4.1]) and by
classical known results in potential theory (cf. Miranda [23, Chap. II, §14]),
we know that $\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\in
C^{0,\alpha}(\partial\Omega^{o})$ and
$v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}},\,\nu_{\Omega^{i}}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}},\,V_{\partial\Omega^{i}}[\eta^{i}],\,V_{\partial\Omega^{i}}[\mu^{i}]\in
C^{0,\alpha}(\partial\Omega^{i})$. Hence, by (18) and by the membership of
$A\in M_{2}(C^{0,\alpha}(\partial\Omega^{i}))$ (cf. first condition in (14)),
we obtain that
$\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{o}}\right)[\mu^{o}]\in
C^{0,\alpha}(\partial\Omega^{o})$ and
$\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\mu^{i}],\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\eta^{i}]\in
C^{0,\alpha}(\partial\Omega^{i})$. Then Theorem 3.8 implies
$(\mu^{o},\mu^{i},\eta^{i})\in C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times C^{0,\alpha}(\partial\Omega^{i})_{0}$.
By the jump relations (cf. Theorem 3.4 (iii)), by Lemma 4.1, and by (18), we
deduce that the pair
$(U^{o}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])$
defined by (7) is a solution of the boundary value problem (16). Then by Lemma
4.2, we have that

$(U^{o}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])=(0,0),$

which implies $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})=(0,0,0,0,0)$, by
the uniqueness of the representation provided by Lemma 4.1. We now prove
statement (ii). First we note that the integral operators that appear in the
definition (17) of $J_{A}$ have either a weakly singular or a real analytic
kernel. It follows that $J_{A}$ is continuous from
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ (cf. Lanza de
Cristoforis and Musolino [20, Prop. 4.1] for the properties of integral
operators with real analytic kernels). Then we observe that, by Theorems 3.7
and 3.8, if we have $J_{A}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]\in
C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ for some
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
L^{2}(\partial\Omega^{o})_{0}\times L^{2}(\partial\Omega^{i})\times
L^{2}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$, then
$(\mu^{o},\mu^{i},\eta^{i})\in C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times C^{0}(\partial\Omega^{i})_{0}$ (see also the
argument used after (18) to prove that $(\mu^{o},\mu^{i},\eta^{i})$ belongs to
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}$). Then, by statement (i) we deduce that $J_{A}$
is a bijective continuous linear map from $C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$. By the Open
Mapping Theorem it follows that $J_{A}$ is a linear homeomorphism from
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$. The proof of
statement (iii) is similar to that of statement (ii) and we leave it to the
zealous reader (see also the argument used after (18) to prove that
$(\mu^{o},\mu^{i},\eta^{i})$ belongs to
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times C^{0,\alpha}(\partial\Omega^{i})_{0}$).
∎

We are now ready to convert (3) into a system of integral equations.

######  Proposition 4.4.

Let $A$ be as in (14). Let $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. Let
$(U^{o}_{\Omega^{i}}[\cdot,\cdot,\cdot,\cdot,\cdot],U^{i}_{\Omega^{i}}[\cdot,\cdot,\cdot,\cdot,\cdot])$
be defined by (7). Let $J_{A}$ be as in Proposition 4.3. Then
$(U^{o}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])$
is a solution of (3) if and only if

(19) $\displaystyle\begin{pmatrix}\mu^{o}\\\ \mu^{i}\\\ \eta^{i}\\\
\rho^{o}\\\ \rho^{i}\end{pmatrix}=J_{A}^{(-1)}$
$\displaystyle\left[\begin{pmatrix}f^{o}\\\
\mathcal{N}_{F_{1}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\\\
\mathcal{N}_{F_{2}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\end{pmatrix}\right.$
$\displaystyle\left.-\begin{pmatrix}0&0&0\\\ 0&A_{11}&A_{12}\\\
0&A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}0\\\
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o}\\\
V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i}\end{pmatrix}\right].$

###### Proof.

By Lemma 4.1 and by the jump relations of Theorem 3.4, we know that if
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ then the pair

$(U^{o}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}],U^{i}_{\Omega^{i}}[\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}])$

defined by (7) is a solution of problem (3) if an only if

(20)
$\begin{split}&\begin{pmatrix}\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{o}}\right)[\mu^{o}]+\nu_{\Omega^{o}}\cdot\nabla
v^{-}_{\Omega^{i}}[\mu^{i}]_{|\partial\Omega^{o}}\\\
\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\mu^{i}]+\nu_{\Omega^{i}}\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}\\\
\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}}\right)[\eta^{i}]\end{pmatrix}\\\
&\qquad=\begin{pmatrix}f^{o}\\\
\mathcal{N}_{F_{1}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\\\
\mathcal{N}_{F_{2}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\end{pmatrix}.\end{split}$

Then, by subtracting in both sides of (20) the term

$\begin{pmatrix}0&0&0\\\ 0&A_{11}&A_{12}\\\
0&A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}0\\\
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o}\\\
V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i}\end{pmatrix}\in
C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$

and by the invertibility of $J_{A}$ from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$
provided by Proposition 4.3 (iii), the validity of the statement follows. ∎

We now introduce an auxiliary map. If $A$ is as in (14) and $J_{A}$ is as in
Proposition 4.3, we denote by $T_{A}$ the map from
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ defined by

(21) $\displaystyle T_{A}$
$\displaystyle(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$
$\displaystyle\equiv J_{A}^{(-1)}\left[\begin{pmatrix}f^{o}\\\
\mathcal{N}_{F_{1}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\\\
\mathcal{N}_{F_{2}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\end{pmatrix}\right.$
$\displaystyle\left.-\begin{pmatrix}0&0&0\\\ 0&A_{11}&A_{12}\\\
0&A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}0\\\
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o}\\\
V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i}\end{pmatrix}\right]\,.$

We study the continuity and compactness of $T_{A}$ in the following
proposition.

######  Proposition 4.5.

Let $A$ be as in (14). Let $T_{A}$ be as in (21). Then $T_{A}$ is a continuous
(nonlinear) operator from $C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ and is
compact.

###### Proof.

By the properties of integral operators with real analytic kernel and no
singularities (cf. Lanza de Cristoforis and Musolino [20, Prop. 4.1]) and by
the compactness of the embedding of $C^{0,\alpha}(\partial\Omega^{i})$ into
$C^{0}(\partial\Omega^{i})$, $v^{+}_{\Omega^{o}}[\cdot]_{|\partial\Omega^{i}}$
is compact from $C^{0}(\partial\Omega^{o})_{0}$ into
$C^{0}(\partial\Omega^{i})$. By mapping properties of the single layer
potential (cf. Miranda [23, Chap. II, §14, III]) and by the compactness of the
embedding of $C^{0,\alpha}(\partial\Omega^{i})$ into
$C^{0}(\partial\Omega^{i})$, $V_{\partial\Omega^{i}}$ is compact from
$C^{0}(\partial\Omega^{i})$ into itself. Hence, by the bilinearity and
continuity of the product of continuous functions, the map from
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ that takes
the quintuple $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the triple
given by

$\begin{pmatrix}0&0&0\\\ 0&A_{11}&A_{12}\\\
0&A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}0\\\
v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o}\\\
V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i}\end{pmatrix}$

is continuous and maps bounded sets into sets with compact closure, i.e., is
compact. Moreover, by assumption (2), one readily verifies that the operators
$\mathcal{N}_{F_{1}}$ and $\mathcal{N}_{F_{2}}$ are continuous from
$(C^{0}(\partial\Omega^{i}))^{2}$ into $C^{0}(\partial\Omega^{i})$. Hence the
map from $C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ that takes
the quintuple $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the triple

$\begin{pmatrix}f^{0}\\\
\mathcal{N}_{F_{1}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\\\
\mathcal{N}_{F_{2}}(v^{+}_{\Omega^{o}}[\mu^{o}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}]+\rho^{o},V_{\partial\Omega^{i}}[\eta^{i}]+\rho^{i})\end{pmatrix}$

is compact. Finally, by Proposition 4.3 (ii), $J_{A}$ is a linear isomorphism
from $C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ and,
accordingly, $T_{A}$ is compact. ∎

In what follows we will assume the following growth condition on the pair
$(F_{1},F_{2})$ with respect to the matrix function $A$ defined as in (14):

(22) $\begin{split}\bullet\,&\mbox{There exist two constants
}C_{F}\in]0,+\infty[\mbox{ and }\delta\in]0,1[\mbox{ such that}\\\
&\qquad\left|\begin{pmatrix}F_{1}(x,\zeta_{1},\zeta_{2})\\\
F_{2}(x,\zeta_{1},\zeta_{2})\end{pmatrix}-A(x)\begin{pmatrix}\zeta_{1}\\\
\zeta_{2}\end{pmatrix}\right|\leq C_{F}(1+|\zeta_{1}|+|\zeta_{2}|)^{\delta}\\\
&\mbox{for all
}(x,\zeta_{1},\zeta_{2})\in\partial\Omega^{i}\times\mathbb{R}^{2}.\end{split}$

In Proposition 4.7 below we prove the existence of a solution in
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ of (19). Our argument
exploits the Leray-Schauder Fixed-Point Theorem (cf. Gilbarg and Trudinger
[12, Thm. 11.3]).

######  Theorem 4.6 (Leray-Schauder Theorem).

Let $X$ be a Banach space. Let $T$ be a continuous operator from $X$ into
itself. If $T$ is compact and there exists a constant $M\in]0,+\infty[$ such
that $\|x\|_{X}\leq M$ for all $(x,\lambda)\in X\times[0,1]$ satisfying
$x=\lambda T(x)$, then $T$ has at least one fixed point $x\in X$ such that
$\|x\|_{X}\leq M$.

Then we have the following.

######  Proposition 4.7.

Let $A$ be as in (14). Let assumption (22) holds. Let $J_{A}$ be as in
Proposition 4.3. Then the nonlinear system (19) has at least one solution

$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}.$

###### Proof.

We plan to apply the Leray-Schauder Theorem 4.6 to the operator $T_{A}$
defined by (21) in the Banach space $C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. By Proposition 4.5 we
already know that $T_{A}$ is a continuous operator from
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ and maps
bounded sets into sets with compact closure. So in order to apply the Leray-
Schauder Theorem 4.6, we are left to show that if $\lambda\in]0,1[$ and if

(23) $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})=\lambda
T_{A}(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$

with $(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$, then there exists a
constant $C\in]0,+\infty[$ (which does not depend on $\lambda$ and
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$), such that

(24)
$\|(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\|_{C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}}\leq
C.$

By (23) and by $|\lambda|<1$, we readily deduce that

(25)
$\begin{split}&\|(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\|_{C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}}\\\
&\qquad\leq\|T_{A}(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\|_{C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}}\,.\end{split}$

By the growth condition (22), we can show that

(26)
$\begin{split}&\left\|\begin{pmatrix}\mathcal{N}_{F_{1}}(h^{i}_{1},h^{i}_{2})\\\
\mathcal{N}_{F_{2}}(h^{i}_{1},h^{i}_{2})\end{pmatrix}-A\begin{pmatrix}h^{i}_{1}\\\
h^{i}_{2}\end{pmatrix}\right\|_{(C^{0}(\partial\Omega^{i}))^{2}}\\\
&\qquad\leq
C_{F}(1+\|h^{i}_{1}\|_{C^{0}(\partial\Omega^{i})}+\|h^{i}_{2}\|_{C^{0}(\partial\Omega^{i})})^{\delta}\end{split}$

for all pair of functions
$(h^{i}_{1},h^{i}_{2})\in(C^{0}(\partial\Omega^{i}))^{2}$. Hence, by (25) and
by the definition of $T_{A}$ in (21), we deduce that there exist two constants
$C_{1},C_{2}\in]0,+\infty[$, which depend only on the operator norm of
$J_{A}^{(-1)}$ from
$C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}$ to
$C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ (cf. Theorem 4.3 (ii)), on
$\|f^{o}\|_{\partial\Omega^{o}}$, on the constant $C_{F}\in]0,+\infty[$
provided by the growth condition (22) (cf. (26)), on the norm of the bounded
linear operator $v^{+}_{\Omega^{o}}[\cdot]_{|\partial\Omega^{i}}$ from
$C^{0}(\partial\Omega^{o})$ to $C^{0}(\partial\Omega^{i})$, and on the norm of
the bounded linear operator $V_{\partial\Omega^{i}}$ from
$C^{0}(\partial\Omega^{i})$ into itself, such that

$\begin{split}&\|(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\|_{C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}}\\\
&\qquad\leq
C_{1}(C_{2}+\|(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\|_{C^{0}(\partial\Omega^{o})\times(C^{0}(\partial\Omega^{i}))^{2}\times\mathbb{R}^{2}})^{\delta}\,.\end{split}$

Then, by a straightforward calculation, we can show the existence of a
constant $C>0$ such that inequality (24) holds true (cf. Lanza de Cristoforis
[17, proof of Thm. 7.2]). Hence, by the Leray-Schauder Theorem 4.6 there
exists at least one solution

$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$

of
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})=T_{A}(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$.
Finally, by the definition of $T_{A}$ (cf. (21)), we conclude that
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})$ is a
solution in $C^{0}(\partial\Omega^{o})_{0}\times
C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ of the nonlinear system
(19). ∎

In what follows we will exploit a continuity condition on the superposition
operators generated by $F_{1}$ and $F_{2}$, namely

(27) $\begin{split}\bullet&\,\mbox{The superposition operators
}\mathcal{N}_{F_{1}}\mbox{ and }\mathcal{N}_{F_{2}}\mbox{ are continuous from
}\\\ &(C^{0,\alpha}(\partial\Omega^{i}))^{2}\mbox{ into
}C^{0,\alpha}(\partial\Omega^{i}).\end{split}$

For conditions on $F_{1}$ and $F_{2}$ which imply the validity of assumption
(27), we refer to Appell and Zabrejko [1, Ch. 8] and to Valent [32, Chap. II].
Then we can prove a regularity result for the fixed point provided by
Proposition 4.7, and, thus, an existence result for problem (3).

######  Proposition 4.8.

Let A be as in (14). Let assumptions (22) and (27) hold. Then the nonlinear
system (19) has at least one solution
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. In particular,
problem (3) has at least one solution $(u^{o}_{0},u^{i}_{0})\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i})\times
C^{1,\alpha}(\overline{\Omega^{i}})$ given by

(28)
$(u^{o}_{0},u^{i}_{0})\equiv(U^{o}_{\Omega^{i}}[\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}],U^{i}_{\Omega^{i}}[\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}])$

where the pair
$(U^{o}_{\Omega^{i}}[\cdot,\cdot,\cdot,\cdot,\cdot],U^{i}_{\Omega^{i}}[\cdot,\cdot,\cdot,\cdot,\cdot])$
is defined by (7).

###### Proof.

Let $T_{A}$ be as in (21). By Proposition 4.7, we deduce the existence of a
quintuple $(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0}(\partial\Omega^{o})_{0}\times C^{0}(\partial\Omega^{i})\times
C^{0}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ such that
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})=T_{A}(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}).$
By the mapping properties of integral operators with real analytic kernel and
no singularities (cf. Lanza de Cristoforis and Musolino [20, Prop. 4.1]),
$v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}$ belongs to
$C^{0,\alpha}(\partial\Omega^{i})$. By classical results in potential theory
(cf. Miranda [23, Chap. II, §14, III]), $V_{\Omega^{i}}[\mu^{i}_{0}]$ and
$V_{\Omega^{i}}[\eta^{i}_{0}]$ belong to $C^{0,\alpha}(\partial\Omega^{i})$.
Then, by condition (27) and by the membership of $A\in
M_{2}(C^{0,\alpha}(\partial\Omega^{i}))$ and of $f^{o}\in
C^{0,\alpha}(\partial\Omega^{o})$, we obtain that

$\begin{split}&\begin{pmatrix}f^{o}\\\
\mathcal{N}_{F_{1}}(v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}_{0}]+\rho^{o}_{0},V_{\partial\Omega^{i}}[\eta^{i}_{0}]+\rho^{i}_{0})\\\
\mathcal{N}_{F_{2}}(v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}_{0}]+\rho^{o}_{0},V_{\partial\Omega^{i}}[\eta^{i}_{0}]+\rho^{i}_{0})\end{pmatrix}\\\
&\qquad-\begin{pmatrix}0&0&0\\\ 0&A_{11}&A_{12}\\\
0&A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}0\\\
v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}}[\mu^{i}_{0}]+\rho^{o}_{0}\\\
V_{\partial\Omega^{i}}[\eta^{i}_{0}]+\rho^{i}_{0}\end{pmatrix}\end{split}$

belongs to the product space
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$.
Finally, by the invertibility of the operator $J_{A}$ from
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$,
we obtain that
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. In particular, by
Proposition 4.4 we deduce that the pair given by (28) is a solution of (3)
(cf. (21)). ∎

##  5\. The perturbed transmission problem (6)

This section is devoted to the study of the perturbed transmission problem
(6). We introduce the map $M=(M_{1},M_{2},M_{3})$ from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$
defined by

(29) $\displaystyle M_{1}[\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}](x)$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{o}}\right)[\mu^{o}](x)+\nu_{\Omega^{o}}(x)\cdot\nabla
v^{-}_{\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}](x)-f^{o}(x)\quad\forall
x\in\partial\Omega^{o}$ $\displaystyle
M_{2}[\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}](t)$
$\displaystyle\equiv\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}[\phi]}\right)[\mu^{i}\circ\phi^{(-1)}](\phi(t))+\nu_{\Omega^{i}[\phi]}(\phi(t))\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}](\phi(t))$ $\displaystyle\qquad\qquad-
F_{1}\bigg{(}t,v^{+}_{\Omega^{o}}[\mu^{o}](\phi(t))+V_{\partial\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}](\phi(t))+\rho^{o},$
$\displaystyle\qquad\qquad\qquad\qquad
V_{\partial\Omega^{i}[\phi]}[\eta^{i}\circ\phi^{(-1)}](\phi(t))+\rho^{i}\bigg{)}\quad\forall
t\in\partial\Omega^{i}$ $\displaystyle
M_{3}[\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}](t)$
$\displaystyle\equiv\left(-\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}[\phi]}\right)[\eta^{i}\circ\phi^{(-1)}](\phi(t))$
$\displaystyle\qquad\qquad-
F_{2}\bigg{(}t,v^{+}_{\Omega^{o}}[\mu^{o}](\phi(t))+V_{\partial\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}](\phi(t))+\rho^{o},$
$\displaystyle\qquad\qquad\qquad\qquad
V_{\partial\Omega^{i}[\phi]}[\eta^{i}\circ\phi^{(-1)}](\phi(t))+\rho^{i}\bigg{)}\quad\forall
t\in\partial\Omega^{i}$

for all
$(\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. We incidentally
observe that by the definition of $\Omega^{i}[\phi]$ we have that
$\partial\Omega^{i}[\phi]=\phi(\partial\Omega^{i})$. Then, by the definition
of $M$, we can deduce the following result.

######  Proposition 5.1.

Let A be as in (14). Let assumptions (22) and (27) hold. Let

$(\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}.$

Then the pair of functions

$(U^{o}_{\Omega^{i}[\phi]}[\mu^{o},\mu^{i}\circ\phi^{(-1)},\eta^{i}\circ\phi^{(-1)},\rho^{o},\rho^{i}],U^{i}_{\Omega^{i}[\phi]}[\mu^{o},\mu^{i}\circ\phi^{(-1)},\eta^{i}\circ\phi^{(-1)},\rho^{o},\rho^{i}])$

defined by (7) is a solution of problem (6) if and only if

(30) $M[\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]=(0,0,0).$

In particular, equation

(31) $M[\phi_{0},\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i}]=(0,0,0)$

is equivalent to the system (19) and has a solution
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ (recall that
$\phi_{0}\equiv\text{id}_{\partial\Omega^{i}}$).

###### Proof.

We first observe that, by the regularity of
$\phi\in\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}$, if
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$, then

$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}.$

Moreover, since $\overline{\Omega^{i}[\phi]}\subset\Omega^{o}$, we can apply
Lemma 4.1 with $\Omega=\Omega^{i}[\phi]$. Then by the jump relations for the
single layer potential (cf. Theorem 3.4 (iii)), by a change of variable on
$\phi(\partial\Omega^{i})$ and by the definition of $M$ (cf. (29)), we obtain
that the pair of functions

$\displaystyle
U^{o}_{\Omega^{i}[\phi]}[\mu^{o},\mu^{i}\circ\phi^{(-1)},\eta^{i}\circ\phi^{(-1)},\rho^{o},\rho^{i}]=(v^{+}_{\Omega^{o}}[\mu^{o}]+v^{-}_{\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}]+\rho^{o})_{|\overline{\Omega^{o}}\setminus\Omega^{i}[\phi]},$
$\displaystyle
U^{i}_{\Omega^{i}[\phi]}[\mu^{o},\mu^{i}\circ\phi^{(-1)},\eta^{i}\circ\phi^{(-1)},\rho^{o},\rho^{i}]=v^{+}_{\Omega^{i}[\phi]}[\eta^{i}\circ\phi^{(-1)}]+\rho^{i}$

is a solution of problem (6) if and only if (30) is satisfied. Finally, since
$\phi_{0}\equiv\text{id}_{\partial\Omega^{i}}$ (cf. (5)) and by the definition
of $J_{A}$ (cf. (17)), we obtain that, for all
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$, equation (31) is
equivalent to the system (19). Then the existence of a solution
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})$ of (31)
follows by Proposition 4.7. ∎

By Proposition 5.1, the study of problem (6) is reduced to that of equation
(30). We now wish to apply the Implicit Function Theorem for real analytic
maps in Banach spaces (cf. Deimling [9, Thm. 15.3]) to equation (30) around
the value $\phi_{0}$. As a first step we have to analyse the regularity of the
map $M$.

In what follows we will assume the following:

(32) $\begin{split}\bullet&\,\mbox{The superposition operators
}\mathcal{N}_{F_{1}}\mbox{ and }\mathcal{N}_{F_{2}}\mbox{ are real analytic
from }\\\ &(C^{0,\alpha}(\partial\Omega^{i}))^{2}\mbox{ into
}C^{0,\alpha}(\partial\Omega^{i}).\end{split}$

For conditions on $F_{1}$ and $F_{2}$ which imply the validity of assumption
(32), we refer to Valent [32, Chap. II]. We now show that $M$ is real
analytic.

######  Proposition 5.2.

Let assumption (32) holds. Then the map $M$ is real analytic from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$.

###### Proof.

We only prove the analyticity of $M_{2}$. The analyticity of $M_{1}$ and of
$M_{3}$ can be proved similarly and it is left to the reader. Therefore, we
now analyse $M_{2}$. The map from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{i})$ to $C^{0,\alpha}(\partial\Omega^{i})$ that
takes $(\phi,\mu^{i})$ to the function of the variable
$t\in\partial\Omega^{i}$ defined by

$\begin{split}&\left(\frac{1}{2}I+W^{\ast}_{\partial\Omega^{i}[\phi]}\right)[\mu^{i}\circ\phi^{(-1)}](\phi(t))=\frac{1}{2}\mu^{i}(t)+W^{\ast}_{\partial\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}](\phi(t))\\\
&=\frac{1}{2}\mu^{i}(t)+\int_{\partial\Omega^{i}}(\nu_{\Omega^{i}[\phi]}(\phi(t)))\cdot\nabla
S_{n}(\phi(t)-\phi(s)))\,\mu^{i}(s)\,\tilde{\sigma}_{n}[\phi](s)\,d\sigma_{s}\end{split}$

is real analytic by the real analyticity result for the dependence of layer
potentials upon perturbation of the support and of the density of Lanza de
Cristoforis and Rossi [21, Thm. 3.12] and Lanza de Cristoforis [18, Prop. 7]
(see also Lemma 2.1). The map from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})$ to $C^{0,\alpha}(\partial\Omega^{i})$ that
takes $(\phi,\mu^{o})$ to the function of the variable
$t\in\partial\Omega^{i}$ defined by

$\nu_{\Omega^{i}[\phi]}(\phi(t))\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}](\phi(t))=\int_{\partial\Omega^{o}}(\nu_{\Omega^{i}[\phi]}(\phi(t))\cdot\nabla
S_{n}(\phi(t)-y))\,\mu^{o}(y)\,d\sigma_{y}$

can be proven to be real analytic by the properties of integral operators with
real analytic kernels and no singularities (see Lanza de Cristoforis and
Musolino [20, Prop. 4.1]). For the third term of $M_{2}$ we proceed in this
way. The map from $\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})$ to $C^{0,\alpha}(\partial\Omega^{i})$ that
takes $(\phi,\mu^{o})$ to the function of the variable
$t\in\partial\Omega^{i}$ defined by

$v^{+}_{\Omega^{o}}[\mu^{o}](\phi(t))=\int_{\partial\Omega^{o}}S_{n}(\phi(t)-y)\,\mu^{o}(y)\,d\sigma_{y}$

can be proven to be real analytic by the properties of integral operators with
real analytic kernels and no singularities (see Lanza de Cristoforis and
Musolino [20, Prop. 4.1]). The map from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{i})$ to $C^{0,\alpha}(\partial\Omega^{i})$ that
takes $(\phi,\mu^{o})$ to the function of the variable
$t\in\partial\Omega^{i}$ defined by

$V_{\partial\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}](\phi(t))=\int_{\phi(\partial\Omega^{i})}S_{n}(\phi(t)-y)\,\mu^{i}\circ\phi^{(-1)}(y)\,d\sigma_{y}$

is real analytic by a result of real analytic dependence for the single layer
potential upon perturbation of the support and of the density (see Lanza de
Cristoforis and Rossi [21, Thm. 3.12], Lanza de Cristoforis [18, Prop. 7]).
Similarly we can treat
$V_{\partial\Omega^{i}[\phi]}[\eta^{i}\circ\phi^{(-1)}](\phi(\cdot))$. Hence,
by the real analyticity of the composition of real analytic maps and by (32),
we conclude that the map from
$\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}\times
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{i})$ that takes a sextuple
$(\phi,\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ to the function

$\begin{split}\mathcal{N}_{F_{1}}\bigg{(}v^{+}_{\Omega^{o}}[\mu^{o}](\phi(\cdot))_{|\partial\Omega^{i}}+V_{\partial\Omega^{i}[\phi]}[\mu^{i}\circ\phi^{(-1)}]&(\phi(\cdot))+\rho^{o},\\\
&V_{\partial\Omega^{i}[\phi]}[\eta^{i}\circ\phi^{(-1)}](\phi(\cdot))+\rho^{i}\bigg{)}\end{split}$

is real analytic. As a consequence $M_{2}$ is real analytic. ∎

It will be convenient to consider $F_{1}$, $F_{2}$ as two components of a
vector field on $\partial\Omega^{i}\times\mathbb{R}^{2}$. We denote by $F$ the
function from $\partial\Omega^{i}\times\mathbb{R}^{2}$ to $\mathbb{R}^{2}$
defined by

$F(t,\zeta_{1},\zeta_{2})=(F_{1}(t,\zeta_{1},\zeta_{2}),F_{2}(t,\zeta_{1},\zeta_{2}))\quad\forall(t,\zeta_{1},\zeta_{2})\in\partial\Omega^{i}\times\mathbb{R}^{2}\,.$

Clearly, we can extend the definition of the superposition operator (cf.
Section 2) in a natural way, i.e., by setting

$\mathcal{N}_{F}:(C^{0,\alpha}(\partial\Omega^{i}))^{2}\to(C^{0,\alpha}(\partial\Omega^{i}))^{2},\,\mathcal{N}_{F}\equiv(\mathcal{N}_{F_{1}},\mathcal{N}_{F_{2}})\,.$

Now let $(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ be as in Proposition
4.8. By standard calculus in Banach space, we have the following formula
regarding the first order differential of $\mathcal{N}_{F}$:

$d\mathcal{N}_{F}(v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}+V_{\Omega^{i}}[\mu^{i}_{0}]+\rho^{o}_{0},V_{\Omega^{i}}[\eta^{i}_{0}]+\rho^{i}_{0}).(h_{1},h_{2})=A_{\mathcal{N}_{F},0}\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}$

for all $(h_{1},h_{2})\in(C^{0,\alpha}(\partial\Omega^{i}))^{2}$, where

(33)
$A_{\mathcal{N}_{F},0}\equiv\begin{pmatrix}\mathcal{N}_{\partial_{\zeta_{1}}F_{1}}(\alpha^{1}_{0},\alpha^{2}_{0})&\mathcal{N}_{\partial_{\zeta_{2}}F_{1}}(\alpha^{1}_{0},\alpha^{2}_{0})\\\
\mathcal{N}_{\partial_{\zeta_{1}}F_{2}}(\alpha^{1}_{0},\alpha^{2}_{0})&\mathcal{N}_{\partial_{\zeta_{2}}F_{2}}(\alpha^{1}_{0},\alpha^{2}_{0})\end{pmatrix}$

and $\alpha^{1}_{0}$ and $\alpha^{2}_{0}$ are the functions from
$\partial\Omega^{i}$ to $\mathbb{R}$ defined by

(34) $\begin{split}\alpha^{1}_{0}\equiv
v^{+}_{\Omega^{o}}[\mu^{o}_{0}]_{|\partial\Omega^{i}}+V_{\Omega^{i}}[\mu^{i}_{0}]+\rho^{o}_{0},\quad\alpha^{2}_{0}\equiv
V_{\Omega^{i}}[\eta^{i}_{0}]+\rho^{i}_{0}.\end{split}$

We will require that the matrix $A_{\mathcal{N}_{F},0}$ given by (33)-(34)
satisfies assumption (14). In particular, we notice that assumption (32)
implies the validity of the first of the three conditions of (14) for the
matrix $A_{\mathcal{N}_{F},0}$. In order to apply the Implicit Function
Theorem (cf. Deimling [9, Thm. 15.3]) to equation (30) we need to prove the
invertibility of the partial differential of $M$.

######  Proposition 5.3.

Let assumptions (22) and (32) hold. Let
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ be as in Proposition
4.8. Let $A_{\mathcal{N}_{F},0}$ be as in (33)-(34) and assume that satisfies
assumption (14). Then the partial differential of $M$ with respect to
$(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})$ evaluated at the point
$(\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})$,
which we denote by

(35)
$\partial_{(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})}M[\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}],$

is an isomorphism from $C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ to
$C^{0,\alpha}(\partial\Omega^{o})\times(C^{0,\alpha}(\partial\Omega^{i}))^{2}$.

###### Proof.

By standard calculus in Banach spaces, we can verify that the partial
differential (35) is the linear and continuous operator defined by

$\displaystyle\partial_{(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})}M_{1}[\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}].(\tilde{\mu}^{o},\tilde{\mu}^{i},\tilde{\eta}^{i},\tilde{\rho}^{o},\tilde{\rho}^{i})(x)$
$\displaystyle\qquad=\left(-\frac{1}{2}I+W^{\ast}_{\Omega^{o}}\right)[\tilde{\mu}^{o}](x)+\nu_{\Omega^{o}}(x)\cdot\nabla
v^{-}_{\Omega^{i}}[\tilde{\mu}^{i}](x)\qquad\forall x\in\partial\Omega^{o}$
$\displaystyle\partial_{(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})}M_{2}[\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}].(\tilde{\mu}^{o},\tilde{\mu}^{i},\tilde{\eta}^{i},\tilde{\rho}^{o},\tilde{\rho}^{i})(t)$
$\displaystyle\qquad=\left(\frac{1}{2}I+W^{\ast}_{\Omega^{i}}\right)[\tilde{\mu}^{i}](t)+\nu_{\Omega^{i}}(t)\cdot\nabla
v^{+}_{\Omega^{o}}[\mu^{o}](t)$
$\displaystyle\qquad-\partial_{\zeta_{1}}F_{1}\left(t,v^{+}_{\Omega^{o}}[\mu^{o}_{0}](t)+V_{\Omega^{i}}[\mu^{i}_{0}](t)+\rho^{o}_{0},V_{\Omega^{i}}[\eta^{i}_{0}](t)+\rho^{i}_{0}\right)\,$
$\displaystyle\qquad\qquad\times\left(v^{+}_{\Omega^{o}}[\tilde{\mu}^{o}](t)+V_{\Omega^{i}}[\tilde{\mu}^{i}](t)+\tilde{\rho}^{o}\right)$
$\displaystyle\qquad-\partial_{\zeta_{2}}F_{1}\left(t,v^{+}_{\Omega^{o}}[\mu^{o}_{0}](t)+V_{\Omega^{i}}[\mu^{i}_{0}](t)+\rho^{o}_{0},V_{\Omega^{i}}[\eta^{i}_{0}](t)+\rho^{i}_{0}\right)\,$
$\displaystyle\qquad\qquad\times\left(V_{\Omega^{i}}[\tilde{\eta}^{i}](t)+\tilde{\rho}^{i}\right)\qquad\forall
t\in\partial\Omega^{i}$
$\displaystyle\partial_{(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})}M_{3}[\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}].(\tilde{\mu}^{o},\tilde{\mu}^{i},\tilde{\eta}^{i},\tilde{\rho}^{o},\tilde{\rho}^{i})(t)$
$\displaystyle\qquad=\left(-\frac{1}{2}I+W^{\ast}_{\Omega^{i}}\right)[\tilde{\eta}^{i}](t)$
$\displaystyle\qquad-\partial_{\zeta_{1}}F_{2}\left(t,v^{+}_{\Omega^{o}}[\mu^{o}_{0}](t)+V_{\Omega^{i}}[\mu^{i}_{0}](t)+\rho^{o}_{0},V_{\Omega^{i}}[\eta^{i}_{0}](t)+\rho^{i}_{0}\right)\,$
$\displaystyle\qquad\qquad\times\left(v^{+}_{\Omega^{o}}[\tilde{\mu}^{o}](t)+V_{\Omega^{i}}[\tilde{\mu}^{i}](t)+\tilde{\rho}^{o}\right)$
$\displaystyle\qquad-\partial_{\zeta_{2}}F_{2}\left(t,v^{+}_{\Omega^{o}}[\mu^{o}_{0}](t)+V_{\Omega^{i}}[\mu^{i}_{0}](t)+\rho^{o}_{0},V_{\Omega^{i}}[\eta^{i}_{0}](t)+\rho^{i}_{0}\right)\,$
$\displaystyle\qquad\qquad\times\left(V_{\Omega^{i}}[\tilde{\eta}^{i}](t)+\tilde{\rho}^{i}\right)\qquad\forall
t\in\partial\Omega^{i}$

for all
$(\tilde{\mu}^{o},\tilde{\mu}^{i},\tilde{\eta}^{i},\tilde{\rho}^{o},\tilde{\rho}^{i})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$. Then, by
Proposition 4.3 with $A=A_{\mathcal{N}_{F},0}$ (cf. (17)) and since
$A_{\mathcal{N}_{F},0}$ satisfies (14), we conclude that
$\partial_{(\mu^{o},\mu^{i},\eta^{i},\rho^{o},\rho^{i})}M[\phi_{0},\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}]$
is an isomorphism of Banach spaces. ∎

By Propositions 5.1, 5.2, 5.3, and by applying the Implicit Function Theorem
for real analytic functions in Banach spaces (cf. Deimling [9, Thm. 15.3]) to
equation (30), we deduce the following real analyticity result for the
dependence of the densities in the integral representation fomula for the
solutions of problem (6) upon the perturbation of the shape of the inclusion
$\Omega^{i}$.

######  Theorem 5.4.

Let assumptions (22) and (32) hold. Let
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})\in
C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$ be as in Proposition
4.8. Let $A_{\mathcal{N}_{F},0}$ be as in (33)-(34) and assume that satisfies
assumption (14). Then, there exist two open neighbourhoods $Q_{0}$ of
$\phi_{0}$ in $\mathcal{A}^{\Omega^{o}}_{\partial\Omega^{i}}$ and $U_{0}$ of
$(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0})$ in
$C^{0,\alpha}(\partial\Omega^{o})_{0}\times
C^{0,\alpha}(\partial\Omega^{i})\times
C^{0,\alpha}(\partial\Omega^{i})_{0}\times\mathbb{R}^{2}$, and a real analytic
map $\Lambda\equiv(M^{o},M^{i},N^{i},R^{o},R^{i}):Q_{0}\to U_{0}$ such that
the set of zeros of $M$ in $Q_{0}\times U_{0}$ coincides with the graph of the
function $\Lambda$. In particular,

$\Lambda[\phi_{0}]=(M^{o}[\phi_{0}],M^{i}[\phi_{0}],N^{i}[\phi_{0}],R^{o}[\phi_{0}],R^{i}[\phi_{0}])=(\mu^{o}_{0},\mu^{i}_{0},\eta^{i}_{0},\rho^{o}_{0},\rho^{i}_{0}).$

We are now ready to exhibit a family of solutions of problem (6).

######  Definition 5.5.

Let assumptions (22) and (32) hold. Let $A_{\mathcal{N}_{F},0}$ be as in
(33)-(34) and assume that satisfies assumption (14). Let $Q_{0}$ and
$\Lambda\equiv(M^{o},M^{i},N^{i},R^{o},R^{i})$ be as in Theorem 5.4. Then, for
each $\phi\in Q_{0}$ we set

$\displaystyle u^{o}_{\phi}(x)$
$\displaystyle=U^{o}_{\Omega^{i}[\phi]}[M^{o}[\phi],M^{i}[\phi]\circ\phi^{(-1)},N^{i}[\phi]\circ\phi^{(-1)},R^{o}[\phi],R^{i}[\phi]](x)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall
x\in\overline{\Omega^{o}}\setminus\Omega^{i}[\phi],$ $\displaystyle
u^{i}_{\phi}(x)$
$\displaystyle=U^{i}_{\Omega^{i}[\phi]}[M^{o}[\phi],M^{i}[\phi]\circ\phi^{(-1)},N^{i}[\phi]\circ\phi^{(-1)},R^{o}[\phi],R^{i}[\phi]](x)\quad\forall
x\in\overline{\Omega^{i}[\phi]},$

where the pair
$(U^{o}_{\Omega^{i}[\phi]}[\cdot,\cdot,\cdot,\cdot,\cdot],U^{i}_{\Omega^{i}[\phi]}[\cdot,\cdot,\cdot,\cdot,\cdot])$
is defined by (7).

By Propositions 4.8, 5.1, and Theorem 5.4, we deduce the following.

######  Theorem 5.6.

Let assumptions (22) and (32) hold. Let $A_{\mathcal{N}_{F},0}$ be as in
(33)-(34) and assume that satisfies assumption (14). Let $Q_{0}$ be as in
Theorem 5.4 and let $(u^{o}_{\phi},u^{i}_{\phi})$ be as in Definition 5.5.
Then, for all $\phi\in Q_{0}$, $(u^{o}_{\phi},u^{i}_{\phi})\in
C^{1,\alpha}(\overline{\Omega^{o}}\setminus\Omega^{i}[\phi])\times
C^{1,\alpha}(\overline{\Omega^{i}[\phi]})$ is a solution of problem (6). In
particular $(u^{o}_{\phi_{0}},u^{i}_{\phi_{0}})=(u^{o}_{0},u^{i}_{0})$ is a
solution of problem (3).

We are now ready to prove our main result, where we show that suitable
restrictions of the functions $u^{o}_{\phi}$ and $u^{i}_{\phi}$ depend real
analytically on the parameter $\phi$ which determines the domain perturbation.

######  Theorem 5.7.

Let assumptions (22) and (32) hold. Let $A_{\mathcal{N}_{F},0}$ be as in
(33)-(34) and assume that satisfies assumption (14). Let $Q_{0}$ be as in
Theorem 5.4 and let $(u^{o}_{\phi},u^{i}_{\phi})$ be as in Definition 5.5.
Then, the following statements hold.

  1. (i)

Let $\Omega_{\mathtt{int}}$ be a bounded open subset of $\Omega^{o}$. Let
$Q_{\mathtt{int}}\subseteq Q_{0}$ be an open neighbourhood of $\phi_{0}$ such
that

$\overline{\Omega_{\mathtt{int}}}\subset{\Omega^{i}[\phi]}\quad\forall\phi\in
Q_{\mathtt{int}}.$

Then the map from $Q_{\mathtt{int}}$ to
$C^{1,\alpha}(\overline{\Omega_{\mathtt{int}}})$ that takes $\phi$ to
$u^{i}_{\phi|\overline{\Omega_{\mathtt{int}}}}$ is real analytic.

  2. (ii)

Let $\Omega_{\mathtt{ext}}$ be a bounded open subset of $\Omega^{o}$. Let
$Q_{\mathtt{ext}}\subseteq Q_{0}$ be an open neighbourhood of $\phi_{0}$ such
that

$\overline{\Omega_{\mathtt{ext}}}\subset\Omega^{o}\setminus\overline{\Omega^{i}[\phi]}\quad\forall\phi\in
Q_{\mathtt{ext}}.$

Then the map from $Q_{\mathtt{ext}}$ to
$C^{1,\alpha}(\overline{\Omega_{\mathtt{ext}}})$ that takes $\phi$ to
$u^{o}_{\phi|\overline{\Omega_{\mathtt{ext}}}}$ is real analytic.

###### Proof.

We prove (i). By Definition 5.5, by (7) and by Lemma 2.1, we have

$\begin{split}u^{i}_{\phi}(x)&=U^{i}_{\Omega^{i}[\phi]}[M^{o}[\phi],M^{i}[\phi]\circ\phi^{(-1)},N^{i}[\phi]\circ\phi^{(-1)},R^{o}[\phi],R^{i}[\phi]](x)\\\
&=\int_{\partial\Omega^{i}}S_{n}(x-\phi(s))\,N^{i}[\phi](s)\,\tilde{\sigma}_{n}[\phi](s)\,d\sigma_{s}+R^{i}[\phi]\qquad\forall
x\in\overline{\Omega^{i}[\phi]}\end{split}$

and for all $\phi\in Q_{0}$. By the assumption $Q_{\mathtt{int}}\subseteq
Q_{0}$ and Theorem 5.4, we know that the map from $Q_{\mathtt{int}}$ to
$\mathbb{R}$ that takes $\phi$ to $R^{i}[\phi]$ is real analytic. Moreover, by
the real analyticity of $N^{i}[\cdot]$ (cf. Theorem 5.4) and by the properties
of integral operators with real analytic kernels and no singularities (see
Lanza de Cristoforis and Musolino [20, Prop. 4.1]), we can prove that the map
from $Q_{\mathtt{int}}$ to $C^{1,\alpha}(\overline{\Omega_{\mathtt{int}}})$
that takes $\phi$ to the function
$\int_{\partial\Omega^{i}}S_{n}(x-\phi(s))\,N^{i}[\phi](s)\,\tilde{\sigma}_{n}[\phi](s)\,d\sigma_{s}$
of the variable $x\in\overline{\Omega_{\mathtt{int}}}$ is real analytic (see
also Lemma 2.1). Hence, we deduce the validity of (i). The proof of (ii) is
similar and it is left to the reader. ∎

## Acknowledgements

The authors are members of the “Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni” (GNAMPA) of the “Istituto Nazionale di
Alta Matematica” (INdAM). R.M. acknowledges the support of the Project
”Variational methods for stationary and evolution problems with singularities
and interfaces” (PRIN 2017) funded by the Italian Ministry of Education,
University, and Research. P.M. acknowledges the support of the Project
BIRD191739/19 “Sensitivity analysis of partial differential equations in the
mathematical theory of electromagnetism” (University of Padova), of the “INdAM
GNAMPA Project 2020 - Analisi e ottimizzazione asintotica per autovalori in
domini con piccoli buchi”, and of the grant “Challenges in Asymptotic and
Shape Analysis - CASA” (Ca’ Foscari University of Venice).

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