\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 248, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/248\hfil Electromagnetic transmission problems]
{Electromagnetic transmission problems with a large parameter in
 weighted Sobolev spaces}

\author[J. E. Ospino  \hfil EJDE-2013/248\hfilneg]
{Jorge Eli\'ecer Ospino}  % in alphabetical order

\address{Jorge Eli\'ecer Ospino Portillo \newline
 Departamento de Matem\'aticas y Estad\'istica, Fundaci\'on
 Universidad del Norte, Barranquilla, Colombia}
 \email{jospino@uninorte.edu.co}

\thanks{Submitted June 15, 2012. Published November 15, 2013.}
\subjclass[2000]{65N15, 35A35}
\keywords{Electromagnetic transmission problem;
 weighted Sobolev spaces; \hfill\break\indent
a priori estimates}

\begin{abstract}
 We present an a priori estimate for an electromagnetic transmission
 problem in unbounded exterior domains in $\mathbb{R}^3$.
 We consider Maxwell's equations in two sub-domains, the bounded interior
 representing a conducting material (metal) and the unbounded exterior
 representing an insulating material (air). The behavior of the solution
 at infinity is described by means of families of weighted Sobolev spaces,
 so-called Beppo-Levi spaces  \cite{Kufner}.
 We prove the existence and uniqueness of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{The electromagnetic transmission problem}
\label{sec:el1}

Let $\Omega^{\rm cd}$ be a bounded region in $\mathbb{R}^3$
representing a metallic conductor and
$\Omega^{\rm is}:=\mathbb{R}^3\backslash \overline{\Omega^{\rm cd}}$.
Let latter  represents the air. The parameters $\varepsilon_0$,
$\mu_0$, $\sigma$ denote permittivity, permeability, and
conductivity in $\Omega^{\rm cd}$. We assume $\sigma=0$ in
$\Omega^{\rm is}$. All fields are time-harmonic with frequency
$\omega$. As in \cite{MacCamyS} we neglect conduction
currents   in the air and displacement currents in the metal.
Thus we consider
\begin{equation}\label{eq:2.1}
\begin{gathered}
  \operatorname{curl} \mathbf{E}-i\omega\mu_0\mathbf{H}=0,
  \quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is}\quad\text{(Faraday's law)},\\
\operatorname{curl} \mathbf{H}+(i\omega\varepsilon_0-\sigma)\mathbf{E}
=\mathbf{J},\quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is}\quad\text{(Ampere's
law),}
\end{gathered}
\end{equation}
where $\mathbf{E}$ denotes the electric field, $\mathbf{H}$ the magnetic
field and $\mathbf{J}$ the electric current. Across the interface
$\Sigma$ the tangential components of both $\mathbf{E}$ and $\mathbf{H}$
must be continuous; i.e. $\mathbf{E}_{T}^{\rm is}=\mathbf{E}_{T}^{\rm cd}$,
$ \mathbf{H}_{T}^{\rm is}=\mathbf{H}_{T}^{\rm cd}$.
Furthermore the Silver-M\"{u}ller radiation condition is assumed  to hold
at infinity (see \eqref{eq:2.4} below).

Following Peron \cite{Peron} we introduce a large parameter
$\rho=\sqrt{\frac{\sigma}{\omega\varepsilon_0}}>0$ and set
$\mu=\sqrt{\mu_0/\varepsilon_0}$,
$\varepsilon(\rho)=\frac{1}{\mu}(1_{\Omega^{\rm is}}+(1+i\rho^2)1_{\Omega^{\rm cd}})$,
and $\mathbf{F}=i\kappa\mathbf{J}$. Then, defining
$\widehat{\mathbf{x}}=\mathbf{x}/|\mathbf{x}|$, equation \eqref{eq:2.1} and the
Silver-M\"{u}ller radiation condition become
\begin{equation}\label{eq:2.4}
\begin{gathered}
\operatorname{curl} \mathbf{E}_{\rho}-i\kappa\mu\mathbf{H}_{\rho}=0,
\quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is}\quad\text{(Faraday's law)},\\
\operatorname{curl} \mathbf{H}_{\rho}
 +i\kappa\varepsilon(\rho)\mathbf{E}_{\rho}
=\frac{1}{i\kappa}\mathbf{F},\quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is}
\quad\text{(Amper's law)},\\
|\mathbf{H}_{\rho}\times\widehat{\mathbf{x}}-\mathbf{E}_{\rho}|
=o\big( \frac{1}{|\mathbf{x}|}\big),\quad\text{as }
|\mathbf{x}|\to\infty\quad\text{(Silver-M\"{u}ller radiation condition)}.
\end{gathered}
\end{equation}
The first two equations in \eqref{eq:2.4} reduce to
$$
\frac{1}{\mu}\operatorname{curl} \operatorname{curl} \mathbf{E}_{\rho}
-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}
=\mathbf{F},\quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is},
$$
setting $\mathbf{H}_{\rho}=\frac{1}{i\kappa\mu}\operatorname{curl}
\mathbf{E}_{\rho}$. The Silver-M\"{u}ller radiation condition at infinity becomes
\begin{equation}\label{eq:2.2}
|\operatorname{curl} \mathbf{E}_{\rho}\times\widehat{\mathbf{x}}
-\mathbf{E}_{\rho}|=o\big( \frac{1}{|\mathbf{x}|}\big),\quad\text{as }
|\mathbf{x}|\to\infty.
\end{equation}
Peron  \cite{Peron} considers  problem \eqref{eq:2.4} in a bounded
domain $\Omega$ which is split into the conductor $\Omega^{\rm cd}$
and the insulator $\Omega^{\rm is}$, with either Dirichlet or Neumann condition.
 In our case $\Omega$ is unbounded and the boundary conditions are
replaced by the Silver-M\"{u}ller decay condition at infinity.
Problem \eqref{eq:2.4} may be analyzed using $H^{s}_{\rm loc}(\mathbb{R}^3)$
spaces or Beppo-Levi spaces. The reader is referred to Costabel
and Stephan \cite{CostabelSt1} and Giroire \cite{Giroirej}
for applications of these spaces to boundary value problems
involving the Laplacian operator. Nedelec \cite{Nedelec}
uses of $H^{s}_{\rm loc}$ spaces for the study of problems in
electromagnetic theory.
Let
$$
\mathbf{L}^2(\Omega^{\rm cd})=(L^2(\Omega^{\rm cd}))^3
:=\big\{ \mathbf{u}:\Omega^{\rm cd}\to\mathbb{R}^3:
\int_{\Omega^{\rm cd}}|\mathbf{u}|^2dx<\infty  \big\},
$$
with norm
$$
\|\mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}
=\Big( \int_{\Omega^{\rm cd}}|\mathbf{u}|^2dx\Big) ^{1/2}.
$$
Let also
\begin{gather*}
\mathbf{H}(\operatorname{curl},\Omega^{\rm cd})
=\{\mathbf{u}\in\mathbf{L}^2(\Omega^{\rm cd}):
 \operatorname{curl} \mathbf{u}\in\mathbf{L}^2(\Omega^{\rm cd})\},\\
\mathbf{H}(\operatorname{div},\Omega^{\rm cd})
 =\{\mathbf{u}\in\mathbf{L}^2(\Omega^{\rm cd}):
\operatorname{div} \mathbf{u}\in L^2(\Omega^{\rm cd})\},
\end{gather*}
with norms
\begin{gather*}
\|\mathbf{u}\|_{\mathbf{H}(\operatorname{curl},\Omega^{\rm cd})}^2
 =\|\operatorname{curl} \mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2
 +\|\mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2,
\\
\|\mathbf{u}\|_{\mathbf{H}(\operatorname{div},\Omega^{\rm cd})}^2
=\|\operatorname{div} \mathbf{u}\|_{L^2(\Omega^{\rm cd})}^2
+\|\mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2,
\end{gather*}
respectively.
As in \cite{Peron} we define
$$
\mathbf{X}(\Omega^{\rm cd})=\mathbf{H}(\operatorname{curl},\Omega^{\rm cd})
\cap\mathbf{H}(\operatorname{div},\Omega^{\rm cd}),
$$
with norm
$$
\|\mathbf{u}\|_{\mathbf{X}(\Omega^{\rm cd})}^2=\|\operatorname{curl}
 \mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2
+\|\operatorname{div} \mathbf{u}\|_{L^2(\Omega^{\rm cd})}^2
+\|\mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2,
$$
and
\begin{gather*}
\mathbf{X}_T(\Omega^{\rm cd})
 =\{\mathbf{u}\in\mathbf{X}(\Omega^{\rm cd}):
[\mathbf{n}\cdot\mathbf{u}]=0,\text{ on }\Sigma\},\\
\mathbf{X}_N(\Omega^{\rm cd})=\{\mathbf{u}\in\mathbf{X}(\Omega^{\rm cd}):
[\mathbf{n}\times\mathbf{u}]=0, \text{ on }\Sigma\},\\
\mathbf{X}_T(\Omega^{\rm cd},\rho)=\{\mathbf{u}\in\mathbf{H}(\operatorname{curl},
\Omega^{\rm cd}):\varepsilon(\rho)\mathbf{u}\in\mathbf{H}(\operatorname{div},
\Omega^{\rm cd}),\; [\mathbf{n}\cdot\mathbf{u}]=0, \text{ on }\Sigma\},\\
\mathbf{X}_N(\Omega^{\rm cd},\rho)=\{\mathbf{u}\in\mathbf{H}(\operatorname{curl},
\Omega^{\rm cd}):\varepsilon(\rho)\mathbf{u}\in\mathbf{H}(\operatorname{div},
\Omega^{\rm cd}),\; [\mathbf{n}\times\mathbf{u}]=0,\text{ on }\Sigma\}.
\end{gather*}
with norm
$$
\|\mathbf{u}\|_{\mathbf{X}(\Omega^{\rm cd},\rho)}^2
=\|\operatorname{curl} \mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2
+\|\operatorname{div}(\varepsilon(\rho)\mathbf{u})\|_{L^2(\Omega^{\rm cd})}^2
+\|\mathbf{u}\|_{\mathbf{L}^2(\Omega^{\rm cd})}^2,
$$
Note that $\mathbf{X}_T(\Omega^{\rm cd})$,
$\mathbf{X}_N(\Omega^{\rm cd})$,
$\mathbf{X}_T(\Omega^{\rm cd},\rho)$  and
$\mathbf{X}_N(\Omega^{\rm cd},\rho)$ are Hilbert spaces.
Also let $\mathfrak{D}$ denote the space  of all $C^{\infty}$-functions
defined in $\mathbb{R}^3$ with compact support and $\mathfrak{D}'$
its topological dual space (space of distributions, see \cite{Rudin}).

In what follows we define several spaces of distributions which turn out
to be Hilbert spaces. For detailed proof of the corresponding facts the reader
 is referred to \cite{Boulmezaoud,Girault,Kufner} and
\cite[Section 2.5.4]{Nedelec}.

For $\mathbf{x}\in\mathbb{R}^3$, let
$\ell(\|\mathbf{x}\|)=\sqrt{1+x_1^2+x_2^2+x_3^2}$, and
\begin{gather*}
\mathbf{W}(\operatorname{curl},\mathbb{R}^3)
 =\{\mathbf{u}\in\mathfrak{D}'(\mathbb{R}^3)
 :\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\in\mathbf{L}^2(\mathbb{R}^3),\;
 \operatorname{curl} \mathbf{u}\in\mathbf{L}^2(\mathbb{R}^3)\},\\
\mathbf{W}(\operatorname{div},\mathbb{R}^3)
 =\{\mathbf{u}\in\mathfrak{D}'(\mathbb{R}^3)
 :\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\in\mathbf{L}^2(\mathbb{R}^3),\;
 \operatorname{div} \mathbf{u}\in L^2(\mathbb{R}^3)\}.
\end{gather*}
Note that $\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$ and
$\mathbf{W}(\operatorname{div},\mathbb{R}^3)$ are Hilbert spaces equipped
with the norms
$$
\|\mathbf{u}\|_{\mathbf{W}(\operatorname{curl},\mathbb{R}^3)}^2
=\|\operatorname{curl} \mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2
+\|\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2,
$$
and
$$
\|\mathbf{u}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)}^2
=\|\operatorname{div} \mathbf{u}\|_{L^2(\mathbb{R}^3)}^2
+\|\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2.
$$
Furthermore we will use the space
$$
\mathbb{X}(\mathbb{R}^3)=\mathbf{W}(\operatorname{curl},
\mathbb{R}^3)\cap\mathbf{W}(\operatorname{div},\mathbb{R}^3),
$$
subject to the norm
$$
\|\mathbf{u}\|_{\mathbb{X}(\mathbb{R}^3)}^2
=\|\operatorname{curl} \mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2
+\|\operatorname{div} \mathbf{u}\|_{L^2(\mathbb{R}^3)}^2
+\|\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2,
$$
and
\begin{gather*}
\mathbb{X}_T(\mathbb{R}^3)=\{\mathbf{u}\in\mathbb{X}(\mathbb{R}^3)
:[\mathbf{n}\cdot\mathbf{u}]=0,\text{ on }\Sigma\},\\
\mathbb{X}_N(\mathbb{R}^3)=\{\mathbf{u}\in\mathbb{X}(\mathbb{R}^3)
 :[\mathbf{n}\times\mathbf{u}]=0,\text{ on }\Sigma\},\\
\mathbb{X}_T(\mathbb{R}^3,\rho)=\{\mathbf{u}\in\mathbf{W}(\operatorname{curl},
 \mathbb{R}^3):\varepsilon(\rho)\mathbf{u}\in\mathbf{W}(\operatorname{div},
 \mathbb{R}^3), [\mathbf{n}\cdot\mathbf{u}]=0,\text{ on }\Sigma\},\\
\mathbb{X}_N(\mathbb{R}^3,\rho)=\{\mathbf{u}\in\mathbf{W}(\operatorname{curl},
 \mathbb{R}^3):\varepsilon(\rho)\mathbf{u}\in\mathbf{W}(\operatorname{div},
 \mathbb{R}^3), [\mathbf{n}\times\mathbf{u}]=0,\text{ on }\Sigma\},\\
\mathbb{X}_{TN}(\mathbb{R}^3,\rho)
 =\{\mathbf{u}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)
 :\varepsilon(\rho)\mathbf{u}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3),
  [\mathbf{n}\times\mathbf{u}]=[\mathbf{n}\cdot\mathbf{u}]=0,
 \text{ on }\Sigma\},
\end{gather*}
with norm
$$
\|\mathbf{u}\|_{\mathbb{X}_{TN}(\mathbb{R}^3,\rho)}^2
=\|\operatorname{curl} \mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2
+\|\operatorname{div}(\varepsilon(\rho)\mathbf{u})\|_{L^2(\mathbb{R}^3)}^2
+\|\ell(\|\mathbf{x}\|)^{-1}\mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)}^2.
$$
Note that $\mathbb{X}_T(\mathbb{R}^3)$, $\mathbb{X}_N(\mathbb{R}^3)$,
$\mathbb{X}_T(\mathbb{R}^3,\rho)$,  and $\mathbb{X}_N(\mathbb{R}^3,\rho)$
are Hilbert spaces.
For $m$ in $\mathbb{N}\cup\{0\}$ and $k$ in $\mathbb{Z}$, we  define
$$
\mathbf{L}_{m,k}^2(\mathbb{R}^3):=\big\{
\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3:
\forall\alpha\in\mathbb{N}^3, 0\leq|\alpha|\leq m,
\ell(\|\mathbf{x}\|)^{|\alpha|-m+k}\mathbf{u}\in
\mathbf{L}^2(\mathbb{R}^3)\big\} ,
$$
with norm
$$
\|\mathbf{u}\|_{\mathbf{L}_{m,k}^2(\mathbb{R}^3)}
=\|\ell(\|\mathbf{x}\|)^{|\alpha|-m+k}\mathbf{u}\|_{\mathbf{L}^2(\mathbb{R}^3)},
$$
where $\mathbf{L}_{m,k}^2(\mathbb{R}^3)=\big( L^2_{m,k}(\mathbb{R}^3)\big)^3 $.

Next we extend \cite[Theorems 1.2.16 and 1.2.17]{Louer} for unbounded domains.

\begin{lemma}\label{lem:2.9}
Let $\mathbf{F}\in\mathbf{L}^2(\mathbb{R}^3)$. Let $\mathbf{E}_{\rho}$
 and $\mathbf{H}_{\rho}$ in $\mathbf{L}^2(\mathbb{R}^3)$ be a solution
to \eqref{eq:2.4}. Then,
 $\mathbf{E}_{\rho},\mathbf{H}_{\rho}\in\mathbf{W}(\operatorname{curl},
\mathbb{R}^3)$ if and only if
$\mathbf{E}_{\rho}^{\rm cd},\mathbf{H}_{\rho}^{\rm cd}\in\mathbf{H}
(\operatorname{curl},\Omega^{\rm cd})$ and
$\mathbf{E}_{\rho}^{\rm is},\mathbf{H}_{\rho}^{\rm is}\in\mathbf{W}
(\operatorname{curl},\Omega^{\rm is})$ and
$[\mathbf{n}\times\mathbf{E}_{\rho}]_{\Sigma}=0$,
$[\mathbf{n}\times\mathbf{H}_{\rho}]_{\Sigma}=0$,
where $[\mathbf{u}]_{\Sigma}=\mathbf{u}^{\rm is}-\mathbf{u}^{\rm cd}$
denotes the jump across $\Sigma$.
\end{lemma}

\begin{proof}
``$\Rightarrow$'' If $\mathbf{E}_{\rho},\mathbf{H}_{\rho}\in\mathbf{W}
(\operatorname{curl},\mathbb{R}^3)$, then by definition
$\mathbf{E}_{\rho}^{\rm cd},\mathbf{H}_{\rho}^{\rm cd}\in\mathbf{H}
(\operatorname{curl},\Omega^{\rm cd})$ and
$\mathbf{E}_{\rho}^{\rm is},\mathbf{H}_{\rho}^{\rm is}
\in\mathbf{W}(\operatorname{curl},\Omega^{\rm is})$. Thus for
$\mathbf{u}_{\rho}=\mathbf{E}_{\rho}$ or $\mathbf{u}_{\rho}=\mathbf{H}_{\rho}$,
we have
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{v}\cdot\operatorname{curl}
\mathbf{u}_{\rho} dx=\int_{\Omega^{\rm cd}}\mathbf{v}\cdot\operatorname{curl}
\mathbf{u}_{\rho}^{\rm cd} dx+\int_{\Omega^{\rm is}}\mathbf{v}
\cdot\operatorname{curl} \mathbf{u}_{\rho}^{\rm is} dx,
$$
and
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{u}_{\rho}\cdot
\operatorname{curl} \mathbf{v} dx=\int_{\Omega^{\rm cd}}
\mathbf{u}_{\rho}^{\rm cd}\cdot\operatorname{curl}
\mathbf{v} dx+\int_{\Omega^{\rm is}}\mathbf{u}_{\rho}^{\rm is}
\cdot\operatorname{curl} \mathbf{v} dx,
$$
for all $\mathbf{v}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$.

In $\Omega^{\rm cd}$, integrating by parts (see \cite[Theorem 1.2.17]{Louer})
gives
$$
\int_{\Omega^{\rm cd}}[\mathbf{v}\cdot\operatorname{curl}
 \mathbf{u}_{\rho}^{\rm cd} dx-\mathbf{u}_{\rho}^{\rm cd}\cdot\operatorname{curl}
\mathbf{v}] dx=\int_{\Sigma}[\mathbf{n}\times(\mathbf{n}
\times\mathbf{u}_{\rho}^{\rm cd})]\cdot(\mathbf{n}\times\mathbf{v}) ds.
$$
where
$$\mathbf{n}\times(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd})
=\mathbf{n}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm cd})
-\mathbf{u}_{\rho}^{\rm cd}(\mathbf{n}\cdot\mathbf{n})
=\mathbf{n}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm cd})
-\mathbf{u}_{\rho}^{\rm cd}.
$$
Thus
$$
[\mathbf{n}\times(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd})]
\cdot(\mathbf{n}\times\mathbf{v})=[\mathbf{n}(\mathbf{n}
\cdot\mathbf{u}_{\rho}^{\rm cd})-\mathbf{u}_{\rho}^{\rm cd}]
\cdot(\mathbf{n}\times\mathbf{v})=-\mathbf{u}_{\rho}^{\rm cd}
\cdot(\mathbf{n}\times\mathbf{v}),
$$
and
$$
-\mathbf{u}_{\rho}^{\rm cd}\cdot(\mathbf{n}\times\mathbf{v})
=\mathbf{v}\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd}).
$$
Hence
$$
\int_{\Omega^{\rm cd}}\mathbf{v}\cdot\operatorname{curl}
 \mathbf{u}_{\rho}^{\rm cd} dx=\int_{\Omega^{\rm cd}}
\mathbf{u}_{\rho}^{\rm cd}\cdot\operatorname{curl}
\mathbf{v} dx+\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}
\times\mathbf{u}_{\rho}^{\rm cd}) ds.
$$
Let $B_{R}$ be a ball with radius $R>0$ containing $\Omega^{\rm cd}$.
Let $\Omega_{R}=B_{R}\cap\Omega^{\rm is}$. Hence
$\partial\Omega_{R}=\partial B_{R}\cup\Sigma$.

In the domain $\Omega_{R}$, we have, integrating by parts,
(see \cite[Theorem 1.2.17]{Louer}),
\begin{equation}\label{eq:2.3}
\int_{\Omega_{R}}\mathbf{v}\cdot\operatorname{curl} \mathbf{u}_{\rho}^{\rm is} dx
=\int_{\Omega_{R}}\mathbf{u}_{\rho}^{\rm is}\cdot\operatorname{curl}
\mathbf{v} dx-\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}
\times\mathbf{u}_{\rho}^{\rm is}) ds+\int_{\partial B_{R}}\mathbf{v}
\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}) ds.
\end{equation}
Due to the Silver-M\"{u}ller radiation conditions (see \eqref{eq:2.2})
\begin{align*}
|\int_{\partial B_{R}}\mathbf{v}\cdot(\mathbf{n}
\times\mathbf{u}_{\rho}^{\rm is}) ds|
&\leq\int_{\partial B_{R}}|\mathbf{v}|
|\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}| ds \\
&\leq\int_{\partial B_{R}}|\mathbf{v}||\mathbf{n}||\mathbf{u}_{\rho}^{\rm is}|
 |\sin\theta|ds\\
&\leq\int_{\partial B_{R}}\frac{C_1}{R^2}\frac{C_2}{R^2}ds
=\frac{C}{R^2}\to 0,\quad\text{as } R\to\infty.
\end{align*}
Hence,  in \eqref{eq:2.3} taking the limit as $R\to\infty$, we have
$$
\int_{\Omega^{\rm is}}\mathbf{v}\cdot\operatorname{curl}
\mathbf{u}_{\rho}^{\rm is} dx=\int_{\Omega^{\rm is}}
\mathbf{u}_{\rho}^{\rm is}\cdot\operatorname{curl} \mathbf{v} dx
-\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}) ds.
$$
Thus
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{v}\cdot\operatorname{curl}
\mathbf{u}_{\rho} dx
=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}
\mathbf{u}_{\rho}\cdot\operatorname{curl} \mathbf{v} dx
+\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd}
-\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}) ds,
$$
yielding
$$
\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd}
-\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}) ds=0,
$$
and therefore $[\mathbf{n}\times\mathbf{u}_{\rho}]_{\Sigma}=0$.

``$\Leftarrow$'' If $\mathbf{E}_{\rho}^{\rm cd},\mathbf{H}_{\rho}^{\rm cd}
\in\mathbf{H}(\operatorname{curl},\Omega^{\rm cd})$,
$\mathbf{E}_{\rho}^{\rm is},\mathbf{H}_{\rho}^{\rm is}
\in\mathbf{W}(\operatorname{curl},\Omega^{\rm is})$,
$[\mathbf{n}\times\mathbf{E}_{\rho}]_{\Sigma}=0$, and
$[\mathbf{n}\times\mathbf{H}_{\rho}]_{\Sigma}=0$.
Then for $\mathbf{v}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$
and $\mathbf{u}_{\rho}=\mathbf{E}_{\rho}$ or $\mathbf{u}_{\rho}=\mathbf{H}_{\rho}$,
integrating by parts,
\begin{align*}
&\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{u}_{\rho}
 \cdot\operatorname{curl} \mathbf{v} dx\\
&=\int_{\Omega^{\rm cd}}\mathbf{u}_{\rho}^{\rm cd}\cdot\operatorname{curl}
 \mathbf{v} dx+\int_{\Omega^{\rm is}}\mathbf{u}_{\rho}^{\rm is}
 \cdot\operatorname{curl} \mathbf{v} dx,\\
&=\int_{\Omega^{\rm cd}}\mathbf{v}\cdot\operatorname{curl}
 \mathbf{u}_{\rho}^{\rm cd} dx+\int_{\Omega^{\rm is}}\mathbf{v}
 \cdot\operatorname{curl} \mathbf{u}_{\rho}^{\rm is} dx
 +\int_{\Sigma}\mathbf{v}\cdot(\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd}
 -\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}) ds.
\end{align*}
Hence $\mathbf{n}\times\mathbf{u}_{\rho}^{\rm cd}
-\mathbf{n}\times\mathbf{u}_{\rho}^{\rm is}=0$ implies
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{u}_{\rho}
\cdot\operatorname{curl} \mathbf{v} dx=\int_{\Omega^{\rm cd}
\cup\Omega^{\rm is}}\mathbf{v}\cdot\operatorname{curl} \mathbf{u}_{\rho} dx.
$$
\end{proof}

Our next Lemma adapts \cite[Lemmas 2.7 and 2.8]{Peron} for exterior domains
in weighted Sobolev spaces.

\begin{lemma}\label{lem:2.10}
Let $\mathbf{F}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3)$.
Let $\mathbf{E}_{\rho}$ and $\mathbf{H}_{\rho}$ in
 $\mathbf{L}^2(\mathbb{R}^3)$ solutions of \eqref{eq:2.4}.
If $\mathbf{E}_{\rho},\mathbf{H}_{\rho}\in\mathbf{W}(\operatorname{curl},
\mathbb{R}^3)$, then $\varepsilon(\rho)\mathbf{E}_{\rho},
\mathbf{H}_{\rho}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3)$,
$[\mathbf{n}\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})]_{\Sigma}=0$, and
$[\mathbf{n}\cdot\mathbf{H}_{\rho}]_{\Sigma}=0$.
Furthermore,
$$
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
=-\frac{1}{\kappa^2}\operatorname{div}\mathbf{F},\quad
\operatorname{div}\mathbf{H}_{\rho}=0\quad\text{in } \mathbf{L}^2(\mathbb{R}^3).
$$
\end{lemma}

\begin{proof}
If $\mathbf{F}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3)$ and
$\mathbf{E}_{\rho},\mathbf{H}_{\rho}\in\mathbf{L}^2(\mathbb{R}^3)$
are solutions of \eqref{eq:2.4}, then applying divergence operator
in \eqref{eq:2.4}, we have
$$
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
=-\frac{1}{\kappa^2}\operatorname{div} \mathbf{F},\quad
\operatorname{div} \mathbf{H}_{\rho}=0\quad\text{in} \mathbf{L}^2(\mathbb{R}^3),
$$
and $\varepsilon(\rho)\mathbf{E}_{\rho}\in\mathbf{W}(\operatorname{div},
\mathbb{R}^3)$, $\mathbf{H}_{\rho}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3)$.
Now, for $\mathbf{u}_{\rho}=\varepsilon(\rho)\mathbf{E}_{\rho}$ or
$\mathbf{H}_{\rho}$, we have
\begin{gather*}
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\phi \operatorname{div}
\mathbf{u}_{\rho} dx=\int_{\Omega^{\rm cd}}\phi \operatorname{div}
\mathbf{u}_{\rho}^{\rm cd} dx+\int_{\Omega^{\rm is}}\phi \operatorname{div}
\mathbf{u}_{\rho}^{\rm is} dx,
\\
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{u}_{\rho}\cdot\nabla\phi dx
=\int_{\Omega^{\rm cd}}\mathbf{u}_{\rho}^{\rm cd}\cdot\nabla\phi dx
+\int_{\Omega^{\rm is}}\mathbf{u}_{\rho}^{\rm is}\cdot\nabla\phi dx,
\end{gather*}
for all $\phi\in\mathcal{\mathbf{V}}=H_0^{1}(\Omega^{\rm cd})
\cup\mathbb{W}^{1}_0(\Omega^{\rm is})$.
In $\Omega^{\rm cd}$, integrating by parts (see \cite[Theorem 1.2.16]{Louer}),
$$
\int_{\Omega^{\rm cd}}\phi \operatorname{div} \mathbf{u}_{\rho}^{\rm cd} dx
=\int_{\Omega^{\rm cd}}\mathbf{u}_{\rho}^{\rm cd}\cdot\nabla\phi dx
-\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm cd})\phi ds.
$$
Let be a ball $B_{R}$ with radius $R>0$ containing $\Omega^{\rm cd}$.
Let $\Omega_{R}=B_{R}\cap\Omega^{\rm is}$.
Hence $\partial\Omega_{R}=\partial B_{R}\cup\Sigma$.
In the domain $\Omega_{R}$, we have, integrating by parts,
(see \cite[Theorem 1.2.16]{Louer}),
\begin{align*}
&\int_{\Omega_{R}}\phi \operatorname{div} \mathbf{u}_{\rho}^{\rm is} dx\\
&=\int_{\Omega_{R}}\mathbf{u}_{\rho}^{\rm is}\cdot\nabla\phi dx
+\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is})\phi ds
-\int_{\partial B_{R}}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is})\phi ds\\
&=\int_{\Omega_{R}}\mathbf{u}_{\rho}^{\rm is}\cdot\nabla\phi dx
+\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is})\phi ds
-\int_{\partial B_{R}}(\mathbf{n}\cdot[\mathbf{u}_{\rho}^{\rm is}
+\mathbf{U}_{\rho}^{\rm is}\times\mathbf{n}])\phi ds,
\end{align*}
where, if $\mathbf{u}_{\rho}^{\rm is}=\varepsilon(\rho)\mathbf{E}_{\rho}$,
$\mathbf{U}_{\rho}^{\rm is}=-\varepsilon(\rho)\mathbf{H}_{\rho}$ or if
$\mathbf{u}_{\rho}^{\rm is}=\mathbf{H}_{\rho}$,
 $\mathbf{U}_{\rho}^{\rm is}=\mathbf{E}_{\rho}$ and
 $\mathbf{n}\cdot[\mathbf{U}_{\rho}^{\rm is}\times\mathbf{n}]=0$.
Due to the Silver-M\"{u}ller radiation conditions (see \eqref{eq:2.2})
$$
\int_{\partial B_{R}}\mathbf{n}\cdot[\mathbf{u}_{\rho}^{\rm is}
+\mathbf{U}_{\rho}^{\rm is}\times\mathbf{n}]\cdot\phi ds\to 0\quad\text{as}
 R\to\infty.
$$
Hence
$$
\int_{\Omega^{\rm is}}\phi \operatorname{div} \mathbf{u}_{\rho}^{\rm is} dx
=\int_{\Omega^{\rm is}}\mathbf{u}_{\rho}^{\rm is}\cdot\nabla\phi dx+\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is})\phi ds.$$
Altogether we have
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\phi \operatorname{div}
\mathbf{u}_{\rho} dx
=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{u}_{\rho}
\cdot\nabla\phi dx+\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is}
-\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm cd})\cdot\phi ds
$$
then
$$
\int_{\Sigma}(\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm is}
-\mathbf{n}\cdot\mathbf{u}_{\rho}^{\rm cd})\cdot\phi ds=0,
$$
for all $\phi$ implies $[\mathbf{n}\cdot\mathbf{u}_{\rho}]_{\Sigma}=0$.
\end{proof}

For $\mathbf{E}_{\rho},\mathbf{E}'\in\widetilde{\mathbf{W}}(\operatorname{curl},
\mathbb{R}^3):=\{\mathbf{E}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3) :
 \mathbf{E}\in\mathbf{L}^2(\mathbb{R}^3)\}$, set
\begin{equation}\label{eq:2.5}
b_{\rho}(\mathbf{E}_{\rho},\mathbf{E}')
:=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}
\Big( \frac{1}{\mu}\operatorname{curl} \mathbf{E}_{\rho}\cdot\operatorname{curl}
\overline{\mathbf{E}'}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}
\cdot\overline{\mathbf{E}'}\Big) dx.
\end{equation}

\begin{proposition}\label{pro:2.15}
 If $\mathbf{F}\in\mathbf{L}^2(\mathbb{R}^3)$,
$\mathbf{E}_{\rho},\mathbf{H}_{\rho}\in\mathbf{L}^2(\mathbb{R}^3)$
satisfy \eqref{eq:2.4}, then,
 $\mathbf{E}_{\rho}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$
and for all $\mathbf{E}'\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$,
\begin{equation}\label{eq:2.6}
b_{\rho}(\mathbf{E}_{\rho},\mathbf{E}')
=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\mathbf{F}\cdot\overline{\mathbf{E}'}dx.
\end{equation}
\end{proposition}

The proof of Proposition \ref{pro:2.15} is a minor modification of
the proof \cite[Proposition 3]{Peron}, and is left for the reader.

\begin{proposition}\label{pro:2.17}
If $\mathbf{E}_{\rho}\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$
 satisfies \eqref{eq:2.6}, then $\mathbf{E}_{\rho}$ satisfies
(in the sense of distributions):
\begin{equation}\label{eq:2.8}
\begin{gathered}
\operatorname{curl} \operatorname{curl} \mathbf{E}_{\rho}
 -\kappa^2\mathbf{E}_{\rho}
=\mu\mathbf{F}^{\rm is}, \quad\text{in }\Omega^{\rm is},\\
\operatorname{curl} \operatorname{curl} \mathbf{E}_{\rho}
-\kappa^2(1+i\rho^2)\mathbf{E}_{\rho}
=\mu\mathbf{F}^{\rm cd}, \quad\text{in }\Omega^{\rm cd},\\
[ \mathbf{n}\times\mathbf{E}_{\rho}] _{\Sigma}=0, \quad
[ \mathbf{n}\times\operatorname{curl} \mathbf{E}_{\rho}] _{\Sigma}=0,
\quad \text{on } \Sigma,
\end{gathered}
\end{equation}
with Silver-M\"{u}ller condition
\[
|\operatorname{curl} \mathbf{E}_{\rho}\times\widehat{\mathbf{x}}
-\mathbf{E}_{\rho}|=o\big( \frac{1}{|\mathbf{x}|}\big), \quad
\text{as }|\mathbf{x}|\to\infty,
\]
On the other hand, if $\mathbf{E}_{\rho}$  solves \eqref{eq:2.8}, then
\begin{gather}\label{eq:2.9}
\frac{1}{\mu}\operatorname{curl} \operatorname{curl}
\mathbf{E}_{\rho}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}=\mathbf{F},
\quad\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is},
\\
\label{eq:2.10}
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
=-\frac{1}{\kappa^2}\operatorname{div} \mathbf{F},\quad\text{in }
\Omega^{\rm cd}\cup\Omega^{\rm is}.
\end{gather}
\end{proposition}

\begin{proof}
We follow the proof of \cite[Proposition 2.15]{Peron}
to show that $\mathbf{E}_{\rho}$ satisfies the first two equations
 in \eqref{eq:2.8}.

Taking $\mathbf{E}'\in\mathfrak{D}'(\mathbb{R}^3)$ with support in
$\Omega^{\rm is}$ as test function in \eqref{eq:2.6} and using
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\mathbf{E}'}dx
=\langle\operatorname{curl} \operatorname{curl} \mathbf{E}_{\rho},
\mathbf{E}'\rangle_{\Omega^{\rm is}},
$$
we see that the first equation in \eqref{eq:2.8} is satisfied.
Taking $\mathbf{E}'\in\mathfrak{D}'(\mathbb{R}^3)$ with support in
$\Omega^{\rm cd}$ as test function in \eqref{eq:2.6} and using
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl}
\overline{\mathbf{E}'}dx=\langle\operatorname{curl}
\operatorname{curl} \mathbf{E}_{\rho},\mathbf{E}'\rangle_{\Omega^{\rm cd}},
$$
we see that the second equation in \eqref{eq:2.8} is satisfied.
By Lemma \ref{lem:2.9}, the third relation \eqref{eq:2.8} holds.

Let $B_{R}$ be a ball with radius $R>0$ containing $\Omega^{\rm cd}$.
Let $\Omega=\Omega^{\rm cd}\cup(\Omega^{\rm is}\cap B_{R})$. Hence
$\partial(\Omega^{\rm is}\cap B_{R})=\Sigma\cup\partial B_{R}=\partial\Omega$,
for all $\mathbf{E},\mathbf{H}\in\mathbf{H}(\operatorname{curl},\Omega)$,
\begin{equation}\label{eq:2.11}
\int_{\Omega}\operatorname{curl} \mathbf{E}\cdot\overline{\mathbf{H}}
-\mathbf{E}\cdot\operatorname{curl} \overline{\mathbf{H}})dx
=\langle\mathbf{n}\times\mathbf{E},\mathbf{H}_{\tau}\rangle_{\partial\Omega},
\end{equation}
where $\mathbf{H}_{\tau}=(\mathbf{n}\times\mathbf{H})\times\mathbf{n}$.
From the first equation in \eqref{eq:2.8} we have
$\operatorname{curl} \mathbf{E}^{\rm is}\in\mathbf{H}(\operatorname{curl},
\Omega^{\rm is}\cap B_{R})$. Applying formula \eqref{eq:2.11} in
$\Omega^{\rm is}\cap B_{R}$ to
$\mathbf{E}=\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}$ and
$\mathbf{H}=\mathbf{E}'\in\mathbf{H}(\operatorname{curl},\Omega)$, we have
\begin{equation}\label{eq:2.11a}
\begin{aligned}
&\int_{\Omega^{\rm is}\cap B_{R}}\operatorname{curl}
\mathbf{E}_{\rho}^{\rm is}\cdot\operatorname{curl}
\overline{(\mathbf{E}')^{\rm is}}dx\\
&=
\int_{\Omega^{\rm is}\cap B_{R}}\operatorname{curl}
\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}
\cdot\overline{(\mathbf{E}')^{\rm is}}dx+\langle\operatorname{curl}
\mathbf{E}_{\rho}^{\rm is}\times\mathbf{n},(\mathbf{E}')^{\rm is}_{\tau}
\rangle_{\partial(\Omega^{\rm is}\cap B_{R})}.
\end{aligned}
\end{equation}
Applying formula \eqref{eq:2.11} in $\Omega^{\rm cd}$ to
$\mathbf{E}=\operatorname{curl} \mathbf{E}_{\rho}^{\rm cd}$ and
$\mathbf{H}=\mathbf{E}'\in\mathbf{H}(\operatorname{curl},\Omega)$, we have
\begin{equation}\label{eq:2.11b}
\begin{aligned}
&\int_{\Omega^{\rm cd}}\operatorname{curl} \mathbf{E}_{\rho}^{\rm cd}
\cdot\operatorname{curl} \overline{(\mathbf{E}')^{\rm cd}}dx\\
&= \int_{\Omega^{\rm cd}}\operatorname{curl} \operatorname{curl}
\mathbf{E}_{\rho}^{\rm cd}\cdot\overline{(\mathbf{E}')^{\rm cd}}dx
+\langle\operatorname{curl} \mathbf{E}_{\rho}^{\rm cd}\times\mathbf{n},
(\mathbf{E}')^{\rm cd}_{\tau}\rangle_{\Sigma}.
\end{aligned}
\end{equation}
In \eqref{eq:2.11a},
$$
\langle\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}\times\mathbf{n}
,(\mathbf{E}')^{\rm is}_{\tau}\rangle_{\partial(\Omega^{\rm is}\cap B_{R})}
=\langle\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}\times\mathbf{n},
(\mathbf{E}')^{\rm is}_{\tau}\rangle_{\Sigma}+\langle\operatorname{curl}
 \mathbf{E}_{\rho}^{\rm is}\times\mathbf{n},(\mathbf{E}')
^{\rm is}_{\tau}\rangle_{\partial B_{R}}.
$$
Applying Silver-M\"{u}ller radiation condition yields
\begin{align*}
|\langle\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}\times\mathbf{n},
(\mathbf{E}')^{\rm is}_{\tau}\rangle_{\partial B_{R}}|
&=\big|\int_{\partial B_{R}}\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}
\times\mathbf{n}\cdot(\mathbf{E}')^{\rm is}_{\tau}ds\big|
\\
&\leq\int_{\partial B_{R}}|\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}
\times\mathbf{n}||(\mathbf{n}\times\mathbf{E}')\times\mathbf{n}|ds
\\
&\leq\int_{\partial B_{R}}|\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}|
|\mathbf{n}||\sin\theta_1||\mathbf{n}||\mathbf{n}\times\mathbf{E}'|
|\sin\theta_2|ds
\\
&\leq\int_{\partial B_{R}}C_1|\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}|
|\mathbf{n}||\mathbf{E}'||\sin\theta_3|ds
\\
&\leq\int_{\partial B_{R}}C_1|\operatorname{curl}
\mathbf{E}_{\rho}^{\rm is}||\mathbf{E}'|ds
\\
&=C_1\frac{C_2}{R^{4}}R^2=\frac{C}{R^2}\to 0,\quad\text{as } R\to\infty.
\end{align*}
Then, by the dominated convergence Theorem,
\begin{equation}\label{eq:2.11c}
\begin{aligned}
&\int_{\Omega^{\rm is}}\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}
\cdot\operatorname{curl} \overline{(\mathbf{E}')^{\rm is}}dx \\
&= \int_{\Omega^{\rm is}}\operatorname{curl} \operatorname{curl}
\mathbf{E}_{\rho}^{\rm is}\cdot\overline{(\mathbf{E}')^{\rm is}}dx
+\langle\operatorname{curl} \mathbf{E}_{\rho}^{\rm is}\times\mathbf{n},
(\mathbf{E}')^{\rm is}_{\tau}\rangle_{\Sigma}.
\end{aligned}
\end{equation}
From \eqref{eq:2.11b} and \eqref{eq:2.11c},
\begin{align*}
&\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\mathbf{E}'}dx\\
&=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\operatorname{curl}
\operatorname{curl} \mathbf{E}_{\rho}\cdot\overline{\mathbf{E}'}dx
+\langle[\operatorname{curl} \mathbf{E}_{\rho}\times\mathbf{n}]_{\Sigma},
\mathbf{E}'_{\tau}\rangle_{\Sigma},
\end{align*}
for all $\mathbf{E}'\in\mathbf{W}(\operatorname{curl},\mathbb{R}^3)$.
From \eqref{eq:2.6} and the first two equations in \eqref{eq:2.8},
$$
\langle[\operatorname{curl} \mathbf{E}_{\rho}\times\mathbf{n}]_{\Sigma},
\mathbf{E}'_{\tau}\rangle_{\Sigma}=0,
$$
which proofs the fourth equation in \eqref{eq:2.8}.
\end{proof}

Next we consider a \emph{regularized version} of problem \eqref{eq:2.6}.
Namely: We consider finding $\mathbf{E}_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$,
such that, for all $\mathbf{E}_{\rho}'\in\mathbb{X}_T(\mathbb{R}^3,\rho)$,
\begin{equation}\label{eq:2.18}
\begin{aligned}
&\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\mathbf{E}'_{\rho}}
+\alpha \operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\operatorname{div}(\overline{\varepsilon(\rho)\mathbf{E}'_{\rho}})
-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot\overline{\mathbf{E}'_{\rho}}
\Big) dx\\
&=\langle f,\mathbf{E}_{\rho}'\rangle,
\end{aligned}
\end{equation}
where
\begin{equation}\label{eq:2.19}
\langle f,\mathbf{E}_{\rho}'\rangle=\int_{\Omega^{\rm cd}
\cup\Omega^{\rm is}}\Big( \mathbf{F}\cdot\overline{\mathbf{E}_{\rho}'}
-\frac{\alpha}{\kappa^2}\operatorname{div} \mathbf{F}\cdot\operatorname{div}
(\overline{\varepsilon(\rho)\mathbf{E}_{\rho}'})\Big) dx,
\end{equation}
and where $\alpha>0$.

The next theorem, extends \cite[Theorem 2.21]{Peron}
(see also Costabel et al.~\cite{Costabeld}), which corresponds
to Peron's theorem \cite[Theorem 2.21]{Peron} and is its modification
for an unbounded exterior domain and weighted spaces.

\begin{theorem}\label{th:2.23}
There exists $\alpha>0$, independent of $\rho$, such that if
$\mathbf{E}_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$
is a solution of \eqref{eq:2.18}--\eqref{eq:2.19} for
$\mathbf{F}\in\mathbf{W}_0(\operatorname{div},\mathbb{R}^3)$, then
\begin{equation}\label{eq:2.20}
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
+\frac{1}{\kappa^2}\operatorname{div} \mathbf{F}=0,\quad
\text{in }\Omega^{\rm cd}\cup\Omega^{\rm is}.
\end{equation}
Furthermore $\mathbf{E}_{\rho}$ and
$\mathbf{H}_{\rho}=\frac{1}{i\omega\varepsilon_0}\operatorname{curl}
\mathbf{E}_{\rho}$ satisfy Maxwell's equations \eqref{eq:2.4}.
\end{theorem}

\begin{proof}
Let us define the operator
$\Delta_{\varepsilon(\rho)}^N$ from $\mathbb{W}^{1}_0(\mathbb{R}^3)$ to
$\mathbb{W}^{1}_0(\mathbb{R}^3)'$ mapping $\varphi$ to
$\operatorname{div}(\varepsilon(\rho)\nabla\varphi)$,
where $\operatorname{div}(\varepsilon(\rho)\nabla\varphi)
\in\mathbb{W}^{1}_0(\mathbb{R}^3)'$ defined for any
$\psi\in\mathbb{W}^{1}_0(\mathbb{R}^3)$ by
$$
\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}\varepsilon(\rho)\nabla\varphi\cdot
\overline{\nabla\psi}dx.
$$
see \cite[Theorem 2.21]{Peron}.
Defining the domain of $\Delta_{\varepsilon(\rho)}^N$ by
$$
\textbf{D}(\Delta_{\varepsilon(\rho)}^N)=\{\varphi\in\mathbb{W}^{1}_0
(\mathbb{R}^3) |\quad\operatorname{div}(\varepsilon(\rho)\nabla\varphi)
\in L^2(\mathbb{R}^3)\},
$$
 $\nabla\varphi\in\mathbb{X}_T(\mathbb{R}^3,\rho)$ for
$\varphi\in\textbf{D}(\Delta_{\varepsilon(\rho)}^N)$.

Let $\mathbf{E}_{\rho}$ satisfy \eqref{eq:2.18}. Choosing
$\mathbf{E}'=\nabla\varphi$ with
$\varphi\in\textbf{D}(\Delta_{\varepsilon(\rho)}^N)$, \eqref{eq:2.18} gives
\begin{equation}\label{eq:2.21}
\begin{aligned}
&\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}(\alpha \operatorname{div}
(\varepsilon(\rho)\mathbf{E}_{\rho})\cdot\operatorname{div}
(\overline{\varepsilon(\rho)\nabla\varphi})
-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot\overline{\nabla\varphi}) dx\\
&=\int_{\Omega^{\rm cd}\cup\Omega^{\rm is}}
\Big( \mathbf{F}\cdot\overline{\nabla\varphi}
 -\frac{\alpha}{\kappa^2}\operatorname{div}
 \mathbf{F}\cdot\operatorname{div}(\overline{\varepsilon(\rho)\nabla\varphi})
\Big) dx.
\end{aligned}
\end{equation}
Since $\varepsilon(\rho)\mathbf{E}_{\rho},\mathbf{F}
\in\mathbf{W}_0(\operatorname{div},\mathbb{R}^3)$ and
$\varphi\in\mathbb{W}^{1}_0(\mathbb{R}^3)$ we have
$$
\int_{\mathbb{R}^3}-\kappa^2\varepsilon(\rho)
\mathbf{E}_{\rho}\cdot\overline{\nabla\varphi}) dx
=\int_{\Omega^{\rm cd}}-\kappa^2\varepsilon(\rho)
\mathbf{E}_{\rho}\cdot\overline{\nabla\varphi}) dx
+\int_{\Omega^{\rm is}}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot
\overline{\nabla\varphi}) dx.
$$
Green's formula in $\Omega^{\rm cd}$, yields
$$
\int_{\Omega^{\rm cd}}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot
\overline{\nabla\varphi}) dx=\int_{\Omega^{\rm cd}}\kappa^2
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\overline{\varphi}) dx+\int_{\Sigma}\kappa^2\mathbf{n}
\cdot(\varepsilon(\rho)\mathbf{E}_{\rho}\overline{\nabla\varphi})) dS.
$$
Let $B_{R}$ be a ball with radius $R>0$ containing $\Omega^{\rm cd}$,
with $\partial\Omega_{R}=\partial B_{R}\cup\Sigma$.
Let $\Omega_{R}$ be as above. Integrating by parts,
\begin{equation}\label{eq:2.21a}
\int_{\Omega_{R}}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}
\cdot\overline{\nabla\varphi}) dx=\int_{\Omega_{R}}\kappa^2
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\overline{\varphi}) dx+\int_{\partial\Omega_{R}}
\kappa^2\mathbf{n}\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})
\overline{\varphi}) ds,
\end{equation}
and
$$
\int_{\partial\Omega_{R}}\kappa^2\mathbf{n}\cdot(\varepsilon(\rho)
\mathbf{E}_{\rho})\overline{\varphi}) ds
=\int_{\Sigma}\kappa^2\mathbf{n}\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})
\overline{\varphi}) ds+\int_{\partial B_{R}}\kappa^2\mathbf{n}
\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})\overline{\varphi}) ds.
$$
As in the proof of Lemma \ref{lem:2.10}, applying the Silver-M\"{u}ller
condition (see \eqref{eq:2.8}),
$$
\int_{\partial B_{R}}\kappa^2\mathbf{n}\cdot(\varepsilon(\rho)
\mathbf{E}_{\rho})\overline{\varphi} ds\to 0,\quad\text{as } R\to\infty,
$$
Hence, by \eqref{eq:2.21a},
\begin{align*}
&\int_{\Omega^{\rm is}}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}
\cdot\overline{\nabla\varphi}) dx\\
&=\int_{\Omega^{\rm is}}\kappa^2\operatorname{div}(\varepsilon(\rho)
\mathbf{E}_{\rho})\cdot\overline{\varphi}dx-\int_{\Sigma}\kappa^2\mathbf{n}
\cdot(\varepsilon(\rho)\mathbf{E}_{\rho}\overline{\nabla\varphi})) ds
-\int_{\Sigma}\kappa^2\mathbf{n}\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})
\overline{\varphi} ds=0,
\end{align*}
for all $\varphi\in\textbf{D}(\Delta_{\varepsilon(\rho)}^N)$ yielding
$[\mathbf{n}\cdot(\varepsilon(\rho)\mathbf{E}_{\rho})]=0$ on $\Sigma$.
Now
$$
\int_{\mathbb{R}^3}\mathbf{F}\cdot\overline{\nabla\varphi}dx
=\int_{\Omega^{\rm cd}}\mathbf{F}\cdot\overline{\nabla\varphi}dx
+\int_{\Omega^{\rm is}}\mathbf{F}\cdot\overline{\nabla\varphi}dx,
$$
Green's formula in $\Omega^{\rm cd}$, yields
$$
\int_{\Omega^{\rm cd}}\mathbf{F}\cdot\overline{\nabla\varphi}dx
=-\int_{\Omega^{\rm cd}}\operatorname{div} \mathbf{F}\cdot
\overline{\varphi}dx-\int_{\Sigma}(\mathbf{n}\cdot\mathbf{F})
\overline{\varphi}ds.
$$
Again, we choose $R>0$ such that $B_{R}$ contains $\Omega^{\rm cd}$.
Using again that  $\Omega^{\rm is}=\bigcup_{R>0}\Omega_{R}$ and that
$\partial\Omega_{R}=\partial B_{R}\cup\Sigma$.

Applying the divergence theorem to
$\mathbf{F}=i\kappa\operatorname{curl}\mathbf{H}
-\kappa^2\varepsilon(\rho)\mathbf{E}$ in $\Omega_{R}$, and the
Silver-M\"{u}ller condition, we have
$$
\int_{\partial B_{R}}(\mathbf{n}\cdot\mathbf{F})\overline{\varphi}ds\to 0,
\quad\text{as }R\to\infty.
$$
Hence
$$
\int_{\Omega^{\rm is}}\mathbf{F}\cdot\overline{\nabla\varphi}dx
=-\int_{\Omega^{\rm is}}\operatorname{div} \mathbf{F}
\cdot\overline{\varphi}dx+\int_{\Sigma}(\mathbf{n}\cdot\mathbf{F})
\overline{\varphi}ds.
$$
Then, we have
\begin{align*}
\int_{\mathbb{R}^3}-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}
\cdot\overline{\nabla\varphi}) dx
&=\int_{\mathbb{R}^3}\kappa^2\operatorname{div}(\varepsilon(\rho)
 \mathbf{E}_{\rho})\cdot\overline{\varphi}dx
 +\int_{\Sigma}\kappa^2\mathbf{n}\cdot[\varepsilon(\rho)
 \mathbf{E}_{\rho}]\overline{\varphi}ds\\
&=\int_{\mathbb{R}^3}\kappa^2\operatorname{div}(\varepsilon(\rho)
\mathbf{E}_{\rho})\cdot\overline{\varphi}dx
\end{align*}
and
$$
\int_{\mathbb{R}^3}\mathbf{F}\cdot\overline{\nabla\varphi}dx
=-\int_{\mathbb{R}^3}\operatorname{div} \mathbf{F}\cdot\overline{\varphi}dx.
$$
Similarly, according to \eqref{eq:2.21} there holds
\begin{align*}
&\int_{\mathbb{R}^3}\Big(\alpha \operatorname{div}(\varepsilon(\rho)
\mathbf{E}_{\rho})\cdot\operatorname{div}(\overline{\varepsilon
 (\rho)\nabla\varphi})+\frac{\alpha}{\kappa^2}\operatorname{div}
\mathbf{F}\cdot\operatorname{div}(\overline{\varepsilon(\rho)\nabla\varphi})\\
& -\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot\overline{\nabla\varphi}
-\mathbf{F}\cdot\overline{\nabla\varphi}\Big) dx=0.
\end{align*}
Then
\begin{align*}
&\int_{\mathbb{R}^3}\Big(\alpha \operatorname{div}(\varepsilon(\rho)
\mathbf{E}_{\rho})\cdot\operatorname{div}(\overline{\varepsilon
(\rho)\nabla\varphi})+\frac{\alpha}{\kappa^2}\operatorname{div}
\mathbf{F}\cdot\operatorname{div}(\overline{\varepsilon(\rho)\nabla\varphi})\\
&+\kappa^2\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\overline{\varphi}+\operatorname{div}
\mathbf{F}\cdot\overline{\varphi}\Big) dx=0.
\end{align*}
Therefore, for all $\varphi\in\textbf{D}(\Delta_{\varepsilon(\rho)}^N)$,
\begin{equation}\label{eq:2.22}
\int_{\mathbb{R}^3}\big( \operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
+\frac{1}{\kappa^2}\operatorname{div} \mathbf{F}\big)
 \cdot(\alpha \operatorname{div}(\overline{\varepsilon(\rho)\nabla\varphi})
+\kappa^2\overline{\varphi})dx=0.
\end{equation}

The sesquilinear form associated with the operator
 $-\Delta_{\varepsilon(\rho)}^N$ is uniformly coercive on
$\mathbb{W}^{1}_0(\mathbb{R}^3)$, because (see Giroire \cite{Giroirej})
\begin{equation}\label{eq:2.23}
\operatorname{Re}\Big( \int_{\mathbb{R}^3}\varepsilon(\rho)
\nabla\varphi\cdot\overline{\nabla\varphi}dx\Big)
=\frac{1}{\mu}|\varphi|^2_{\mathbb{W}^{1}_0(\mathbb{R}^3)}
\geq C\|\varphi\|^2_{\mathbb{W}^{1}_0(\mathbb{R}^3)}.
\end{equation}

Next, we follow again Peron \cite{Peron} and examine the real
non-zero eigenvalues $\lambda$ of $-\Delta_{\varepsilon(\rho)}^N$;
i.e,
\begin{equation}\label{eq:2.24}
-\Delta_{\varepsilon(\rho)}^N\varphi=\lambda\varphi\quad\text{in }\mathbb{R}^3,
\end{equation}
which, after integration by parts, gives
$$
\int_{\mathbb{R}^3}\varepsilon(\rho)\nabla\varphi\cdot
\overline{\nabla\varphi}dx=\lambda\int_{\mathbb{R}^3}
\varphi\cdot\overline{\varphi}dx.
$$
Now \eqref{eq:2.23} gives $\lambda\geq C$ and we take $\alpha>0$
large enough such that $\frac{\kappa^2}{\alpha}<C$. Then
$\frac{\kappa^2}{\alpha}$ is not an eigenvalue of
$-\Delta_{\varepsilon(\rho)}^N$. Consequently  \eqref{eq:2.22} implies
\begin{equation*}
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
+\frac{1}{\kappa^2}\operatorname{div} \mathbf{F}=0,\quad\text{in }\mathbb{R}^3.
\end{equation*}
This way, from \eqref{eq:2.18} and \eqref{eq:2.19},
\begin{equation*}
\int_{\mathbb{R}^3}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl}
\overline{\mathbf{E}'_{\rho}}-\kappa^2\varepsilon(\rho)
\mathbf{E}_{\rho}\cdot\overline{\mathbf{E}'_{\rho}}\Big) dx
=\int_{\mathbb{R}^3}\mathbf{F}\cdot\overline{\mathbf{E}_{\rho}'} dx.
\end{equation*}
for all $\mathbf{E}_{\rho}'\in\mathbb{X}_T(\mathbb{R}^3,\rho)$.

We define $\mathbf{H}_{\rho}$ from Faraday's law by
$\mathbf{H}_{\rho}=\frac{1}{i\omega\varepsilon_0}\operatorname{curl}
\mathbf{E}_{\rho}$ in $\mathbb{R}^3$. Then from Proposition 2, it follows that
\begin{equation*}
\int_{\mathbb{R}^3}\Big( \frac{i\omega\varepsilon_0}{\mu}\operatorname{curl}
\mathbf{H}_{\rho}\cdot\overline{\mathbf{E}'_{\rho}}-\kappa^2\varepsilon(\rho)
\mathbf{E}_{\rho}\cdot\overline{\mathbf{E}'_{\rho}}\Big) dx
=\int_{\mathbb{R}^3}\mathbf{F}\cdot\overline{\mathbf{E}_{\rho}'} dx,
\end{equation*}
for all $\mathbf{E}_{\rho}'\in\mathbb{X}_T(\mathbb{R}^3,\rho)$, implying
$$
\operatorname{curl} \mathbf{H}_{\rho}
+i\kappa\varepsilon(\rho)\mathbf{E}_{\rho}=\frac{1}{i\kappa}\mathbf{F},
\quad\text{in }\mathbb{R}^3.
$$
\end{proof}

\begin{remark}\label{th:2.24} \rm
An easy modification of the proof of the Theorem 1 shows that there
exists $\beta>0$, independent of $\rho$, such that if
$\mathbf{E}_{\rho}\in\mathbb{X}_N(\mathbb{R}^3,\rho)$ is a solution to
\begin{equation}\label{eq:2.25}
\int_{\mathbb{R}^3}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\mathbf{E}'_{\rho}}
+\beta \operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\operatorname{div}(\overline{\varepsilon(\rho)\mathbf{E}'_{\rho}})
-\kappa^2\varepsilon(\rho)\mathbf{E}_{\rho}\cdot\overline{\mathbf{E}'_{\rho}}
\Big) dx=\langle f,\mathbf{E}_{\rho}'\rangle,
\end{equation}
and \eqref{eq:2.19} for some $\mathbf{F}\in\mathbf{W}(\operatorname{div},
\mathbb{R}^3)$, then
\begin{equation}\label{eq:2.26}
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
+\frac{1}{\kappa^2}\operatorname{div} \mathbf{F}=0,\quad\text{in }\mathbb{R}^3.
\end{equation}
Furthermore, $\mathbf{E}_{\rho}$ and
$\mathbf{H}_{\rho}=\frac{1}{i\omega\varepsilon_0}\operatorname{curl}
\mathbf{E}_{\rho}$ solve \eqref{eq:2.4}. This results corresponds directly
to \cite[Theorem 2.22]{Peron}.
\end{remark}

In this part, we give a variational formulation for the term
$\varphi_{\rho}\in\mathcal{V}$, with
$\mathcal{V}=H^{1}_0(\Omega_{-})\cup\mathbb{W}^{1}_0(\Omega_{+})$,
 (see \cite[Chapter 2]{Ospino} and \cite{Ospino1}), which appears
in the decomposition of the electrical field,
to see  Theorem \ref{th:2.3.4.1}. Again, we extend the ideas of Peron
\cite{Peron} to prove Lemmas \ref{lem:2.34} and \ref{lem:2.35}
for the unbounded exterior domain.
Our Lemma \ref{lem:2.34} corresponds to \cite[Lemma 2.33]{Peron}
and gives the appropriate setting for an unbounded exterior domain.

\begin{lemma}\label{lem:2.34}
Let $\mathbf{E}_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$ satisfy
\eqref{eq:2.18}-\eqref{eq:2.19} for
$\mathbf{F}\in\mathbf{W}_0(\operatorname{div},\mathbb{R}^3)$,
and let $(\mathbf{w}_{\rho},\varphi_{\rho})\in
\mathcal{\mathbf{W}}^{1}_0(\mathbb{R}^3)\times\mathcal{V}$
with $\operatorname{div}\mathbf{w}_{\rho}=0$ given by Theorem \ref{th:2.3.4.1}.
Then, $\varphi_{\rho}$ solves the variational problem:
Find $\varphi_{\rho}\in\mathcal{V}$, such that for all $\psi\in\mathcal{V}$,
\begin{equation}\label{eq:2.47}
\int_{\mathbb{R}^3}\varepsilon(\rho)\nabla\varphi_{\rho}\cdot
\overline{\nabla\psi}dx=\frac{1}{\kappa^2}\int_{\mathbb{R}^3}
\operatorname{div} \mathbf{F}\cdot\overline{\psi}dx+\frac{1}{\mu}
i\rho^2\int_{\Sigma}\mathbf{w}_{\rho}\cdot\mathbf{n}|_{\Sigma}
\overline{\psi}ds.
\end{equation}
\end{lemma}

\begin{proof}
Due to Theorem \ref{th:2.3.4.1} there exists an unique couple
 $(\mathbf{w}_{\rho},\varphi_{\rho})\in\mathbb{W}_0^{1}(\mathbb{R}^3)
\times\mathcal{V}$ such that
 $\mathbf{E}_{\rho}=\mathbf{w}_{\rho}+\nabla\varphi_{\rho}$. Thus  we have
\begin{equation*}
\int_{\mathbb{R}^3}\varepsilon(\rho)\nabla\varphi_{\rho}\cdot
\overline{\nabla\psi}dx=\int_{\mathbb{R}^3}\varepsilon(\rho)
\mathbf{E}_{\rho}\cdot\overline{\nabla\psi}dx-\int_{\mathbb{R}^3}
\varepsilon(\rho)\mathbf{w}_{\rho}\cdot\overline{\nabla\psi}dx,
\quad \forall\psi\in\mathcal{V}.
\end{equation*}
Then, since $\varepsilon(\rho)\mathbf{E}_{\rho}\in\mathbf{W}_0(\operatorname{div},
\mathbb{R}^3)$, there holds
\begin{equation*}
\int_{\mathbb{R}^3}\varepsilon(\rho)\mathbf{E}_{\rho}\cdot
\overline{\nabla\psi}dx=-\int_{\mathbb{R}^3}\operatorname{div}
(\varepsilon(\rho)\mathbf{E}_{\rho})\cdot\overline{\psi}dx,
\end{equation*}
so, due to Theorem \ref{th:2.23},
\begin{equation*}
\int_{\mathbb{R}^3}\varepsilon(\rho)\mathbf{E}_{\rho}
\cdot\overline{\nabla\psi}dx=\frac{1}{\kappa^2}
\int_{\mathbb{R}^3}\operatorname{div} \mathbf{F}\cdot\overline{\psi}dx.
\end{equation*}
Next, we have
$$
\int_{\mathbb{R}^3}\varepsilon(\rho)\mathbf{w}_{\rho}\cdot\overline{\nabla\psi} dx
=\int_{\Omega^{\rm is}}\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho}\cdot
\overline{\nabla\psi} dx+\int_{\Omega^{\rm cd}}\varepsilon(\rho)^{\rm cd}
\mathbf{w}_{\rho}\cdot\overline{\nabla\psi} dx,
$$
and, by integration by parts,
$$
\int_{\Omega^{\rm cd}}\varepsilon(\rho)^{\rm cd}\mathbf{w}_{\rho}
\cdot\overline{\nabla\psi} dx=\int_{\Omega^{\rm cd}}\operatorname{div}
(\varepsilon(\rho)^{\rm cd}\mathbf{w}_{\rho})\overline{\psi} dx
-\int_{\Sigma}(\varepsilon(\rho)^{\rm cd}\mathbf{w}_{\rho}\cdot\mathbf{n})
\overline{\psi} ds.
$$
Let $B_{R}$ be a ball with radius $R>0$ containing $\Omega^{\rm cd}$. We have
\begin{align*}
\int_{\Omega_{R}}\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho}
\cdot\overline{\nabla\psi} dx
&=\int_{\Omega_{R}}\operatorname{div}
(\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho})\overline{\psi} dx
+\int_{\Sigma}(\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho}\cdot\mathbf{n})
\overline{\psi} ds\\
&\quad +\int_{\partial B_{R}}(\varepsilon(\rho)^{\rm is}
\mathbf{w}_{\rho}\cdot\mathbf{n})\overline{\psi} ds.
\end{align*}
Applying the Silver-M\"{u}ller condition,
$$
\int_{\partial B_{R}}(\varepsilon(\rho)^{\rm is}
\mathbf{w}_{\rho}\cdot\mathbf{n})\overline{\psi} ds
=\int_{\partial B_{R}}(\varepsilon(\rho)^{\rm is}[\mathbf{w}_{\rho}
-\mathbf{w}_{\rho}\times\mathbf{n}]\cdot\mathbf{n})\overline{\psi} ds\to 0
\quad\text{as } R\to\infty,
$$
Hence
$$
\int_{\Omega^{\rm is}}\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho}\cdot
\overline{\nabla\psi} dx=\int_{\Omega^{\rm is}}\operatorname{div}
(\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho})\overline{\psi} dx
+\int_{\Sigma}(\varepsilon(\rho)^{\rm is}\mathbf{w}_{\rho}\cdot\mathbf{n})
\overline{\psi} ds.
$$
Thus
$$
\int_{\mathbb{R}^3}\varepsilon(\rho)\mathbf{w}_{\rho}\cdot\overline{\nabla\psi} dx
=\int_{\mathbb{R}^3}\operatorname{div}(\varepsilon(\rho)\mathbf{w}_{\rho})
\overline{\psi} dx+\int_{\Sigma}(\varepsilon(\rho)^{\rm is}
-\varepsilon(\rho)^{\rm cd})\mathbf{w}_{\rho}\cdot\mathbf{n}|_{\Sigma}
\overline{\psi} ds.
$$
Since $\operatorname{div} \mathbf{w}_{\rho}=0$ in $\mathbb{R}^3$,
we obtain \eqref{eq:2.47} because (see \eqref{eq:2.4})
$$\varepsilon(\rho)^{\rm is}-\varepsilon(\rho)^{\rm cd}
=-\frac{1}{\mu}i\rho^2.
$$
\end{proof}

Analogously we obtain the following counterpart of \cite[Lemma 2.34]{Peron}.

\begin{lemma}\label{lem:2.35}
Let $\mathbf{E}_{\rho}\in\mathbb{X}_N(\mathbb{R}^3,\rho)$ solution of
\eqref{eq:2.25}-\eqref{eq:2.19} associated with
$\mathbf{F}\in\mathbf{W}(\operatorname{div},\mathbb{R}^3)$, and
let $(\mathbf{w}_{\rho},\varphi_{\rho})\in\mathbb{X}_N(\mathbb{R}^3)
\times\mathcal{V}$ given by Theorem \ref{th:2.3.4.1}. Then,
 $\varphi_{\rho}$ is solution of following variational problem:
Find $\varphi\in\mathcal{V}$, such that for all $\psi\in\mathcal{V}$,
\begin{equation}\label{eq:2.48}
\int_{\mathbb{R}^3}\varepsilon(\rho)\nabla\varphi\cdot
\overline{\nabla\psi}dx=\frac{1}{\kappa^2}
\int_{\mathbb{R}^3}\operatorname{div} \mathbf{F}
\cdot\overline{\psi}dx+\frac{1}{\mu}i\rho^2\int_{\Sigma}
\mathbf{w}_{\rho}\cdot\mathbf{n}|_{\Sigma}\overline{\psi}ds.
\end{equation}
\end{lemma}

\section{Decomposition of vector fields and compact embedding in weighted spaces}
\label{sec:el3}

In this section we collect the tools needed in the proof of our a priori
estimate (Theorem \ref{th:2.26}), namely a vector Helmholtz decomposition
in $\mathbb{R}^3$ and a compactness results (Lemma \ref{lemnew}) for the embedding
in weighted spaces.

First we consider the vector potential of divergence-free vector
fields and present results for a Helmholtz decomposition by
 Girault \cite{Girault}. The weighted Sobolev spaces used here were
introduced and studied by Hanouzet in \cite{Hanouzet}.
For any multi-index $\alpha$ in $\mathbb{N}^3$, we denote by
$\partial^{\alpha}$ the differential operator of order $\alpha$:
$$
\partial^{\alpha}=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}
\partial x_2^{\alpha_2}\partial x_3^{\alpha_3}},\quad
\text{with } |\alpha|=\alpha_1+\alpha_2+\alpha_3.
$$
Then, for all $m$ in $\mathbb{N}$ and all $k$ in $\mathbb{Z}$, we
define the weighted Sobolev space
\begin{equation}\label{eq:2.3.2.0}
\mathbb{W}_{k}^{m}(\Omega^{\rm is})
:=\big\{v\in\mathfrak{D}'(\Omega^{\rm is}):
\forall\alpha\in\mathbb{N}^3,\; 0\leq|\alpha|\leq m,\;
 \ell(r)^{|\alpha|-m+k}\partial^{\alpha}v\in
 L^2(\Omega^{\rm is})\big\},
\end{equation}
which is a Hilbert space with the norm:
$$
\|v\|_{\mathbb{W}_{k}^{m}(\Omega^{\rm is})}
=\Big\{\sum_{|\alpha|=0}^{m}\|\ell(r)^{|\alpha|-m+k}\partial^{\alpha}v\|_{L^2
(\Omega^{\rm is})}^2\Big\}^{1/2}.
$$
Hence
$$
\mathbb{W}_0^{0}(\Omega^{\rm is})=L^2(\Omega^{\rm is}),\quad\
\mathbb{W}_{-1}^{0}(\mathbb{R}^3)=L^2_{0,-1}(\mathbb{R}^3).
$$
For all $n\in\mathbb{Z}$, $\mathbf{P}_{n}$ denotes the space of all
polynomials (in three variables) of degree at most $n$, with the convention
that the space is reduced to zero when $n$ is negative.

We denote by $\mathcal{P}_{n}$ is the subspace of all harmonic polynomials
of $\mathbf{P}_{n}$, again with the convention that the space is reduced
to zero when $n$ is negative.
For all integers $k\geq 0$, we define the following subspace of
$(\mathcal{P}_{k})^3$,
$$
\mathcal{G}_{k}:=\{\nabla q:q\in\mathcal{P}_{k+1}\}.
$$
Note that $\mathcal{G}_0=\mathbb{R}^3$.
The following result is based on the paper by Girault
\cite{Girault}. In the case of a bounded domain, there are two
classical orthogonal decompositions of vector fields: a
decomposition in $\mathbf{L}^2$ and a decomposition in
$H_0^{1}$ (cf. for example \cite{GiraultR}). The following
theorem establishes the analogue of the decomposition in
$\mathbf{L}^2$ for vector fields in $\mathbb{R}^3$.
Let
\begin{gather*}
\mathbf{V}_{k}^{m}(\mathbb{R}^3):=\{ \mathbf{v}\in\mathbb{W}_{k}^{m}
(\mathbb{R}^3)^3 :\operatorname{div} \mathbf{v}=0\} , \\
\mathcal{C}_{k}:=\{\operatorname{curl} \mathbf{q}:\mathbf{q}
\in(\mathcal{P}_{k+1})^3\},
\end{gather*}
with the usual convention that $\mathcal{C}_{k}=\{0\}$, when $k<0$,
 observe that $\mathcal{C}_0=\mathbb{R}^3=\mathcal{G}_0$.
In addition, for all $k\geq 1$,
$\mathcal{G}_{k}\subset\mathcal{C}_{k}$, but the inverse inclusion
is false.

\begin{theorem}[{Girault \cite[Theorem 5.1]{Girault}}]   \label{th:2.3.4.1}
Let the integers $m$ and
$k$ belong to $\mathbb{Z}$ and let $\mathbf{u}$ be a vector field
in $\mathbb{W}_{m+k}^{m}(\mathbb{R}^3)^3$.

(1) If $k\leq 1$, $\mathbf{u}$ may be decomposed as
\begin{equation}\label{eq:2.3.4.1}
\mathbf{u}=\nabla p+\operatorname{curl} \Phi,
\end{equation}
where $\Phi$ is unique in
$\mathbf{V}_{m+k}^{m+1}(\mathbb{R}^3)/\mathcal{C}_{-k-1}$ and
$p$ is uniquely determined by $\mathbf{u}$ and $\Phi$ in
$\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)/\mathbb{R}$, or
$\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)$ if $k=0$ or $1$. They
satisfy the bounds:
\begin{equation}\label{eq:2.3.4.2}
\|\Phi\|_{\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)^3/\mathcal{C}_{-k-1}}
+\|p\|_{\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)/\mathbb{R}}\leq
C\|\mathbf{u}\|_{\mathbb{W}_{m+k}^{m}(\mathbb{R}^3)^3},
\end{equation}
with the convention that the quotient norm of $p$ is replaced by
$\|p\|_{\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)}$ when $k=0$ or
$1$.

(2) If $k\geq 2$ has the decomposition \eqref{eq:2.3.4.1}
with a unique $p$ in $\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)$ and
a unique $\Phi$ in $\mathbf{V}_{m+k}^{m+1}(\mathbb{R}^3)$ if and
only if $\mathbf{u}$ is orthogonal to $\mathcal{C}_{k-2}$ (for the
duality paring). The analogue of \eqref{eq:2.3.4.2} holds:
\begin{equation}\label{eq:2.3.4.3}
\|\Phi\|_{\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)^3}+\|p\|_{\mathbb{W}_{m+k}^{m+1}(\mathbb{R}^3)}\leq
C\|\mathbf{u}\|_{\mathbb{W}_{m+k}^{m}(\mathbb{R}^3)^3},
\end{equation}

(3) When both $m$ and $k$ belong to $\mathbb{N}$, the
decomposition is orthogonal for the scalar product of
$\mathbf{L}^2(\mathbb{R}^3)$.
\end{theorem}

Now, this part is concerned with compact embedding in weighted Sobolev
spaces for unbounded domains, and is based on Avantaggiati and Troisi
\cite{Avantaggiati}.
Let $\Omega$ be an unbounded domain of $\mathbb{R}^{n}$, satisfying
the cone property, and $\delta\in C^{0}(\overline{\Omega})$,
a positive continuous function divergent for
$|\mathbf{x}|\to\infty$, satisfying also:
\begin{enumerate}
\item There exist two open and separated subsets $\Omega_1$ and
$\Omega_2$ of $\mathbb{R}^{n}$, such that\\
$\overline{\Omega}=\overline{\Omega_1}\cup\overline{\Omega_2}$
and
$$
\delta(\mathbf{x})\leq 1, \quad \forall\mathbf{x}\in\Omega_1, \;
\delta(\mathbf{x})\geq 1,\; \forall\mathbf{x}\in\Omega_2.
$$
We will put also, $\Omega_0=\Omega$.

\item For each $\mathbf{x}_0\in\Omega_{i}$, $i=0,1,2$, let
$$
A_{i}(\mathbf{x}_0)=\Omega_{i}\cap\{ \mathbf{x}:|\mathbf{x}
-\mathbf{x}_0|<\delta(\mathbf{x}_0)\}.
$$
We assume that there $c_1$ and $c_2$ are two positive constants
independent of $\mathbf{x}_0$ and $\mathbf{x}$, and
$$
c_1\delta(\mathbf{x}_0)\leq\delta(\mathbf{x})
\leq c_2\delta(\mathbf{x}_0),\quad  \forall\mathbf{x}\in A_{i}(\mathbf{x}_0),
$$

\item If $\varphi_{i}(\mathbf{x},\mathbf{x}_0)$ is the characteristic
function of the set $A_{i}(\mathbf{x}_0)$, then the inequalities
$$
c_3\delta^{n}(\mathbf{x})\leq\int_{\Omega_{i}}\varphi_{i}(\mathbf{x},
\mathbf{x}_0)d\mathbf{x}_0\leq c_{4}\delta^{n}(\mathbf{x}), \quad
\forall\mathbf{x}\in A_{i}(\mathbf{x}_0),
$$
hold, where $c_3$ and $c_{4}$ are two positive constants
independent of $\mathbf{x}$.
\end{enumerate}
If $s,\lambda\in\mathbb{R}$ and $0<p\leq\infty$, we will denote by
$\widetilde{L}^{p}_{s,\lambda}(\Omega)$ the space of the functions
$u(\mathbf{x})$, such that
$\delta^{s}\big(\frac{\delta}{1+\delta^2}\big) ^{\lambda}u\in L^{p}(\Omega)$,
with norm
\begin{equation}\label{n1}
\|u\|_{\widetilde{L}^{p}_{s,\lambda}(\Omega)}:=\|\delta^{s}\big(
\frac{\delta}{1+\delta^2}\big)
^{\lambda}u\|_{L^{p}(\Omega)}.
\end{equation}
If $s,\lambda\in\mathbb{R}$, $r\in\mathbb{N}_0$ and
$p\in(1,\infty)$, we will denote by $W^{r,p}_{s,\lambda}(\Omega)$
the space of the distributions $u$ on $\Omega$, such that
 $\partial^{\alpha}u\in \widetilde{L}^{p}_{s+|\alpha|-r,\lambda}(\Omega)$
for $|\alpha|\leq r$, with norm
\begin{equation}\label{n2}
\|u\|_{W^{r,p}_{s,\lambda}(\Omega)}:=\Big[
\sum_{k=0}^{r}\|\partial^{k}u\|^{p}_{\widetilde{L}^{p}_{s+|\alpha|-r,\lambda}
(\Omega)}\Big]^{1/p}.
\end{equation}
We observe that $W^{r,p}_{s,\lambda}(\Omega)$ is continuously embedded in
\begin{equation}\label{eq:2.5.1.1}
W^{k,p}_{s+k-r+t,\lambda+\tau}(\Omega),\quad\text{for } k\leq r,\tau\geq 0,\;
t\in[-\tau,\tau].
\end{equation}
Therefore,
$$
W^{0,p}_{s,\lambda}(\Omega)=\widetilde{L}^{p}_{s,\lambda}(\Omega).
$$
We have also
$L^2_{0,-1}(\Omega)=\widetilde{L}^2_{-1,1}(\Omega)$.

The next theorem is due to Avantaggiati and Troisi \cite[Theorem 6.1]{Avantaggiati}.

\begin{theorem}\label{teo:2.5.1}
There are real numbers $s,\lambda,r,p$, where $r\in\mathbb{Z}_{+}$ and
$p>1$, such that for each non-negative integer $k<r$, for each real number
$\tau>0$, and for each $t\in(-\tau,\tau)$, the injection
\begin{equation}\label{eq:2.5.1.2}
W^{r,p}_{s,\lambda}(\Omega)\hookrightarrow
W^{k,p}_{s+k-r+t,\lambda+\tau}(\Omega)
\end{equation}
is compact.
\end{theorem}

As a consequence of the above results, we have the following Lemma.

\begin{lemma}\label{lemnew}
The emmbeding of $\mathbf{PH}^{1}(\mathbb{R}^3)$ into
$\mathbf{L}^2_{0,-1}(\mathbb{R}^3)$ is compact.
\end{lemma}

\begin{proof}
First, we observe that by definition
$\widetilde{L}^2_{-1,1}(\Omega)=L^2_{0,-1}(\Omega)=W^{0,2}_{-1,1}(\Omega)$.
 On the other hand choosing $t=s=\lambda=k=0$, $\tau=r=1$, $p=2$
in \eqref{eq:2.5.1.2} gives the compact embedding
$W^{1,2}_{0,0}(\Omega)\subset\subset W^{0,2}_{-1,1}(\Omega)$.
Hence $W^{1,2}_{0,0}(\Omega)\subset\subset L^2_{0,-1}(\Omega)$
where we can set $\Omega=\mathbb{R}^3$.

Furthermore $\varphi\in PH^{1}(\mathbb{R}^3):=\{
\varphi=(\varphi^{\rm is},\varphi^{\rm cd}):
\varphi^{\rm is}\in\mathbb{W}^{1}_0(\Omega^{\rm is}), \varphi^{\rm cd}\in
H^{1}(\Omega^{\rm cd}) \}$, due to the definition of
$\mathbb{W}^{1}_0$, gives  that $\nabla\varphi\in L^2$ and hence
$\nabla\varphi\in\widetilde{L}^2_{0,0}$ with $s=\lambda=0$ in
\eqref{n1}.
Therefore, $\varphi\in W^{1,2}_{0,0}(\Omega)$ with
$r=1$, $p=2$, $s=\lambda=0$ in  \eqref{n2} because with
$s=\lambda=0=\lvert\alpha\rvert$, $r=1$ there holds
$$
\lVert\varphi\rVert_{\widetilde{L}^2_{-1,0}}(\Omega)
=\lVert\delta^{-1}\varphi\rVert_{L^2(\Omega)}
\leq c\lVert\frac{\varphi}{\sqrt{1+x^2}}\rVert_{L^2(\Omega)}<\infty
$$
by taking $\delta$ proportional to $\sqrt{1+x^2}$.
\end{proof}

\section{A priori estimate for the electrical field}
\label{sec:el2}

Next we give an existence and uniqueness result for the
solution of \eqref{eq:2.18}-\eqref{eq:2.19}.
The proof uses an a priori estimate. The ideas of this section are
 based on those of Peron \cite{Caloz,Peron}, but using compactness
 results for the embedding of weighted spaces with unbounded domains.
This is a crucial difference of our proof compared to Peron's proof.
An alternative proof may be obtained using \cite[Theorem 2.1]{Hiptmair}.

For the rest of this article we assume the following condition.

\noindent\textbf{Spectral hypothesis}:
We assume $\kappa^2$ is not an eigenvalue of the limit problem.
That is, we assume that if $\mathbf{E}_0\in\mathbf{W}(\operatorname{curl},
\Omega^{\rm is})$ is such that for all
$\mathbf{E}'\in\mathbf{W}(\operatorname{curl},\Omega^{\rm is})$,
\begin{equation}\label{eq:2.30}
\int_{\Omega^{\rm is}}(\operatorname{curl} \mathbf{E}_0\cdot\operatorname{curl}
\overline{\mathbf{E}'}-\kappa^2\mathbf{E}_0\cdot\overline{\mathbf{E}'})dx=0,
\quad \mathbf{n}\times\mathbf{E}=0 \quad\text{on } \Sigma,
\end{equation}
then $\mathbf{E}_0=0$.

Now, we can formulate our main theorem of this section.

\begin{theorem}\label{th:2.26}
Under the spectral hypothesis \eqref{eq:2.30}, there exists a constant
 $\rho_0>0$, such that for all $\rho>\rho_0$, problem
\eqref{eq:2.18}-\eqref{eq:2.19} admits an unique solution
$\mathbf{E}_{\rho}\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$ for
$\mathbf{F}\in\mathbf{W}_0(\operatorname{div},\mathbb{R}^3)$, satisfying
\begin{equation}\label{eq:2.31}
\begin{aligned}
&\|\operatorname{curl} \mathbf{E}_{\rho}\|_{\mathbf{L}^2_{0,-1}
(\mathbb{R}^3)}+\|\operatorname{div}(\varepsilon(\rho)
\mathbf{E}_{\rho})\|_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}
+\|\mathbf{E}_{\rho}\|_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}
+\rho\|\mathbf{E}_{\rho}\|_{\mathbf{L}^2(\Omega^{\rm cd})}\\
&\leq C\|\mathbf{F}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)},
\end{aligned}
\end{equation}
with a constant $C>0$, independent of $\rho$.
\end{theorem}

The proof of the above theorem  is given in various steps, below.
The estimate \eqref{eq:2.31} is based on the a priori estimate
\eqref{eq:2.49}.

\begin{theorem}\label{th:2.36}
 If \eqref{eq:2.30} holds, then there exists a constant $\rho_0>0$,
such that, for all $\rho>\rho_0$,
if $\mathbf{E}_{\rho}\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$ satisfies
\eqref{eq:2.18}-\eqref{eq:2.19} for
$\mathbf{F}\in\mathbf{W}_0(\operatorname{div},\mathbb{R}^3)$, then
\begin{equation}\label{eq:2.49}
\|\mathbf{E}_{\rho}\|_{\mathbf{L}^2_{0,-1}
(\mathbb{R}^3)}\leq C\|\mathbf{F}\|_{\mathbf{W}
(\operatorname{div},\mathbb{R}^3)},
\end{equation}
where $C>0$ is a constant independent of $\rho$.
\end{theorem}

\begin{proof}
The proof is similar to the one given by Peron \cite[Theorem 2.35]{Peron}.
Here we use a compact embedding of $\mathbf{PH}^{1}(\mathbb{R}^3)$ into
$\mathbf{L}^2_{0,-1}(\mathbb{R}^3)$ where
$$
\mathbf{PH}^{1}(\mathbb{R}^3)=\{\varphi : \varphi^{\rm is}
\in(\mathbb{W}^{1}_0(\Omega^{\rm is}))^3, \;
\varphi^{\rm cd}\in (H^{1}(\Omega^{\rm cd}))^3\}.
$$
Let $\mathbf{E}_{\rho}\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$,
be a solution of \eqref{eq:2.18}-\eqref{eq:2.19}.
For $\Phi\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$,
\begin{equation}\label{eq:2.51}
\begin{aligned}
&\int_{\mathbb{R}^3}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\Phi}
+\alpha \operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\operatorname{div}(\overline{\varepsilon(\rho)\Phi})
-\frac{\kappa^2}{\mu}\mathbf{E}_{\rho}\overline{\Phi}\Big) dx\\
&-\frac{1}{\mu}i\rho^2\int_{\Omega^{\rm cd}}\mathbf{E}_{\rho}
\overline{\Phi}dx\\
&=\int_{\mathbb{R}^3}\Big( \mathbf{F}\cdot\overline{\Phi}
-\frac{\alpha}{\kappa^2}\operatorname{div}
\mathbf{F}\cdot\operatorname{div}(\overline{\varepsilon(\rho)\Phi})\Big) dx.
\end{aligned}
\end{equation}
By Theorem \ref{th:2.23} there holds
$$
\operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
+\frac{1}{\kappa^2}\operatorname{div} \mathbf{F}=0,\quad\text{in }\mathbb{R}^3.
$$
For $\Phi\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$,
\begin{equation}\label{eq:2.52}
\int_{\mathbb{R}^3}(\operatorname{curl} \mathbf{E}_{\rho}\cdot\operatorname{curl}
\overline{\Phi}-\kappa^2\mathbf{E}_{\rho}\overline{\Phi})dx
-i\rho^2\int_{\Omega^{\rm cd}}\mathbf{E}_{\rho}\overline{\Phi}dx
=\mu\int_{\mathbb{R}^3}\mathbf{F}\cdot\overline{\Phi}dx,
\end{equation}
for all $\Phi\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$.
Just as in Peron \cite{Peron} we prove the theorem by contradiction argument,
but we crucially apply a compactness result for the embedding in
weighted Sobolev spaces by Avantaggiati and Troisi \cite{Avantaggiati}.

Since Peron \cite{Peron} considers only bounded domains, he can, in contrary,
apply standard embedding arguments (Rellich's theorem).
Suppose that exists a sequence $\{\mathbf{F}_{\rho_{n}}\}_{n\geq 1}$ in
$\mathbf{W}(\operatorname{div},\mathbb{R}^3)$ with $\rho_{n}\to\infty$,
 $\|\mathbf{F}_{\rho_{n}}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)}=1$,
$\mathbf{F}_{\rho_{n}}\cdot\mathbf{n}=0$ in $\Sigma$, and such that for the
 corresponding solutions $\mathbf{E}_{\rho_{n}}\in\mathbb{X}_{TN}(\mathbb{R}^3,
\rho_{n})$ satisfy
$$
\lim_{n\to\infty}\|\mathbf{E}_{\rho_{n}}\|_{\mathbb{X}_{TN}
(\mathbb{R}^3,\rho_{n})}=\infty.
$$
Letting $\widetilde{\mathbf{E}}_{\rho_{n}}
=(\|\mathbf{E}_{\rho_{n}}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)})^{-1}
\mathbf{E}_{\rho_{n}}$ we have
\begin{equation}\label{eq:2.53}
\|\widetilde{\mathbf{E}}_{\rho_{n}}\|_{\mathbf{W}(\operatorname{div},
\mathbb{R}^3)}=1,\quad
\lim_{n\to\infty}\|\widetilde{\mathbf{F}}_{\rho_{n}}\|_{\mathbf{W}
(\operatorname{div},\mathbb{R}^3)}=0.
\end{equation}
With $\Phi=\widetilde{\mathbf{E}}_{\rho_{n}}$, equality \eqref{eq:2.52} becomes
\begin{equation}\label{eq:2.54}
\|\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}}\|^2_{\mathbf{L}^2
(\mathbb{R}^3)}-\kappa^2\|\widetilde{\mathbf{E}}_{\rho_{n}}
\|^2_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}-i\rho^2_{n}
\|\widetilde{\mathbf{E}}_{\rho_{n}}\|^2_{\mathbf{L}^2(\Omega^{\rm cd})}
=\mu(\widetilde{\mathbf{F}}_{\rho_{n}},\widetilde{\mathbf{E}}_{\rho_{n}}
)_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}.
\end{equation}
Taking imaginary parts we have
\begin{equation}\label{eq:2.55}
\rho^2_{n}\|\widetilde{\mathbf{E}}_{\rho_{n}}\|^2_{\mathbf{L}^2(\Omega^{\rm cd})}
=-\mu\operatorname{Im}(\widetilde{\mathbf{F}}_{\rho_{n}},
\widetilde{\mathbf{E}}_{\rho_{n}})_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}.
\end{equation}
By the Cauchy-Schwartz inequality we obtain,
$$
|\operatorname{Im}(\widetilde{\mathbf{F}}_{\rho_{n}},
\widetilde{\mathbf{E}}_{\rho_{n}})_{\mathbf{L}^2_{0,-1}
(\mathbb{R}^3)}|\leq\|\widetilde{\mathbf{F}}_{\rho_{n}}\|_{\mathbf{W}
(\operatorname{div},\mathbb{R}^3)}\|\widetilde{\mathbf{E}}_{\rho_{n}}
\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)}.
$$
Hence \eqref{eq:2.53} yields
\begin{equation}\label{eq:2.56}
\lim_{n\to\infty}\|\widetilde{\mathbf{E}}_{\rho_{n}}\|_{\mathbf{L}^2
(\Omega^{\rm cd})}=0.
\end{equation}
Also, taking real parts in \eqref{eq:2.54},
\begin{equation}\label{eq:2.57}
\|\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}}
\|^2_{\mathbf{L}^2(\mathbb{R}^3)}-\kappa^2\|\widetilde{\mathbf{E}}_{\rho_{n}}
\|^2_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}
=\mu\operatorname{Re}(\widetilde{\mathbf{F}}_{\rho_{n}},
\widetilde{\mathbf{E}}_{\rho_{n}})_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}.
\end{equation}
Hence due to Cauchy-Schwartz inequality and \eqref{eq:2.53}, there are
constants $C_1$ and $C_2$ independent of $n$, such that
\begin{equation}\label{eq:2.58}
\|\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}}\|^2_{\mathbf{L}^2
(\mathbb{R}^3)}\leq C_1+C_2\|\widetilde{\mathbf{F}}_{\rho_{n}}
\|_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}.
\end{equation}
Therefore, $\{\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}}\}_{n\geq 1}$
is bounded in $\mathbf{L}^2_{0,-1}(\mathbb{R}^3)$.

Let $(\mathbf{w}_{\rho_{n}},\varphi_{\rho_{n}})\in\mathbb{W}_0^{1}
(\mathbb{R}^3)\times\mathcal{V}$, (for definition of $\mathcal{V}$
see \cite[Chapter 2]{Ospino} and \cite{Ospino1}), be given by Girault
\cite[Theorems 3.2 and 5.1]{Girault}, such that
$$
\widetilde{\mathbf{E}}_{\rho_{n}}=\widetilde{\mathbf{w}}_{\rho_{n}}
+\nabla\widetilde{\varphi}_{\rho_{n}},\quad
\operatorname{div} \widetilde{\mathbf{w}}_{\rho_{n}}=0,\quad\text{in }
\mathbb{R}^3,
$$
and
\begin{equation}\label{eq:2.59}
\|\widetilde{\mathbf{w}}_{\rho_{n}}\|_{\mathbb{W}^{1}_0(\mathbb{R}^3)}
\leq C\|\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}}\|_{\mathbf{L}^2
(\mathbb{R}^3)},
\end{equation}
where $C>0$ is a constant independent of $n$. Therefore,
$\{\widetilde{\mathbf{w}}_{\rho_{n}}\}_{n\in\mathbb{N}}$ is bounded in
$\mathbb{W}_0^{1}(\mathbb{R}^3)$.
According to Lemma \ref{lem:2.34} and \eqref{eq:2.20},
$\widetilde{\varphi}_{\rho_{n}}$ satisfies
\begin{equation}\label{eq:2.60}
\int_{\mathbb{R}^3}\varepsilon(\rho)\nabla\widetilde{\varphi}_{\rho_{n}}
\cdot\overline{\nabla\psi}dx=\frac{1}{\kappa^2}\int_{\mathbb{R}^3}
\operatorname{div} \widetilde{\mathbf{F}}_{\rho_{n}}
\cdot\overline{\psi}dx+\frac{1}{\mu}i\rho^2
\int_{\Sigma}\widetilde{\mathbf{w}}_{\rho_{n}}
\cdot\mathbf{n}|_{\Sigma}\overline{\psi}ds.
\end{equation}
for all $\psi\in\mathcal{V}$.

Let $\rho_0>0$ and the constant $C_{\rho_0}>0$ be given by
\cite[Theorem 3]{Ospino} and \cite[Teorema 1]{Ospino1}.
We set $\delta_{n}=1+i\rho^2_{n}$. Then there exists
$n_0\in\mathbb{N}$, such that for all $n\geq n_0$ we have
$|\delta_{n}|\geq\rho_0$. Note that
$\operatorname{div} \widetilde{\mathbf{F}}_{\rho_{n}}$ and
$\widetilde{\mathbf{w}}_{\rho_{n}}\cdot\mathbf{n}$ verify the
hypotheses of \cite[Theorem 3]{Ospino} and \cite[Teorema 1]{Ospino1}.
Also, problem \eqref{eq:2.60} is
coercive on $\mathcal{V}$. Hence the solution of
\eqref{eq:2.60} belongs to $PH^2(\mathbb{R}^3)$ and there holds
\begin{equation*}
\Arrowvert\widetilde{\varphi}^{\rm cd}_{\rho_{n}}
\Arrowvert_{H^2(\Omega^{\rm cd})}
+\Arrowvert\widetilde{\varphi}^{\rm is}_{\rho_{n}}
\Arrowvert_{\mathbb{W}^2_1(\Omega^{\rm is})}
\leq C_{\delta_0}\Big(\|\operatorname{div} \widetilde{\mathbf{F}}_{\rho_{n}}
\|_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}+\|\widetilde{\mathbf{w}}_{\rho_{n}}
\cdot\mathbf{n}\|_{H^{1/2}(\Sigma)} \Big).
\end{equation*}
for any $n\geq n_0$.
Thus $\{\nabla\widetilde{\varphi}_{\rho_{n}}\}_{n\geq 1}$ is bounded
in $\mathbf{PH}^{1}(\mathbb{R}^3)$, and
$\{\widetilde{\mathbf{E}}_{\rho_{n}}\}_{n\geq 1}$ is bounded in
$\mathbf{H}^{1}(\Omega^{\rm cd})\cup\left( \mathbb{W}^{1}_0(\Omega^{\rm is})
\right) ^3$.

According to Lemma \ref{lemnew}, the embedding of $\mathbf{PH}^{1}(\mathbb{R}^3)$
in $\mathbf{L}^2_{0,-1}(\mathbb{R}^3)$ is compact. This implies that there
exists a subsequence $\{\widetilde{\mathbf{E}}_{\rho_{n}}\}_{n\geq 1}$ and
$\widetilde{\mathbf{E}}\in\mathbf{L}^2_{0,-1}(\mathbb{R}^3)$, such that
\begin{equation}\label{eq:2.61}
\widetilde{\mathbf{E}}_{\rho_{n}}\rightharpoonup\widetilde{\mathbf{E}}
\quad\text{in }\left( \mathbf{PH}^{1}(\mathbb{R}^3)\right) ^3,\quad
 \widetilde{\mathbf{E}}_{\rho_{n}}\to\widetilde{\mathbf{E}}\quad\text{in }
\mathbf{L}^2_{0,-1}(\mathbb{R}^3).
\end{equation}
By \eqref{eq:2.53}, we have
\begin{equation}\label{eq:2.62}
\|\widetilde{\mathbf{E}}\|_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}=1.
\end{equation}
To obtain a contradiction, we show that
$\widetilde{\mathbf{E}}=0$ in $\Omega^{\rm is}\cup\Omega^{\rm cd}$.
Due to \eqref{eq:2.56},
$\|\widetilde{\mathbf{E}}\|_{\mathbf{L}^2(\Omega^{\rm cd})}=0$. Hence
\begin{equation}\label{eq:2.63}
\widetilde{\mathbf{E}}=0,\quad\text{in }\Omega^{\rm cd}.
\end{equation}
Next, we take $\Phi\in\mathbb{X}_{TN}(\mathbb{R}^3,\rho)$
with support in $\Omega^{\rm is}$. Then $\mathbf{n}\cdot\Phi=0$,
$\mathbf{n}\times\Phi=0$ on $\Sigma$ and due to \eqref{eq:2.52}, we have
\begin{equation}\label{eq:2.64}
(\operatorname{curl} \widetilde{\mathbf{E}}_{\rho_{n}},
\operatorname{curl} \Phi)_{\mathbf{L}^2_{0,-1}(\Omega^{\rm is})}
-\kappa^2(\widetilde{\mathbf{E}}_{\rho_{n}},\Phi)_{\mathbf{L}^2_{0,-1}
(\Omega^{\rm is})}=\mu(\widetilde{\mathbf{F}}_{\rho_{n}},\Phi)
_{\mathbf{L}^2_{0,-1}(\Omega^{\rm is})}.
\end{equation}
Letting $n\to\infty$ in \eqref{eq:2.64} and using \eqref{eq:2.61} we obtain
\begin{equation}\label{eq:2.65}
(\operatorname{curl} \widetilde{\mathbf{E}},\operatorname{curl} \Phi
)_{\mathbf{L}^2_{0,-1}(\Omega^{\rm is})}-\kappa^2(\widetilde{\mathbf{E}},
\Phi)_{\mathbf{L}^2_{0,-1}(\Omega^{\rm is})}=0.
\end{equation}
Now \eqref{eq:2.30} gives $\widetilde{\mathbf{E}}=0,\quad\text{in }
\Omega^{\rm is}$, and therefore $\widetilde{\mathbf{E}}=0$, in
$\mathbb{R}^3$, which is a contradiction to \eqref{eq:2.62}
and therefore \eqref{eq:2.49} holds.
\end{proof}

Now with the help of Theorem \ref{th:2.36} we can prove Theorem \ref{th:2.26}.

\begin{proof}[Proof of Theorem \ref{th:2.26}]
 Let $\rho_0>0$ be given by Theorem \ref{th:2.36}. Let us
 assume $\mathbf{E}_{\rho}$ satisfies \eqref{eq:2.18}-\eqref{eq:2.19}.
Then $\mathbf{E}_{\rho}$ satisfies  \eqref{eq:2.52} and
taking $\Phi=\mathbf{E}_{\rho}$ we obtain
\begin{equation}\label{eq:2.68}
\|\operatorname{curl} \mathbf{E}_{\rho}\|^2_{\mathbf{L}^2(\mathbb{R}^3)}
-\kappa^2\|\mathbf{E}_{\rho}\|^2_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}
-i\rho^2\|\mathbf{E}_{\rho}\|^2_{\mathbf{L}^2(\Omega^{\rm cd})}
=\mu(\mathbf{F},\mathbf{E}_{\rho})_{\mathbf{L}^2_{0,-1}(\mathbb{R}^3)}.
\end{equation}
Taking real and imaginary parts as in the proof of Theorem \ref{th:2.36}
we obtain the a priori estimate \eqref{eq:2.31} from \eqref{eq:2.20} and
\begin{gather}\label{eq:2.69}
\rho\|\mathbf{E}_{\rho}\|_{\mathbf{L}^2(\Omega^{\rm cd})}
\leq C_1\|\mathbf{F}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)},\\
\label{eq:2.70}
\|\operatorname{curl} \mathbf{E}_{\rho}\|_{\mathbf{L}^2(\mathbb{R}^3)}
\leq C_2\|\mathbf{F}\|_{\mathbf{W}(\operatorname{div},\mathbb{R}^3)}.
\end{gather}

Next, note that the a priori estimate \eqref{eq:2.31} implies the
injectivity of the solution operator to the variational problem \eqref{eq:2.18}.
Therefore to show existence of the solution it suffices to demonstrate
that this operator is surjective. We
introduce the sesquilinear form $c_{\rho}$ defined by
\begin{equation}\label{eq:2.72}
c_{\rho}(\mathbf{E}_{\rho},\mathbf{E}'_{\rho})
=\int_{\mathbb{R}^3}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl} \overline{\mathbf{E}'_{\rho}}
+\alpha \operatorname{div}(\varepsilon(\rho)\mathbf{E}_{\rho})
\cdot\operatorname{div}(\overline{\varepsilon(\rho)\mathbf{E}'_{\rho}})\Big) dx.
\end{equation}
for all $\mathbf{E}_{\rho},\mathbf{E}'_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$.
The bilinear form $c_{\rho}$ is coercive on
$\mathbb{X}_T(\mathbb{R}^3,\rho)$. By the Lax-Milgram Theorem there
exist a bounded linear operator $\mathbf{M}$ such that
$c_{\rho}(\mathbf{E}_{\rho},\mathbf{E}'_{\rho})
=\langle\mathbf{M}\mathbf{E}_{\rho},\mathbf{E}'_{\rho}\rangle$.
Since the embedding $I_{\rho}(\mathbf{E}_{\rho})
=\varepsilon(\rho)\mathbf{E}_{\rho}$ for
$\mathbf{E}_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$ from
$\mathbb{X}_T(\mathbb{R}^3,\rho)$ into $\mathbb{X}_T(\mathbb{R}^3,\rho)'$
is compact. Hence $\mathbf{M}-\kappa^2I_{\rho}$ is a Fredholm operator.
In particular, it is surjective if and only if his adjoint
$\mathbf{M}^{*}-\kappa^2I_{\rho}^{*}$ is injective where
 $I_{\rho}^{*}=\overline{\varepsilon(\rho)}I_{\rho}$.
Let $c_{\rho}^{*}$ be the sesquilinear form associated with the operator
$c_{\rho}$; i.e.,
\begin{equation}\label{eq:2.73}
c_{\rho}^{*}(\mathbf{E}_{\rho},\mathbf{E}'_{\rho})
=\int_{\mathbb{R}^3}\Big( \frac{1}{\mu}\operatorname{curl}
\mathbf{E}_{\rho}\cdot\operatorname{curl}
\overline{\mathbf{E}'_{\rho}}+\alpha \operatorname{div}
(\overline{\varepsilon(\rho)}\mathbf{E}_{\rho})
\cdot\operatorname{div}(\varepsilon(\rho)\overline{\mathbf{E}'_{\rho}})\Big) dx,
\end{equation}
for all $\mathbf{E}_{\rho},\mathbf{E}'_{\rho}\in\mathbb{X}_T(\mathbb{R}^3,\rho)$.

As in Theorem \ref{th:2.26}, an a priori estimate for
$\mathbf{M}^{*}-\kappa^2I_{\rho}^{*}$ is proven, yielding its
injectivity. Hence $\mathbf{M}-\kappa^2I_{\rho}$ is a subjectivity
of the operator. Proving the Theorem.
\end{proof}

\subsection*{Acknowledgements}
This research was  supported by the
Progama ALECOL-DAAD and Fundaci\'{o}n Universidad del Norte,
Colombia. The author wants to thank the anonymous referees
for their valuable suggestions.


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\end{document}