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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 68, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/68\hfil An equation for the limit state]
{An equation for the limit state of a superconductor with pinning sites}
\author[J. Sun\hfil EJDE-2005/68\hfilneg]
{Jianzhong Sun}  

\address{Jianzhong Sun \hfill\break
457 Racine Dr., Apt 210, Wilmington, NC 28403, USA}
\email{sunj\_999\_1999@yahoo.com; lsjason@gmail.com}

\date{}
\thanks{Submitted March 15, 2005. Published June 27, 2005.}
\subjclass[2000]{35B45, 35J55, 35J50, 82D55}
\keywords{Ginzburg-Landau; limit state; degree; pinning}

\begin{abstract}
 We study the limit state of the inhomogeneous Ginzburg-Landau
 model as the Ginzburg-Landau parameter
 $\kappa=1/\epsilon\to \infty$, and derive an equation to
 describe the limit state. We analyze the properties of solutions
 of the limit equation, and investigate the convergence of (local)
 minimizers of the Ginzburg-Landau energy with large $\kappa$.
 Our results verify the pinning effect of an inhomogeneous
 superconductor with large $\kappa$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}

\section{Introduction}

Since the presence of vortices is inevitable for high temperature
superconductors in high magnetic fields, it is desirable to pin
the vortices to some specific locations, so that the supercurrent
pattern around the vortices will be stable under the influence of
the applied magnetic field and thermal fluctuation, which are
important in applications (see \cite{GP,DL,DGP}).
One of the pinning mechanisms is to add  normal impurities to the
superconductors to attract the vortices, however, this procedure
destroys the homogeneity of the superconductors, introduces an
inhomogeneous structure inside the superconductor. The  analysis
of the behavior of inhomogeneous superconductors provides a good
help for the understanding of such pinning mechanism.

Inhomogeneous models of superconductor under Ginzburg-Landau frame
work have been discussed in both physics and mathematical
literatures (see \cite{ASS,ABP,CR,DG}
\cite{L} etc.). We consider a Ginzburg-Landau system describing an
inhomogeneous superconducting material used in \cite{ABP}, through
the study of the limit case of such system,we derive an equation
to describe the limit system, which is useful to understand the
pinning effect. The following is the energy of the inhomogeneous
superconductor with the parameter $\epsilon$:
\begin{equation}\label{ENERGY}
J_{\epsilon}({\psi},A)=
   \int_{\Omega} (\,\left| ({\nabla - i\, A}) \psi \right| ^ {2} +
    \frac {1 } {2 \epsilon^2} ( a - \left|\psi \right| ^2)^2 +
    \left|{ \mathop{\rm curl} A - H_e} \right|^2 ) d x,
\end {equation}
where the parameter $\epsilon = 1 / \kappa$  is a nonnegative
number, and $\kappa$ is the Ginzburg-Landau parameter of the
superconductor material;  $\Omega \subset {\mathbb R}^2$ is a
bounded simply  connected domain with a smooth $(C^{2,\beta})$
boundary, represents the cross-section of an infinite cylindrical
body
 with $ {\rm\bf e_3} $ as its generator;
$ H_e = h_e {\rm\bf e_3} $ is the applied magnetic field with
 $h_e $ being a constant;
$ A \in  H^1 (\Omega;\mathbb{R}^2) $ is the magnetic potential and
$\mathop{\rm curl} A = \nabla\times (A_1,A_2,0)$ is the induced
magnetic field in the cylinder;
 $\psi \in  H^1(\Omega;\mathbb{C}) $ is complex-valued,
 with $|\psi|^2=\psi^*\psi$ represents the density of
 superconducting electron pairs and
$ j =\dfrac {i} {2} (\psi^*\nabla\psi-\psi\nabla\psi^*)-|\psi|^2 A
$ denotes the superconducting current density circulating in
$\Omega$; $a:\Omega\to\mathbb [0,1]$ is a bounded continuous
function, describing the inhomogeneities of the material, the zero
set of $a(x)$ corresponds to normal regions in the material.

In order to analyze the limit problem as $\epsilon \to 0$, we
define the energy
 \begin{equation}\label{EZERO}
J_0({\psi},A)=\int_{\Omega} \left| ({\nabla - i\, A}) \psi \right|
^ {2} +
    \left|{ \mathop{\rm curl} A - H_e}\right| ^2 d x,
\end{equation}
where  $(\psi, A) \in H^1_a\times H^1 (\Omega;\mathbb{R}^2)$,
$a(x)$ is the same as in \eqref{ENERGY},
\begin{equation} \label{SpaceHa}
H^1_a \equiv\{\psi\in H^1 (\Omega;\mathbb{C}) \mbox{ such that }
|\psi|^2=a \mbox{ almost everywhere}\}.
\end{equation}
In Lemma~\ref{Lemma:Degree}, we show that for each $u \in H^1_a$,
there is a unique well-defined degree $ D\equiv(d_1,\ldots,d_n)\in
{\mathbb Z}^n $  around ${\overline\Omega}_H$, denote the homotopy
class in $H^1_a $ corresponding to $D$ as  $ H^1_{a,D}$, then
$$
H^1_a=\bigcup_{D\in {\mathbb Z}^n} H^1_{a,D}\,.
$$
 Since $ H^1_{a,D}$ is a nonempty open and (sequentially
weakly) closed  subspace of $H^1_a$ (Theorem~\ref{Thm:1.5}), we can
find the minimizer of $J_0$ in $ H^1_{a,D} \times H^1
(\Omega;\mathbb{R}^2)$, and call it  the local minimizer of $J_0$
in $ H^1_a \times H^1 (\Omega;\mathbb{R}^2)$.

In~\cite{ABP} Andre, Bauman and Phillips have considered the
case $a(x)$ vanishes at a finite number of points
$\{x_1,\dots ,x_n\}$,
 and showed that for sufficiently large
$\kappa=1/\epsilon$  the local minimizers of $J_{\epsilon}$ in
\eqref{ENERGY} have nontrivial vortex structures, which are pinned
near the zero points of $a(x)$  with  any prescribed vortex
pattern. In this paper we consider the case where $a(x)$ vanishes
in subdomains (holes), which is more realistic in the presence of
normal inclusions.

Our situation is different from the cases studied in \cite{JM} or
\cite{RS}, where they have considered the energy $J_{\epsilon}$
with $a \equiv 1$ in a multiply connected domain without applied
magnetic field, they have shown the existence of the local
minimizers of $J_{\epsilon}$ with prescribed vortex structures
within certain homotopy class. In a recent paper~\cite{AB} by
Alama and Bronsard, they studied the energy $J_{\epsilon}$ with $a
\equiv 1$ in a multiply connected domain with applied magnetic
field, and achieved deeply results related to the pinning
phenomena. They proved the interior vortex will not be shown until
the applied magnetic field exceeds $H_{c_1}$ of order $\left| \ln
\epsilon \right|$, when the applied magnetic filed exceeds
$H_{c_1}$, the vortices are nucleated strictly inside the multiply
connected domain. Their techniques and results are similar to
those from \cite{AAB}, \cite{ASS}, \cite{ABP}, \cite{SS1} and
\cite{SS2}.

We analyze the limit state through the investigation of the
structure of local minimizers of $J_0$, and derive an equation to
describe the limit state. Our methods and results are similar to
those of \cite{ABP}. While, we concentrate more on the analysis of
the properties of the solutions of the limit equation, especially
various nontrivial properties of the base functions of the
solutions, which is a consequence of the setting of $a(x)$.

 In detail, $a(x)$ satisfies the following conditions:\\
  $a\in C^1({\overline\Omega} \setminus \Omega_H ),
\sqrt a\in H^1(\Omega)$, $a(x)\geq 0$  for all $x$ in
${\overline\Omega}$, and $a(x)=0$ iff $x\in {\overline\Omega}_H
\subset \Omega$, where $ \Omega_H = \cup_{j=1}^n \Omega_j $
 corresponds to the inhomogeneities of the superconductor,
$ n\in \mathbb N$,   and $\Omega_j, j = 1,\dots, n$, are  simply
connected Lipschitz subdomains with
  ${\overline\Omega}_j \subset \Omega$.
 There also exists a constant  $ 0 < r_1 < 1 $ such that
$$
\mathop{\rm dist}\{ {\Omega_i}, \, {\Omega_j} \} > r_1,  i\neq j,
  1 \leq i, j \leq n, \mbox{  and  }
 \mathop{\rm dist}\{ { \Omega_H }, \, { \partial \Omega }\} > r_1.
 $$
In addition,  for $x \in \Omega\setminus {\overline\Omega}_j$,
with $d_j(x) = \mathop{\rm dist}\{ { x},\, {\Omega_j }\} < r_1, $ there
are positive constants $C_0, C_1,\alpha_j$, such that
\begin{equation}\label{Eq:ApproxAx}
C_0 d_j^{\alpha_j}(x)\leq a(x)\leq C_1 d_j^{\alpha_j}(x),\quad
 \big|{ d_j(x) \frac {\nabla a(x)} {a(x)} }\big|\leq C_1  \,\quad
 j = 1,2, \dots, n.
\end{equation}
Choose one $x_j \in \Omega_j$, and fix it, for any $ x\in \Omega
\setminus \Omega_j$,  write
\begin{equation} \label{NormPoint}
{\rm\bf n}_j (x) = \frac {x - x_j} {\left|{x-x_j}\right|}\, ,
\end{equation}
 then  $ {\rm\bf n}_j \in C^{\infty}(\Omega \setminus \Omega_j,{\mathbb R}^2)$
 with $\left|{{\rm\bf n}_j }\right| = 1$.
Moreover, we can rewrite $ {\rm\bf n}_j(x) $ in term of its
azimuthal angle $\theta_j(x)$, so that
\begin{equation} \label{Thetaj}
 {\rm\bf n}_j(x) = (\cos \theta_j(x), \sin \theta_j(x)),
1\leq j \leq n.
\end{equation}
Note that $e^{ i\theta_j(x)}$ and $\nabla \theta_j(x) $ are
single-valued with $e^{ i\theta_j(x)} \in
 H^1(\Omega \setminus {\overline\Omega}_j, {\mathbb C})$.

Set
$$
{\mathcal{M}} \equiv H^1(\Omega;\mathbb{C})\times H^1
(\Omega;\mathbb{R}^2), \quad {\mathcal{M}}_0 \equiv H^1_a \times
H^1(\Omega;\mathbb{R}^2),
$$
then ${\mathcal{M}} $ and
${\mathcal{M}}_0$ are the domains of the functional $J_\epsilon$
and $J_0$ respectively.

 If $(\psi,A)\in{{\mathcal{M}}} ({{\mathcal{M}}}_0)$ and
 $\phi\in H^2(\Omega)$, the gauge transformation of
 $(\psi,A)$ under $\phi$ is defined by
\begin{equation}\label{GaugeDef}
(\psi',A')=G_{\phi}(\psi,A)\equiv  (\psi e^{i\phi}, A+\nabla\phi)
\in  { {\mathcal{M}} } ( {{\mathcal{M}}}_0 ) \,.
\end{equation}
$(\psi',A')$  is  {\bf gauge equivalent} to $ (\psi,A)$
 whenever \eqref{GaugeDef} has been satisfied for
some $\phi\in H^2(\Omega)$.
 As is well-known that $J_\epsilon$,
$\epsilon\geq 0$, are gauge invariant, i.e. $J_\epsilon
(\psi,A)=J_\epsilon(\psi',A')$. Hence if $(\psi,A)$ is a (local)
minimizer of $J_\epsilon$ in ${{\mathcal{M}}}_0$,  so is
$(\psi',A')$.

In this paper, we fix a gauge by requiring that $A$ satisfy
\begin{equation}\label{Gauge0}
\begin{gathered}
 \mathop{\rm div} A  = 0 \quad \mbox{ in }\Omega,\\
 A\cdot {\rm\bf n}  = 0\quad \mbox{ on }\partial\Omega \,.
\end{gathered}
\end{equation}
This can be done by choosing  a gauge $\phi$ such that
\begin{equation}\label{GaugeTrans}
\begin{gathered}
 \triangle  \phi = -\mathop{\rm div} A \quad \mbox{ in }\Omega,\\
 \frac {\partial \phi} {\partial{\rm\bf n}} = - A \cdot {\rm\bf n}
 \quad \mbox{ on }\partial\Omega.
\end{gathered}
\end{equation}
Apply \eqref{GaugeDef} to $(\psi,A)$, we get  $(\psi',A') =
G_{\phi}(\psi,A)$ satisfies \eqref{Gauge0}.

Since $ J_{\epsilon}(\sqrt{a}, 0) = J_0 (\sqrt{a}, 0)
 = \int_\Omega \left|{\nabla \sqrt{a}}\right|
    + h_e^2 \left|{\Omega}\right| < \infty$,  it makes sense to talk about the
minimizers and local minimizers of $J_0 $ and $J_{\epsilon}$ in
${\mathcal{M}}_0$ and ${\mathcal{M}}$ respectively. In Section II,
we derive a few preliminary results. In Section \ref{Sec:E0}, we
analyze the local minimizers of $J_0$ in ${\mathcal{M}}_0$, and
establish the following equation to describe them.


\begin{thm}[see Theorem~\ref{Thm:H}] \label{Thm:H:intro}
Fix $h_e$. Let $(\psi_D,A_D)$ be a minimizer of $J_0$ in
$H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2)$
 under gauge  (\ref{Gauge0}),
define $h_D$ by $\mathop{\rm curl} A_D=h_D {\rm\bf e_3} $, then
$h_D \in V $ is the unique solution of
\begin{equation}\label{Heq:intro}
\begin{gathered}
\int_{\Omega\setminus {\overline\Omega}_H}
 a^{-1} \nabla h \cdot \nabla f d x
+  \int_{\Omega} h f d x = \sum_{j=1}^n 2\pi d_j f_j \, ,\\
 \forall f(x) \in V \cap H_0^1(\Omega),  \quad
 \mbox{and} \quad h - h_e \in V \cap H^1_0(\Omega)\, ,
\end{gathered}
\end{equation}
 where the space
\begin{equation}\label{Eq:SpaceV}
\begin{array}{rl}
V \equiv \{ f\in  H^1(\Omega) &|\ f|_{\Omega_j}
 = f_j = \mbox{constant,} 1 \leq j \leq n,\\
& \int_{\Omega \setminus {\overline\Omega}_H} a^{-1}(x)
\left|{\nabla f(x)}\right|^2 d\, x < \infty \},
\end{array}
\end{equation}
and $f_j$ is the constant for $f$ on $\Omega_j, j = 1, 2, \dots,
n$. \end{thm}

Note that $V$ is nontrivial, since $a \in V $ by $\sqrt{a}
\in H^1(\Omega)$.
We further reveal the relation between
the local minimizers and critical points of $J_0$ in
${{\mathcal{M}}}_0$, namely  critical points are the same as local
minimizers, as below.

\begin{thm}[see Theorem~\ref{Thm:Structure}]\label{Thm:Structure:intro}
Fix $h_e$.  For each $D = (d_1, \dots, d_n)\in {\mathbb Z}^n$,  $
J_0$ has a unique minimizer in $H^1_{a,D}\times
H^1(\Omega;{\mathbb R}^2) \subset   H^1_a\times
H^1(\Omega;{\mathbb R}^2)$ in sense of gauge equivalence; moreover
for any two such minimizers, say  $(\psi,A)$ and $(\psi',A')$,
under gauge  (\ref{Gauge0}), then $ A = A'$ and $ \psi = \psi'
e^{ic} $ for some $c\in {\mathbb R}$.
\end{thm}

Combine Theorem~\ref{Thm:H:intro} and Theorem~\ref{Thm:Structure:intro},
 we can see there
is a one to one relation between the solutions of
\eqref{Heq:intro} and the gauge equivalent minimizers of $J_0$ in
${{\mathcal{M}}}_0$.

In Section IV, we study the properties of solutions of
\eqref{Heq:intro}, where we show its solution can be represented
by a linear combination of $n+1$ independent functions in
$C({\overline\Omega})\cap V$ (see Theorem~\ref{thm:regularity}), we
derive more detailed properties of the independent functions.
Under a slightly stronger assumption on $a(x)$, we also achieve
higher regularity of the solution.

In Section V, we discuss the motivation of our analysis of the
limit problem. We show (see Theorem~\ref{thm:4.2}) the minimizers of
$J_{\epsilon}$ converge to the minimizer of $J_0$ in
${\mathcal{M}}$. Moreover, for $\epsilon$ sufficiently small, all
vortices of minimizers of $J_{\epsilon}$ are pinned near
${\overline\Omega}_H$, the zero set of $a(x)$. Since the zero set
of $a(x)$ corresponds to the normal regions, the result confirms
the effectiveness of the pinning mechanism by adding normal
impurities to a superconductor to attract vortices.

Consider  the local minimizers of $J_\epsilon$ in the neighborhood
of a local minimizer of $J_0$,  similar to the above result, we
have the following theorem.

\begin{thm}\label{thm:4.6}
Fix $h_e$ and $D \in {\mathbb Z}^n$. Let $(\psi_D,A_D)$ be a
minimizer for $J_0$ in $H^1_{a,D}\times H^1(\Omega;\mathbb{R}^2)$
under gauge (\ref{Gauge0}). Choose $r>0$ such that
${\mathcal{B}}_r\cap{\mathcal{M}}_0={\mathcal{B}}_r\cap
[H^1_{a,D}\times H^1(\Omega;{\mathbb R}^2)]$. Then for all
$\epsilon > 0$ sufficiently small, ${\mathcal{B}}_r(\psi_D,A_D)$
contains a local minimizer, $(\psi_\epsilon,A_\epsilon)$, of
$J_\epsilon$ in ${\mathcal{M}}$, such that, $|\psi_{\epsilon}|\to
\sqrt a$  in $C(\overline\Omega)$, and
$(\psi_{\epsilon},A_{\epsilon})\to (\psi_D,A_D)$ in
${\mathcal{M}}$ as $\epsilon \to 0$. In addition, for each $0<
\sigma < r_1$ and all $\epsilon$ sufficiently small,
$|\psi_\epsilon|$ is uniformly positive outside $\bigcup_{j=1}^{n}
{\overline\Omega}^{\sigma}_j$ and the degree of $\psi_\epsilon$
around
 $\overline{\Omega^{\sigma}_j } $ is $d_j$,  $j=1,2,\dots,n$.
 \end{thm}
Where
${\mathcal{B}}_r\equiv{\mathcal{B}}_r(\psi_D,A_D)=\{(\psi,A)\in{\mathcal{M}}
| \|(\psi,A)-(\psi_D,A_D)\|_{H^1(\Omega)}\leq r\}$.


\section{Preliminaries}\label{Sec:Prelim}

In this section, we describe some  properties of two Sobolev
spaces to be used for our later analysis. Section~\ref{SubSec:H}
is about the properties of $H^1_a$ defined in (\ref{SpaceHa}),
Section~\ref{SubSec:Space} is about the properties of the weighted
Sobolev space $V$ in \eqref{Eq:SpaceV}, where we generalize the
space ideas from \cite{ABP}.

\subsection{Space $H^1_a$} \label{SubSec:H}
Recall that we have defined
$$
H^1_a \equiv\{\psi\in H^1
(\Omega;\mathbb{C}), \mbox{ such that } |\psi|^2=a \mbox{ almost
everywhere}\}.
$$
 By the assumption on  $a(x)$, $\sqrt a \in
H^1_a$, $ H^1_a $ is nonempty. The following lemma justifies
 the existence of the degree for any $u \in H^1_a$.

\begin{lem}\label{Lemma:Degree}
For every $u\in H_a^1$,   there is
 a unique $ D\equiv(d_1,\ldots,d_n)\in {\mathbb Z}^n$,  depending only
 on $u$,   such that for
 any subdomain $G_j$ and any function $f_{G_j}$,
 $1 \leq j \le n$,  satisfying
\begin{equation}\label{Eq:Gj}
G_j\subset \Omega, \mbox{ be a  simply connected smooth subdomain
with }  G_j \cap {\overline\Omega}_H = {\overline\Omega}_j,
\end{equation}
\begin{equation}\label{Eq:FGj}
f_{G_j} \in C^{\infty}({\overline\Omega}), 0 \leq f_{G_j} \leq 1,
f_{G_j}= 1 \mbox{ on } {\overline\Omega}\setminus G_j, \mbox{ and
} supp\{f_{G_j}\}\subset {\overline\Omega}
\setminus{\overline\Omega}_j,
\end{equation}
then we have the representation
\begin{equation}\label{DegU}
d_{j} = \deg ( u/\sqrt{a}, \partial G_j) =
 \frac 1 {\pi}\int_{G_j\setminus {\overline\Omega}_j}
 {\mathbb\bf J} ( \frac {u f_{G_j}} {\sqrt{a}} ) dx \,.
\end{equation}
Where  ${\mathbb\bf J} ({\bf w}) $ is the Jacobian of
the map $ { \bf w }: G_j \to \mathbb{C}$. Write $ x=(x_1, x_2) \in
G_j$, ${\bf w} = w_1 + i w_2, $ then
$$ {\mathbb\bf J} ({\bf w}) = \frac {\partial (w_1, w_2)}
     {\partial( x_1, x_2)} = \det \begin{bmatrix}
   \frac {\partial w_1} {\partial x_1}  &
   \frac {\partial w_1} {\partial x_2} \\[4pt]
   \frac {\partial w_2} {\partial x_1} &
   \frac {\partial w_2} {\partial x_2}
\end{bmatrix}\,.
$$
\end{lem}

\begin{proof}  Fix $u\in H_a^1$,   set $v(x)= u(x) / \sqrt{a(x)}
=u(x)/|u(x)|$ in $\Omega\setminus{\overline\Omega}_H$, and
$v(x) =0 $ for other case. By the assumption on $a(x), \ v \in
H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H;S^1)$, where
$S^1 =\{z\in\mathbb C:|z|=1\}$.

 Let $G_j$ be as in
(\ref{Eq:Gj}) and  $f_{G_j}$ as in (\ref{Eq:FGj}), we have   $v
f_{G_j} $ is well defined  on $ G_j$,  in addition, $supp\{v
f_{G_j}\}\cap{\overline G}_j\subset {\overline G}_j
\setminus{\overline\Omega}_j$, $v f_{G_j} \in H^1(G_j)$, and
$\left|{v f_{G_j}}\right|= \left|{ v }\right| = 1$ a.e. on
$\partial G_j$. From \cite{BN0} (Property~5 at page~220 and
 lemma~11 at page~337),
\begin{equation}\label{Eq:BNDef}
\deg(v,\partial G_j) = \deg ( v f_{G_j}, \partial G_j)= \frac 1
{\pi}\int_{G_j}  {\mathbb\bf J}(v f_{G_j} ) d x
 = \frac 1 {\pi}\int_{G_j\setminus {\overline\Omega}_j}
 {\mathbb\bf J} ( \frac {u f_{G_j}} {\sqrt{a}} ) dx\,,
\end{equation}
$\deg(v,\partial G_j)$ is well-defined,  integer-valued and
independent  of $f_{G_j}$,

 Now to show $\deg(v,\partial G_j)$ is independent of the choice of
 $G_j$.
 \\ {\bf Claim: } If two subdomains $G_j^1, G_j^2 $
satisfy (\ref{Eq:Gj}) with
  $G_j^2 \subset G_j^1 \subset \Omega$,  then
$ \deg ( v,\partial G_j^1) = \deg ( v, \partial G_j^2 )$.
\\
{\bf Proof of the Claim:} By $v \in  H^1({\overline G}_j^1
\setminus G_j^2)$,  there is a constant $ \delta = \delta(G_j^1,
G_j^2, v)$, such that for any set $ A \subset G_j^1 \setminus
G_j^2 $ and $\mathop{\rm meas}\{A\}  < \delta$,  $\| v \|_{H^1(A)}^2 < 1$.
Then for any two simply connected smooth subdomains
$B^1, B^2 $  with $G_j^2 \subset B^2 \subset B^1 \subset G_j^1$
and $\mathop{\rm meas}\{B^1 \setminus B^2 \} < \delta$, from \eqref{Eq:BNDef},
we have
\begin{equation*}
\begin{split}
\left|{\deg(v,\partial B^1)  -  \deg(v,\partial B^2)}\right|
&=\Big| \frac 1 {\pi}\int_{B^1}  {\mathbb\bf J}(v f_{G_j^2} ) d x
- \frac 1
{\pi}\int_{B^2}  {\mathbb\bf J}(v f_{G_j^2} ) d x \Big| \\
&=\big| \frac 1 {\pi}\int_{B^1\setminus B^2}  {\mathbb\bf J}(v )
d x \big| \\
&\leq 2 \| v \|_{H^1(B^1\setminus B^2 )}^2 / \pi < 1\,.
\end{split}
\end{equation*}
Since the left-hand side is integer-valued, $\deg(v,\partial B^1)
=  \deg(v,\partial B^2)$.

 Choose a finite number of
nested simply connected smooth subdomains, say
$ G_j^1 = A^1 \supset\supset A^2 \supset\supset A^2 \supset\supset\dots
\supset\supset A^k = G_j^2$, such that
$\mathop{\rm meas}\{A^{\ell} \setminus
A^{\ell+1}\} < \delta$, $\ell = 1,\dots, k-1$, then
 $\deg ( v,\partial G_j^1) =  \deg ( v, \partial A^1)=
   \deg ( v, \partial A^2)= \dots =
   \deg ( v, \partial G_j^2)$.
From the above claim,  we know for any two subdomains $G_j^1,
G_j^2 $ satisfy (\ref{Eq:Gj}),
$$
\deg ( v, \partial G_j^1 ) =
 \deg ( v, \partial (G_j^1 \cap G_j^2)  ) = \deg ( v, \partial G_j^2 ).
$$
Hence $\deg ( v, \partial G_j)$ is constant  for any $G_j$
satisfying (\ref{Eq:Gj}), i.e.,  \eqref{DegU} is well-defined, and
$d_j \equiv \deg ( v, \partial G_j )
 = \deg ( u/\sqrt{a},\partial G_j )$ depends on $u$ only.
\end{proof}

\begin{lem}\label{Lem:1.4}
 Each $u\in H_a^1$ can be written in the form of
$$
u(x) = \sqrt{ a(x) } e^{i\Theta (x)},
 \, x \in  \Omega\setminus{\overline\Omega}_H,
$$
where $\Theta (x) = \phi (x)+ \underset{j=1}{\overset n\sum}
 d_j\theta_j,\ \theta_j(x) $ is from  (\ref{Thetaj}) defined on
$ \Omega \setminus{\overline\Omega}_j$,
 $D\in{\mathbb Z}^n$ is from
\eqref{DegU}, uniquely decided by $ u\in H^1_a$, and $\phi \in
H_{\rm loc}^1( \Omega \setminus{\overline\Omega}_H )$ is unique up to
an additive constant $2\pi k$ for $k\in {\mathbb Z}$, satisfying
$\int_{ \Omega \setminus{\overline\Omega}_j }
 a|\nabla \phi |^2\le C( \Omega_H, a, D) +
\int_\Omega |\nabla u|^2$.
\end{lem}

 We follow the same idea as in \cite[Theorem 1.4]{ABP} to prove
 the lemma,  please see the proof in the appendix.

For each $D=(d_1,\dots,d_n)\in {\mathbb Z}^n$,  we define the
homotopy class
$$
H_{a,D}^1=\{u\in H_a^1| \mbox{ degree for } u \mbox{ around }
    {\overline\Omega}_j \mbox{ is } d_j, j=1,2, \dots, n \}.
$$
By Lemma~\ref{Lem:1.4}, $u \in H_{a,D}^1$,  if and only if \ $ u =
\sqrt{a} e^{i[ \phi(x) + \underset{j=1}{\overset n\sum}
 d_j\theta_j]},$\
 where $\phi \in H_{\rm loc}^1( \Omega \setminus{\overline\Omega}_j )$
and $\int_{ \Omega \setminus{\overline\Omega}_j }
 a|\nabla \phi |^2\le C( \Omega_H, \Omega,  a, D) +
\int_\Omega |\nabla u|^2$; Lemma~\ref{Lemma:Degree} implies that
$H_a^1={\underset {D\in {\mathbb Z}^n} \bigcup } H_{a,D}^1 $ and
$H_{a,D}^1\cap H_{a,D'}^1=\emptyset$ for $D\not= D'$ in ${\mathbb
Z}^n$; the following theorem further reveals the topology of
$H_a^1$.

\begin{thm}\label{Thm:1.5}
For each $D\in {\mathbb Z}^n$,  $H_{a,D}^1$ is a nonempty, open
and closed subset of $H_a^1$. In addition, $H_{a,D}^1$ is
sequentially weakly closed in $H^1(\Omega;\mathbb C)$, i.e., if
$\{u_k\}_{k=1}^{\infty}\subset H_{a,D}^1$ and $u_k\rightharpoonup
u$ in $H^1(\Omega;\mathbb C)$ as $k\to \infty$, then
$u\in H_{a,D}^1$.
\end{thm}

\begin{proof} Since   $ \sqrt{a}\in H^1(\Omega)$ and
${\rm\bf n}_j \in C^{\infty}({\overline\Omega} \setminus\Omega_j;
{\mathbb R}^2)$,
  $1 \leq j \leq  n$, according to Lemma~\ref{Lem:1.4},
$\sqrt{a} e^{i \sum_{j=1}^n  d_j\theta_j }\in H_{a,D}^1$,  so that
$H_{a,D}^1\neq \emptyset$.

 Assume $u_0  \in H_{a,D}^1$,   let
 $ B_r(u_0)=\big\{ u\in H_a^1: \|u-u_0\|_{H^1(\Omega;\mathbb C)}<r  \big\}$,
 where $r>0$ to be chosen later.
Pick any $ u\in B_r(u_0)$, set $v_0=u_0/|u_0|=a^{-1/2}u_0$,
$v=u/|u|=a^{-1/2}u$.  Fix $G_j$ as in (\ref{Eq:Gj}) and
 $f_{G_j}$ as in (\ref{Eq:FGj}), $1 \le j \le n$,  by (\ref{DegU}),
$$
d_j = \frac 1 {\pi}\int_{G_j\setminus \Omega_j}
 {\mathbb\bf J} (v_0 f_{G_j}) dx
\mbox{\quad  and \quad } \tilde d_j = \frac 1
{\pi}\int_{G_j\setminus \Omega_j}
 {\mathbb\bf J} (v f_{G_j}) dx,
$$
 then
$$
\| {\mathbb\bf J} (v_0 f_{G_j}) - {\mathbb\bf J} (v f_{G_j})
\|_{L^1(G_j)} \le C\cdot
 (1+\|u-u_0\|_{H^1(G_j)} ) \cdot\|u-u_0\|_{H^1(G_j)}
\leq C r(1+r)\,,
$$
where $C=C(a, v_0, G_j)$. It follows that for $r$ small (say
$r=\frac 1 {2C+1}$)
 $d_j=\tilde d_j$ and  $u\in H_{a,D}^1$.
 Thus $B_r(u_0)\subset H_{a,D}^1$ for $r$ small,
 $H_{a,D}^1$ is an open subset of $H_a^1$.

 Since $H_a^1=\bigcup_{D\in {\mathbb Z}^n} H_{a,D}^1$ and $H_{a,D} \cap
H_{a,D'}^1=\emptyset$ for $D \ne D'$ in ${\mathbb Z}^n$,  from the
closeness of $H_a^1$,  we obtain
 that $H_{a,D}^1$ is a closed subset of $H_a^1$.

Now prove $H_{a,D}^1$ is weakly sequentially
 closed in $H_a^1$.
Assume that $\{u_k\}_{k=1}^{\infty} \subset H_{a,D}^1$ and
$u_k\rightharpoonup u$ weakly in $H^1(\Omega;\mathbb C)$
 as $k \to \infty$.  By  compactness,
a subsequence (which we relabel as $\{u_k\}_{k=1}^{\infty}$)
satisfies $u_k\to u$ in $L^2(\Omega)$
 as $k \to \infty$,  so $|u|=a^{1/2}$   a.e. in
$\Omega$,   and  $u\in H_a^1$, according to Lemma~\ref{Lemma:Degree},
$u\in H_{a,\tilde D}^1$ for some $\tilde D\in {\mathbb Z}^n$.
 We show $D = \tilde D$.

Set $v_k(x)=u_k(x)/|u_k(x)|, v(x)=u(x)/|u(x)| $
 in $ \Omega \setminus  \Omega_H$,
then  $ v_k \rightharpoonup v  $ in $ H_{\rm loc}^1(\Omega \setminus
\Omega_H )$ and $v_k \to v $ in $L_{\rm loc}^2(\Omega \setminus
\Omega_H )$,  as $k \to \infty$.

Choose  $ \Omega^1_j \supset \Omega^2_j $ satisfying
(\ref{Eq:Gj}),
 assume $\| v_k \|_{H^1(\Omega^1_j \setminus \Omega^2_j)} < M,
 M \in {\mathbb Z},  k = 1, 2, 3, \dots$.
Partition $\Omega^1_j \setminus \Omega^2_j$ into $4 M^2 + 1$
subdomains enclosing $\Omega^2_j$, say, they are
 $G^{(l)}\setminus G^{(l+1)}$, $l = 1, 2 \dots 4 M^2 + 1$,
where
$$
\Omega^1_j = G^{(1)} \supset\supset  G^{(2)}
\supset\supset \dots \supset\supset G^{(4 M^2 + 2)}
 = \Omega^2_j\,.
$$
For any $v_k$,  at least on one of the
 $G^{(l)} \setminus G^{(l+1)}$,
$\| v_k \|_{H^1(G^{(l)} \setminus G^{(l+1)})} < \frac 1 2$.
Choose $G^{(l)} \setminus G^{(l+1)}$ with infinitely
many $v_k$ such that $\| v_k \|_{H^1(G^{(l)} \setminus G^{(l+1)})}
< \frac 1 2$. Let $ G_j = G^{(l)}, \tilde G_j  = G^{(l+1)}$, and
take the corresponding  subsequence on $G^{(l)} \setminus
G^{(l+1)}$
 (still labelled as $\{ v_k\}$), then
$\Omega_j \subset\subset \tilde G_j \subset\subset G_j$ and $\|
v_k \|_{H^1(G_j\setminus \tilde G_j )} < \frac 1 2$, for all $k
\geq 1$. By the weak convergence, $\| v\|_{H^1(G_j\setminus \tilde
G_j )} < \frac 1 2$, $\int_{G_j\setminus\tilde G_j }
 \left|{{\mathbb\bf J} (v_k )}\right| \leq
\| v_k \|^2_{H^1(G_j\setminus \tilde G_j )}
 <  \frac 1 4$, and $\int_{G_j\setminus\tilde G_j }
 \left| {{\mathbb\bf J} (v )} \right| \leq
 \| v \|^2_{H^1(G_j\setminus \tilde G_j )}<  \frac 1 4$.
Pick $f_{G_j}$ satisfying (\ref{Eq:FGj}).
By (\ref{DegU}),
 \[
d_j = \frac 1 {\pi}\int_{G_j\setminus\tilde G_j }
 {\mathbb\bf J} (v_k f_{G_j}) d x, \quad
 \tilde d_j = \frac 1 {\pi}\int_{G_j\setminus\tilde G_j }
 {\mathbb\bf J} (v f_{G_j}) d x.
 \]
\begin{align*}
&\int_{G_j\setminus\tilde G_j }
 ({\mathbb\bf J} (v_k f_{G_j}) - {\mathbb\bf J}(v f_{G_j} ) )d x\\
&= \int_{G_j\setminus\tilde G_j } f^2_{G_j} ({\mathbb\bf J} ( v_k)
- {\mathbb\bf J} (v)) d x\\
&\quad + \int_{G_j\setminus\tilde G_j } f_{G_j}
\frac {\partial f_{G_j} } {\partial x_1} \mathop{\rm Re} \left( i
v_k (\frac {\partial v_k} {\partial x_2})^*
 - i v (\frac {\partial v} {\partial x_2})^* \right) d x \\
&\quad + \int_{G_j\setminus\tilde G_j }  f_{G_j}  \frac
{\partial f_{G_j} } {\partial x_2} \mathop{\rm Re} \left( i v_k
(\frac {\partial v_k} {\partial x_1})^*
 - i v (\frac {\partial v} {\partial x_1})^* \right) d x\,.
\end{align*}
 Since $ f_{G_j} \frac {\partial f_{G_j} }  {\partial x_1}
\in C^{\infty}(\overline G_j),\  v_k \to v $ in $L^2$, and  $\frac
{\partial v_k} {\partial x_2} \rightharpoonup \frac {\partial v}
{\partial x_2} $ weakly in $L^2$, it follows that
 $\int_{G_j\setminus\tilde G_j } f_{G_j} \frac {\partial f_{G_j}
} {\partial x_1} \mathop{\rm Re} \left( i v_k (\frac {\partial v_k}
{\partial x_2})^* - i v (\frac {\partial v} {\partial x_2})^* \right) d x
\to 0$, as $ k \to \infty$.
Similarly $
\int_{G_j\setminus\tilde G_j } f_{G_j} \, \frac {\partial f_{G_j}
} {\partial x_2} \mathop{\rm Re} \left( i v_k (\frac {\partial v_k}
{\partial x_1})^*
 - i v (\frac {\partial v} {\partial x_1})^* \right) d x
\to 0$,   as $ k \to \infty$.

By $0\leq f_{G_j} \leq 1$,
$\int_{G_j\setminus\tilde G_j } f^2_{G_j} \left| {{\mathbb\bf J} (
v_k) - {\mathbb\bf J} (v)}\right| d x \leq \frac 1 2$, for any
$k$, $\left| { d_j - \tilde d_j } \right| < \frac 1 2$. Since
$d_j, \tilde d_j \in {\mathbb Z}$, $d_j = \tilde d_j, j = 1, 2,
\dots n$. Thus $D=\tilde D$ and $u \in H_{a,D}^1$.
\end{proof}

\subsection{Space V }\label{SubSec:Space}

By \eqref{Eq:SpaceV}, $V\equiv\{ f\in  H^1(\Omega):
f|_{\Omega_j} = f_j = \mbox{ constant,}$ $1 \leq j \leq n$,
$\int_{\Omega \setminus {\overline\Omega}_H} a^{-1}(x) \left|
{\nabla f(x)}\right|^2 d x < \infty \}$
 is a weighted Sobolev space.
Define the norm of $V$ as
\begin{equation}\label{Eq:HILBERT}
\|f\|_V=\Big( \int_{ \Omega \setminus {\overline\Omega}_H }
a^{-1}(x) \left| {\nabla f(x)}\right|^2 d \, x + \int_\Omega f^2
\Big) ^{1/2}.
\end{equation}

\begin{lem}\label{Lemma:HILBERT}
$V$ is a Hilbert space with norm (\ref{Eq:HILBERT}).
\end{lem}

\begin{proof}
Assume $\{f_k\}_{k=1}^{\infty} \subset V$
is a Cauchy sequence under norm  (\ref{Eq:HILBERT}). By
$1/a(x)>c>0$ in ${\overline\Omega}\setminus\Omega_H$ for some
constant $c\in {\mathbb R}$,  we know $\{f_k\}_{k=1}^{\infty}$ is
a Cauchy sequence in $H^1(\Omega)$. Hence there is a $f\in
H^1(\Omega)$, such that $ f_k \to f $ in $H^1(\Omega)$.  Also $f_k
\in V$ implies that $f$ is constant on $\Omega_j, 1\leq j \leq n$.
By $\{ \nabla f_k / \sqrt{a} \}_{k=1}^{\infty}$ is a Cauchy
sequence in $L^2(\Omega\setminus\Omega_H)$,    there are
  $g_1, g_2$  in  $L^2(\Omega\setminus\Omega_H)$,  such that
$\nabla f_k / \sqrt{a} \to (g_1, g_2)$ in
$L^2(\Omega\setminus\Omega_H)$,
 from $ 1/\sqrt{a} $ is bounded away from 0, we get
 $ \nabla f_k \to (\sqrt{a} g_1, \sqrt{a} g_2)  $
 in $L^2(\Omega\setminus\Omega_H)$. Therefore
$(\sqrt{a} g_1, \sqrt{a} g_2) = \nabla f$ by the uniqueness of the
convergence in $L^2(\Omega\setminus\Omega_H)$, i.e. $\nabla f
/\sqrt{a} \in L^2(\Omega\setminus\Omega_H)$,  and we get $f\in V$,
$ f_k \to f $ in $V$.
\end{proof}

Using the same idea as above, we can obtain that $V$ is weakly
closed under the  norm  (\ref{Eq:HILBERT}). In addition, we have
the following lemma  proved in the appendix.

\begin{lem}\label{Lemma:2.2}
$C^1(\Omega)\cap V$ is dense in $V$.
\end{lem}

To go forward, let us first investigate properties of the
Lipschitz domain $\Omega_j \subset {\mathbb R}^d, 1\le j \le n$,
$d$ is the dimension. By saying $\Omega_j $ is Lipschitz, we means
that for every point $p \in
\partial \Omega_j$,  there is a neighborhood $\mathcal{U}_p$ of $p$, and a
function $\phi_p : {\mathbb R}^{d-1}\to {\mathbb R}$,  such that
there is a Cartesian coordinate system in $\mathcal{U}_p$ with
$p$ as the origin,
 satisfying:
\begin{itemize}
\item[(i)]  $\left|{\phi_p(\tilde{x}) -
\phi_p(\tilde{y})}\right| \leq A \left|{\tilde{x} -
\tilde{y}}\right|$, where $A = A(\Omega_j)$, $\tilde{x}, \tilde{y}
\in {\mathbb R}^{d-1}$.
\item[(ii)]  $\Omega_j \cap \mathcal{U}_p = \{ (\tilde{x},
x_d) |
  x_d < \phi_p(\tilde{x}) \} \cap \mathcal{U}_p$,
   and $\mathcal{U}_p \setminus \Omega_j = \{ (\tilde{x}, x_d) |
  x_d > \phi_p(\tilde{x}) \} \cap \mathcal{U}_p, $
where $\tilde{x} \in  {\mathbb R}^{d-1}$.

\item[(iii)] For all $x \in \mathcal{U}_p$, $d(x) = \mathop{\rm dist}\{ { x },\, {\partial
\Omega_j }\} > \left| {x_d -\phi_p({\tilde x})} \right| /g_p$,
for some constant $g_p > 1$.
\end{itemize}
Since $\partial \Omega_j$ is compact, we can choose
$\{ {\mathcal{U}}_k^j \}_{k=1} ^{n_j}$ to cover it, $j = 1,2, \dots, n$, where
 $ \mathcal{U}_k^j = \{ x=(\tilde x, x_d) \in {\mathbb R}^{d} |
\left| {\tilde x} \right| \leq \lambda_k^j, \mbox { and } \left| {
x_d - \phi_k^j(\tilde x)}\right| < \lambda_k^j \}$,
 $ \lambda_k^j $  is constant,
 $\phi_k^j$  is as in (i) and (ii).

 Apply (iii), for any $x = (\tilde x, x_d) \in  \mathcal{U}_k^j$,
there is a constant
  $ g = g(\Omega_H) >1 $,
\begin{equation}\label{Eq:ApproxDx}
\left| {x_d - \phi_k^j({\tilde x})} \right| / g \leq d(x) \leq
\left| {x_d - \phi_k^j({\tilde x})} \right| \,\quad k = 1,2, \dots,
n_j, \; j=1,\dots, n.
\end{equation}
Since $\partial \Omega_j \subset \cup_{k=1}^{n_j}
{\mathcal{U}}_k^j$ and $\mathcal{U}_k^j$ is open, there is a
constant $r_1$, such that for $\sigma <  r_1$,
${\overline\Omega}_j^{\sigma}\setminus \Omega_j \subset
\cup_{k=1}^{n_j}  {\mathcal{U}}_k^j$, where
$\Omega_j^{\sigma} =  \{x\in \Omega |
 \mathop{\rm dist}\{ { x }, \, { \Omega_j }\} < {\sigma} \},\ j = 1, 2,\dots, n$.
Choose a partition  of unity for
${\overline\Omega}_j^{r_1}\setminus \Omega_j$
 subordinate to $\{ {\mathcal{U}}_k^j \}_{k=1} ^{n_j}$,
say, $\{ \beta_k^j \}_{k=1} ^{n_j}$,  such that,
\begin{equation}\label{Eq:unity}
\beta_k^j \in C_0^{\infty}({\mathcal{U}}_k^j ), 0 \leq \beta_k^j
\leq 1, \mbox{  and } \sum_{k=1} ^{n_j} \beta_k^j (x) = 1 \quad
 x \in {\overline\Omega}_j^{r_1}\setminus \Omega_j, \; 1\leq j
\leq n.
\end{equation}

\begin{lem}\label{Lemma:2.3}
Assume $f \in C^1(\Omega)\cap V$,  and
 $f|_{\Omega_j} = f_j$,  $1 \leq j \leq n$,
pick the constant $g$  satisfying \eqref{Eq:ApproxDx}, then for
any $\sigma_0 < r_1 / g$,
$$
\int_{\partial \Omega_j^{\sigma_0} }
a^{-1}(x)\left|{f-f_j}\right|^2 d s \leq c(\Omega_j)\sigma_0
 \int_{\Omega_j^{g \sigma_0} \setminus \Omega_j }
a^{-1}(x) \left| {\nabla f}\right|^2 d x \,.
$$
\end{lem}

\begin{proof}
By $ \partial \Omega_j^{\sigma_0}\subset
 \Omega_j^{r_1}\setminus \Omega_j \subset
 \cup_{k=1}^{n_j} \mathcal{U}_k^j $
 and the partition of unity,
\begin{align*}
\int_{\partial \Omega_j^{\sigma_0} }
a^{-1}(x)\left|{f-f_j}\right|^2 d s
&= \int_{\partial \Omega_j^{\sigma_0} }
 \sum_{k=1}^{n_j}\beta_k^j(x)a^{-1}(x) \left|{f_j - f}\right|^2 ds \\
&=\sum_{k=1}^{n_j} \int_{\partial
\Omega_j^{\sigma_0}\cap \mathcal{U}_k^j }
 \beta_k^j(x)a^{-1}(x) \left| {f_j - f}\right|^2 ds\,.
\end{align*}
Then apply the local coordinate system on $\mathcal{U}_k^j$, we
obtain
\begin{align*}
& \int_{\partial \Omega_j^{\sigma_0}\cap \mathcal{U}_k^j  }
\beta_k^j(x) a^{-1}(x)\left|{f-f_j}\right|^2 d s \\ =&
\int_{\partial \Omega_j^{\sigma_0} \cap \mathcal{U}_k^j }
\beta_k^j(x) a^{-1}(x)\left| \int_{ \phi_k^j(\tilde{x})}^{x_d}
\nabla f \cdot {\rm\bf e_d} d t
 \right|^2 d s\\
\leq &\int_{\partial \Omega_j^{\sigma_0}\cap \mathcal{U}_k^j  }
a^{-1}(x) \left( \int_{ \phi_k^j(\tilde{x})}^{x_d} \left|{\nabla
f}\right|^2 d t \int_{ \phi_k^j(\tilde{x})}^{x_d} d t \right) d s \\
\leq &\int_{\partial \Omega_j^{\sigma_0} \cap \mathcal{U}_k^j  }
   \left|{\phi_k^j(\tilde{x}) -{x_d}}\right|
 \int_{ \phi_k^j(\tilde{x})}^{g \sigma_0}
 a^{-1}(x(s)) \left|{\nabla f}\right|^2 d t  d s\\
\leq& c(\Omega_j) \sigma_0 \int_{\Omega_j^{g \sigma_0} \setminus
\Omega_j } a^{-1}(x) \left|{\nabla f}\right|^2 d x\,.
\end{align*}
Where $\rm\bf e_d$ is the $d-$th unit vector in the local
coordinate system,. In the proof, \eqref{Eq:ApproxAx},
\eqref{Eq:ApproxDx},
 $0 \leq \beta_k^j(x) \leq 1$
and  the Lipschitz property (i) are used. Hence $\int_{\partial
\Omega_j^{\sigma_0} } a^{-1}(x)\left|{f-f_j}\right|^2 d s \leq
c(\Omega_j) \sigma_0 \int_{\Omega_j^{g \sigma_0} \setminus
\Omega_j } a^{-1}(x) \left|{\nabla f}\right|^2 d x$,   $ 1 \leq j
\leq n$.
\end{proof}

\section{ Limit Equation}\label{Sec:E0}

In this section, we prove  Theorem~\ref{Thm:H:intro} and
Theorem~\ref{Thm:Structure:intro} stated in the introduction. First let
us give a result  concerning the existence of
 the minimizers of $J_0$
in $ H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2)$.

\begin{lem}  %[Existence of Minimizers]
\label{Lemma:Existence}
For fixed $h_e$ and $D \in {\mathbb Z}^n$,
there is a  minimizer
 of  $J_0$ in  $ H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2)$
under  gauge (\ref{Gauge0}), which is a local minimizer of $J_0$
in  ${\mathcal{M}}_0$.
\end{lem}

 \begin{proof} By the gauge equivalence in
(\ref{GaugeDef}) and (\ref{GaugeTrans}), we need only to consider
the situation under the fixed gauge  (\ref{Gauge0}), i.e., in the
space
$$\{(\psi,A)\in H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2):\
\text{div }A=0\text{ in }\Omega \text{ and }A\cdot{\rm\bf
n}=0\text{ on }\partial\Omega\}.
$$
According to Theorem~\ref{Thm:1.5},
$H_{a,D}^1$ is sequentially weakly closed in $H^1(\Omega;{\mathbb
C})$,  we can apply direct method in the calculus of variations to
find the minimizer of $J_0$ in  $H_{a,D}^1 \times H^1(\Omega;
{\mathbb R}^2)$. Since  $H_{a,D}^1$ is both open and closed in
$H_{a}^1$,
 the minimizer in  $H_{a,D}^1 \times H^1(\Omega; {\mathbb R}^2)$
is also a local minimize of $J_0$ in  ${\mathcal{M}}_0$.
\end{proof}

 From \eqref{EZERO}, we get the Euler-Lagrange equations of the
  minimizer of $J_0$,
\begin{equation}\label{Phi0}
\begin{gathered}
\mathop{\rm div} \left[-{\frac i 2}(\psi^*\nabla\psi-\psi\nabla\psi^*)
- |\psi|^2A\right] =  0\quad\mbox{ in } \Omega\,,\\
\left[-{\frac i
2}(\psi^*\nabla\psi-\psi\nabla\psi^*)-
|\psi|^2A\right]\cdot{{\rm\bf n}} =  0 \quad\mbox{ on }\partial\Omega\,,
\end{gathered}
\end{equation}
and
\begin{equation}\label{Magnet0}
\begin{gathered}
 \mathop{\rm curl} \mathop{\rm curl} A  =
-\frac i 2 (\psi^*\nabla\psi- \psi\nabla\psi^*)-|\psi|^2 A
 \equiv j_0 \quad\mbox{in }\Omega \,, \\
 \mathop{\rm curl}  A  =  h_e {\rm\bf e_3}  \quad\mbox{on }\partial\Omega.
\end{gathered}
\end{equation}
Where $A=(A_1, A_2)$, ${\mathop{\rm curl} \mathop{\rm curl}} A =
(\partial_{x_2 x_1} A_2 -
\partial_{x_2 x_2} A_1, -\partial_{x_1 x_1} A_2 + \partial_{x_2 x_1} A_1)$.

 Note: Taking divergence on both sides of the second
equation (\ref{Magnet0}) in above, we could get the first equation
of (\ref{Phi0}) in distribution sense.

 Assume $(\psi,A)$ is under gauge (\ref{Gauge0}), from
\eqref{GaugeTrans}, we see that \eqref{Magnet0} becomes
\begin{gather*}
\triangle  A  = -\frac i 2 (\psi^*\nabla\psi-
 \psi\nabla\psi^*)-|\psi|^2 A
  \quad\mbox{in}\quad \Omega.\\
 \mathop{\rm curl}  A  =  h_e {\rm\bf e_3}  \quad\mbox{on}\quad\partial\Omega.\\
  A \cdot {\rm\bf n}  = 0 \quad\mbox{in}\quad\partial\Omega.
\end{gather*}
Since $\mathop{\rm div} A = 0$ in $\Omega$  and $ A\cdot {\rm\bf n} = 0$
on $\partial\Omega$, according to Poincar{\'e}'s lemma, rewrite $
A =(A_1, A_2) $ with $(A_2, - A_1)= \nabla\zeta $ for some  $\zeta
\in H^1_0(\Omega)$, from the above equation, $\zeta \in
W^{3,2}(\Omega) $,
 so that we obtain the following regularity result on $A$:

\begin{lem} \label{Lemma:Regularity}
If $(\psi_D,A_D)$ under gauge
(\ref{Gauge0}) is a minimizer in $ H_{a,D}^1\times
H^1(\Omega;{\mathbb R}^2)$, then $A_D \in  W^{2,2}(\Omega)$.
\end{lem}

Now we prove Theorem~\ref{Thm:H:intro} in the introduction, for the
convenience to read, let us restate it.

\begin{thm}\label{Thm:H}
Fix $h_e$. Let $(\psi_D,A_D)$ be a minimizer of $J_0$ in
$H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2)$
 under gauge  (\ref{Gauge0}),
define $h_D$ by $\mathop{\rm curl} A_D=h_D {\rm\bf e_3} $, then $
h_D \in V $ is the unique solution of
\begin{equation}\label{Heq}
\begin {gathered}
\int_{\Omega\setminus{\overline\Omega}_H}
 a^{-1} \nabla h \cdot \nabla f d x
+  \int_{\Omega} h f d x = \sum_{j=1}^n 2\pi d_j f_j \,. \\
\forall f(x) \in V \cap H_0^1(\Omega), \quad
 \mbox{and} \quad h - h_e \in V \cap H^1_0(\Omega).
\end{gathered}
\end{equation}
\end{thm}

\begin{proof} First we show $h_D \in V$.  From the
boundary condition, $h=h_e$ on $\partial \Omega$. By $\psi_D\in
H_{a,D}^1$ and  $\psi|_{\Omega_H} = 0$, \eqref{Magnet0} implies,
\begin{equation}\label{hIN}
\mathop{\rm curl} ( h_D {\rm\bf e_3} )  = 0 \quad \mbox{in }
\Omega_H.
\end{equation}
Hence  in $\Omega_j$,  $\nabla h_D = 0$, i.e.,
 $h_D = h_{D,j} $ a.e.,  where $ h_{D,j} $ is a constant depending
on $\Omega_j$, $ 1 \leq j \leq n$.  On $\Omega\setminus
{\overline\Omega}_H$, $\left|{\psi_D}\right| = \sqrt{a} \neq 0$,
we can write $\psi_D = \sqrt{a} e^{i \theta_D}$,  so that
\begin{equation}\label{Eq:HTemp}
\mathop{\rm curl} (h_D {\rm\bf e_3} )  = j_D = a(\nabla
\theta_D-A_D)
\quad \mbox{in } \Omega\setminus {\overline\Omega}_H \,.
\end{equation}
Since $\left|{(\nabla - i A_D) \psi_D}\right|^2= \left|{\nabla
\sqrt{a}}\right|^2 +
 \left|{\sqrt{a}(\nabla \theta_D-A_D) }\right|^2 $ and $J_0 (\psi_D,A_D) $
 is bounded,
 $a^{-1/2}|\nabla h_D| =\sqrt a |\nabla\theta_D-A_D|\in
L^2(\Omega)$, so that $h_D \in H^1(\Omega)$,  and $h_D \in V$.

 Now we prove  $h_D$  satisfies \eqref{Heq}.  Divide  on both
sides of \eqref{Eq:HTemp} by $a(x)$,  then take $\mathop{\rm curl}$
to annihilate $\nabla \theta_D$, then
 $\mathop{\rm curl} \frac 1 {a(x)} \mathop{\rm curl} (h_D  {\rm\bf e_3} ) = - \mathop{\rm curl} A_D
 = (0, 0, - h_D)$,  rewriting the equation, we obtain
\begin{equation}\label{hOUT}
 \nabla \cdot \frac 1 {a(x)} \nabla h_D = h_D \quad \mbox{in }
\Omega\setminus {\overline\Omega}_H
\end{equation}
in the sense of distributions.
Set
\[
\Omega_j^{\sigma} = \{x\in \Omega |
 \mathop{\rm dist}\{ { x }, \, {\Omega_j } \} < {\sigma} \},\quad
 \Omega^{\sigma}  = {\underset{j=1}{\overset n \cup }}
\Omega_j^{\sigma}.
\]
Since $a \in C^1({\overline\Omega} \setminus \Omega_H )$ and $a>0$
in ${\overline\Omega} \setminus \Omega_H$, $h_D\in
H_{\text{\rm loc}}^2(\Omega\setminus {\overline\Omega}_H )$, and
$\nabla\theta_D\in H_{\text{\rm loc}}^1(\Omega\setminus
{\overline\Omega}_H )$. Take the test function $f(x) \in
C^1(\Omega) \cap V \cap H_0^1(\Omega)$ for (\ref{hOUT}), and
integrate by parts,
\begin{align*}
&\int_{\Omega\setminus {\overline\Omega}^{\sigma}}
(a^{-1}(x)(\nabla h_D \cdot \nabla f ) + h_D(x) f(x)) d x\\
&= \int_{\partial(\Omega\setminus {\overline\Omega}^{\sigma} )} \frac
{\nu \cdot  \nabla h_D f} {a(x)} d s  \\
&= -\int_{\partial \Omega^{\sigma} }
 \frac {{\rm\bf n} \cdot  \nabla h_D f}{ a(x)} d s\\
&=  -\int_{\partial \Omega^{\sigma}}
 \frac {{\rm\bf n} \cdot  \nabla h_D f_j} {a(x)} d s -
 \int_{\partial \Omega^{\sigma}}
 \frac {{\rm\bf n} \cdot  \nabla h_D (f-f_j)} {a(x)} d s \,.
\end{align*}
Here ${\rm\bf n}$  is the outward normal of $\partial
\Omega^{\sigma}$, $\nu = -{\rm\bf n} $ is the inward normal.
Assume $\tau$ is the counterclockwise tangent vector field of $
\partial \Omega^{\sigma}$, use (\ref{Magnet0}) on $\partial
\Omega^{\sigma}$,
\begin{align*}
 -\int_{\partial \Omega^{\sigma} }
  \frac {{\rm\bf n} \cdot  \nabla h_D f_j} {a(x)} d s
&=  f_j \int_{\partial \Omega^{\sigma} } \frac {\tau \cdot \mathop{\rm curl} h_D} {a(x)}ds =
 f_j  \int_{\partial \Omega^{\sigma} }
 (\tau \cdot \nabla \theta_D - \tau \cdot A_D) ds \\
&= 2\pi d_j f_j -  f_j  \int_{ \Omega^{\sigma} } \mathop{\rm curl}
A_D d x
 = 2\pi d_j f_j -  f_j \int_{\Omega^{\sigma} } h_D dx \\
&=  2\pi d_j f_j - \int_{\Omega^{\sigma}} h_D f dx -
  \int_{\Omega^{\sigma}} h_D (f_j - f) dx \nonumber \\
&=  2\pi d_j f_j - \int_{ \Omega^{\sigma} } h_D f dx - o(1),
\quad \mbox{as }  \sigma \to 0.
\end{align*}
Using the Cauchy inequality,
$$
\big|\int_{\partial \Omega^{\sigma}}
 \frac {{\rm\bf n} \cdot  \nabla h_D (f-f_j)} {a(x)} d s\big|
\leq \Big(\int_{\partial \Omega^{\sigma} } \frac
{|{f-f_j}|^2} {a(x)} d s
     \Big)^{1/2}
 \Big(\int_{\partial \Omega^{\sigma} }
 \frac { \left|{\nabla h_D}\right|^2 } {a(x)} d s \Big)^{1/2}\,.
 $$
By Lemma~\ref{Lemma:2.3}, for $ d(x) \leq r_1$,
$$\int_{\partial \Omega^{\sigma} } \frac {\left|{f-f_j}\right|^2} {a(x)} d s
\leq c(r_1) \sigma \int_{\Omega^{g\sigma} \setminus
{\overline\Omega}_H }
 \frac {\left|{\nabla f}\right|^2} {a(x)} d x \,.
 $$
Because $\int_{\Omega^{\sigma} \setminus {\overline\Omega}_H}
\frac {\left|{\nabla h_D}\right|^2} {a(x)} d x \to 0$, as
$\sigma \to 0$, there is a sequence $ \{ \sigma_m\}_{m = 1}^{\infty}$,
$ \sigma_m \to 0$, $\sigma_{m+1}  < \sigma_m$,  and
 $ \int_{\partial \Omega^{\sigma_m}}
 \frac {  \left|{\nabla h_D}\right|^2 } {a(x)} d s \leq
\frac {c(a, \Omega )} {\sigma_m }$,  as $m \to \infty$, then
$$
\big|\int_{\partial \Omega^{\sigma_m}}
 \frac {{\rm\bf n} \cdot  \nabla h_D (f-f_j)} {a(x)} d s \big|
\leq c(a, \Omega) \Big( \int_{\Omega^{g \sigma_m}\setminus
{\overline\Omega}_H} \frac {\left|{\nabla f}\right|^2} {a(x)} d x
\Big)^{1/2} \to 0,
$$
as $\sigma_m \to 0$. Now we have
$$\int_{\Omega\setminus {\overline\Omega}^{\sigma_m}} a^{-1} (\nabla h_D \nabla f
 + h_D f) d x
= \sum_{j=1}^n  2\pi d_j f_j - \int_{  \Omega^{\sigma_m} } h_D f
dx - o(1)\,,$$
i.e.
$$
\int_{\Omega\setminus {\overline\Omega}^{\sigma_m} } a^{-1} \nabla
h_D \cdot \nabla f d x +  \int_{\Omega} h_D f d x = \sum_{j=1}^n
2\pi d_j f_j - o(1)\,.
$$
If $\sigma\in (\sigma_{m+1}, \sigma_{m})$,
\begin{align*}
&\int_{\Omega\setminus {\overline\Omega}^{\sigma}}
 a^{-1} \nabla h_D \cdot \nabla f d x
+  \int_{\Omega} h_D f d x \\
&= \int_{\Omega\setminus
{\overline\Omega}^{\sigma_m}} a^{-1} \nabla h_D \cdot \nabla f d x
+ \int_{\Omega} h_D f d x + \int_{ \Omega^{\sigma_m}\setminus
{\overline\Omega}^{\sigma}}
 a^{-1} \nabla h_D  \cdot \nabla f d x\,.
\end{align*}
As  $\sigma \to 0$,  $\mathop{\rm  meas}\{\Omega^{\sigma_m} \setminus
{\overline\Omega}^{\sigma} \} \to 0$, $\int_{ \Omega^{\sigma_m}
\setminus {\overline\Omega}^{\sigma} }
 a^{-1} \nabla h_D \cdot \nabla f d x \to 0$,  then
$$
\int_{\Omega\setminus {\overline\Omega}^{\sigma}} a^{-1} \nabla
h_D \cdot \nabla f d x +  \int_{\Omega} h_D f d x \to \sum_{j=1}^n
2\pi d_j f_j, \quad \mbox{as } \sigma\to 0\,.$$
 Hence the weak form of $h_D$ becomes
\begin{gather*}
 \int_{\Omega\setminus {\overline\Omega}_H}
 a^{-1} \nabla h \cdot \nabla f d x
+  \int_{\Omega} h f d x = \sum_{j=1}^n 2\pi d_j f_j\,, \\
  \quad \forall f(x) \in C^1(\Omega)\cap V \cap H_0^1(\Omega) \quad
 \mbox{and} \quad h -  h_e \in V \cap H^1_0(\Omega).
\end{gather*}
By Lemma~\ref{Lemma:2.2}, $ C^1(\Omega) \cap V \cap H_0^1(\Omega)$ is
dense in $  V \cap H_0^1(\Omega)$,  thus the above equation is
true for $\forall f(x)\in V \cap H_0^1(\Omega)$,
 i.e. we get (\ref{Heq}).

Now to prove the solution of (\ref{Heq})  is  unique.
 Assume that $h_1$ and $h_2$ are solutions,
then \mbox{ $h = h_1-h_2\in V\cap H_0^1(\Omega). $ } Apply $h$ as
a test function to the corresponding equations about $h_1$ and
$h_2$ respectively, then take their difference,
$$  \int_{\Omega\setminus {\overline\Omega}_H } a^{-1}
\left|{\nabla h}\right|^2 d x +  \int_{\Omega} h^2 d x = 0 \,,
$$
whence $h_1-h_2=0$ in $V$.
\end{proof}

\begin{lem}%[Existence and Uniqueness of eq~(\ref{Heq})]
\label{Lemma:EU} For fixed $h_e\in{\mathbb R} $ and  $D\in
{\mathbb Z}^n$, there is a unique solution for \eqref{Heq}.
\end{lem}

\begin{proof} Existence: For the given  $h_e$ and $D\in
{\mathbb Z}^n$, by Lemma~\ref{Lemma:Existence}, we can find
$(\psi_D,A_D)$ as the minimizer(i.e. a minimizer)
 of $J_0$  in $H^1_{a,D}\times H^1(\Omega;{\mathbb R}^2)$
under gauge (\ref{Gauge0}), apply Theorem~\ref{Thm:H}, we know
$h_D{\rm\bf e_3} = \mathop{\rm curl} A_D$ satisfying eq~(\ref{Heq}).
\\ Uniqueness is exactly the last part of Theorem~\ref{Thm:H}.
\end{proof}

Note that for any $h_e \in H^1(\Omega)$, Lemma~\ref{Lemma:EU} holds.

\begin{thm}\label{Thm:Structure}
For any fixed $h_e$ and $D = (d_1, \dots, d_n)\in {\mathbb Z}^n$,
$ J_0$ has a unique minimizer in the space $H^1_{a,D}\times
H^1(\Omega;{\mathbb R}^2) \subset   H^1_a\times
H^1(\Omega;{\mathbb R}^2)$ in the sense of gauge equivalence;
moreover, for any two such minimizers, say  $(\psi,A)$ and
$(\psi',A')$,  under gauge  (\ref{Gauge0}), then $ A = A'$ and $
\psi = \psi' e^{ic} $ for some $c\in {\mathbb R}$.
\end{thm}

\begin{proof} The existence follows  from Lemma~\ref{Lemma:Existence}.
 Uniqueness: By (\ref{GaugeDef}) and
(\ref{GaugeTrans}), we only need to  consider the situation under
gauge (\ref{Gauge0}). Without loss of generality, we assume  the
two minimizers $(\psi,A)$ and $(\psi',A')$ in $H^1_{a,D}\times
H^1(\Omega;{\mathbb R}^2)$ are under gauge (\ref{Gauge0}),  so
that $\mathop{\rm div} A = \mathop{\rm div} A' = 0$ and $A\cdot {\rm\bf n}
= A' \cdot {\rm\bf n} = 0$, according to  Poincar{\'e}'s lemma, we
have $ A-A' = (- \frac{\partial \zeta } {\partial y }, \frac
{\partial \zeta} {\partial x} ) $ for some
 $\zeta \in H^1(\Omega)$,  where $(x,y)$ are the coordinates
in $2-$dim. we can also derive that $\zeta $ is constant on
$\partial \Omega$ from  $ (A-A')\cdot {\rm\bf n} = 0$.
 Through Theorem~\ref{Thm:H}, we get $\mathop{\rm curl} A = \mathop{\rm curl} A'$,
which implies $\triangle   \zeta = 0 $ in $\Omega$,  thus $\zeta $
is constant on $\Omega$,  i.e. $A = A'$. Then by \eqref{Magnet0},
we have
 $j_0 = j_0'$, so  $\nabla \theta=\nabla \theta'$ in
$\Omega \setminus {\overline\Omega}_H$,  i.e. $ e^{i\theta -
i\theta'} = e^{ic}$,   for some $c\in {\mathbb R}$,
 hence $\psi=\psi'e^{ic}$,  and We have
 proved the later part of the theorem.
If take $\phi = c = \mbox{constant}$,  we then have  $ (\psi',A')
= G_{\phi} (\psi,A)$, i.e. they are gauge equivalent.
\end{proof}

We need to mention that Theorem~\ref{Thm:Structure} is a generalization
of the Theorem~{3.2} in~\cite{ABP} under our setting.

Now suppose $(\psi_D,A_D)$ is a critical point of $J_0$,  i.e. a
solution of \eqref{Phi0} and \eqref{Magnet0}, then from the first
part of Theorem~\ref{Thm:H}, $ h_D $ in $\mathop{\rm curl} A_D =h_D
{\rm\bf e_3} $ is the unique solution of \eqref{Heq}, hence
$(\psi_D,A_D)$ is a local minimizer of $J_0$ in
${{\mathcal{M}}}_0$. On the other hand, by Lemma~\ref{Lem:1.4}, every
local minimizer $(\psi,A)$ belongs to $ H^1_{a,D}\times
H^1(\Omega;{\mathbb R}^2 ) $ for some $D\in {\mathbb Z}^n$,  hence
it satisfies \eqref{Phi0} and \eqref{Magnet0}, and by
Theorem~\ref{Thm:Structure}, it is gauge equivalent to the minimizer in
$ H^1_{a,D}\times H^1(\Omega;{\mathbb R}^2 )$. Thus we have the
following statement.


\begin{cor}\label{Cor:Critic}
All critical points of $J_0$ are local minimizers in $
H^1_a\times H^1(\Omega;{\mathbb R}^2)$.
\end{cor}

As is easy to see that if we obtain the solution $h_D$ of
\eqref{Heq}, then we can recover $A_D$ with the condition
$\mathop{\rm div} A_D=0$ in $\Omega$, and recover $\psi_D$ from
\eqref{Magnet0}, so that \eqref{Heq} describes the limit system
completely.

\section{Properties Of The Solutions Of The Limit Equation}

Consider the $n+1$ functions in $V\cap H^1_0(\Omega)$,
 $\{ \eta_0, \eta_1, \dots \eta_n\}$ satisfying
\begin{equation}\label{Eta0}
\begin{gathered}
\int_{\Omega\setminus{\overline\Omega}_H} a^{-1} \nabla \eta_0
\cdot \nabla f d x +  \int_{\Omega} \eta_0 \, f d x =
0 \,,\\
 \forall f(x) \in V \cap H_0^1(\Omega)\quad
 \mbox{and} \quad \eta_0 = 1  \mbox{ on } \partial \Omega
\end{gathered}
\end{equation}
and
\begin{equation}\label{Etan}
\begin{gathered}
 \int_{\Omega\setminus{\overline\Omega}_H
} a^{-1} \nabla \eta_j \cdot \nabla f d x +  \int_{\Omega} \eta_j
f d x = 2\pi f_j \,,\\
\forall f(x) \in V \cap H_0^1(\Omega)\quad
 \mbox{and} \quad \eta_j = 0  \mbox{ on } \partial  \Omega\,,
 \; j = 1, \dots, n\,.
\end{gathered}
\end{equation}
The existence and uniqueness of solutions in $V$
 for both \eqref{Eta0} and \eqref{Etan} follows
the result in Lemma~\ref{Lemma:EU}. We can use them to represent the
solution of (\ref{Heq}).

\begin{thm}\label{thm:regularity}
Fix $h_e\in{\mathbb R} $,  $D\in {\mathbb Z}^n$. If $h_D$ solves
\eqref{Heq}, then  $h_D\in C({\overline\Omega})$ and
\begin{equation}\label{Eq:hD}
h_D=\sum^n_{j=1} d_j\eta_j + h_e \eta_0\,.
\end{equation}
Moreover, if $\alpha_k>1, k=1,2,\dots,n$, then $h_D \in
C^{1}({\overline\Omega})$, where $\alpha_k$ is from
\eqref{Eq:ApproxAx}.
\end{thm}

 The proof of Theorem~\ref{thm:regularity} is a consequence
 of properties of $\eta_0$ and $\eta_j, j=1, 2, \dots, n$. We
 will postpone it to the end of this section.
We first discuss some properties of $\{\eta_1, \dots \eta_n\}$.

\noindent {\bf Property(i)} $\eta_j \geq 0 $ in $\Omega$, $1\leq j \leq n$.

\noindent {\bf Property(ii)} $\eta_1, \dots, \eta_n $ are
linear independent in $ V \cap H^1_0(\Omega)$,  i.e.,
$\sum_{j=1}^n w_j \eta_j \equiv 0 $ for $w_j \in {\mathbb R},
1\leq j \leq n$, if and only if $w_j = 0, 1\leq j \leq n$.

\begin{proof}
To prove this property (i), we use the  test function
$f = \min\{\eta_j,0\}$ in (\ref{Etan}) and obtain
$$\int_{\Omega\setminus{\overline\Omega}_H }
a^{-1} \left|{\nabla f}\right|^2 d x +
\int_{\Omega}\left|{f}\right|^2 d x \leq 0.
$$
So that $f \equiv 0$, i.e.,
 $ \eta_j \geq 0 $ in $\Omega$ for $1\leq j \leq n$,
and $(i)$ is proved.

Assume  $g = \sum_{j=1}^n w_j \eta_j \equiv 0 $ for  some  $w_j
\in {\mathbb R},  1\leq j \leq n$.
 From  (\ref{Etan}),  $g$ satisfies  the equation
\begin{equation}\label{Eq:G0}
\begin{gathered}
\int_{\Omega\setminus{\overline\Omega}_H } a^{-1} \nabla g \cdot
\nabla f d x +  \int_{\Omega} g  f d x = 2\pi \sum_{j=1}^n w_j f_j
\equiv 0\,,
 \\
\forall f(x) \in V \cap H_0^1(\Omega),\quad
 \mbox{and} \quad g = 0 \quad \mbox{on } \partial \Omega\,.
\end{gathered}
\end{equation}
Fix $ k \in \{ 1, \dots, n \}$. Choose $ \sigma, m $ such that
$\Omega_k^{\sigma} \cap \Omega_H = \Omega_k $ and $ m > 2/\sigma$.
Set $\chi_k^{\sigma}$ as the characteristic function of
 $\Omega_k^{\sigma}$, $\Omega_k^{\sigma} =
 \{ x\in \Omega | \mathop{\rm dist}\{ { x } ,\, { \Omega_k } \} < \sigma \}$.
Let  $f^{k} \equiv \chi_k^{\sigma} * \rho_m (x)$, where  $ \rho_m
(x) =  m^2 \rho ( m x)$,
 $\rho(x)$ is defined as in
\eqref{Eq:MOLLIFIER}. Then $ f^{k} \in V\cap H_0^1(\Omega)$,
$f^k_k = 1$ and  $f^k_j = 0$ if $k\neq j$,  for $1\leq j \leq n$.
Apply $f^{k}$ as a test function for \eqref{Eq:G0},
 we have $w_k = 0$, for $ k =1, 2, \dots, n$.
Thus  $\eta_1, \dots, \eta_n $ are linear independent in $ V \cap
H^1_0(\Omega)$,  we have $(ii)$.
\end{proof}

\noindent{\bf Property (iii)} Assume  $\eta_j^k $ is the value of
$\eta_j$ on $\Omega_k$, then $ \eta_j^j =
 \mathop{\rm ess\,sup}{}_ {\Omega} \eta_j <  2\pi/\mathop{\rm  meas}
 \{\Omega_j\}$ and $\eta_j^j > \eta_j^k $ for $k\neq j$,  $1 \leq k, j \leq n$.

\begin{proof} Using $ f =  \eta_j $
as a test function  for \eqref{Etan}, then
$$
2 \pi \eta_j^j = \int_{\Omega\setminus{\overline\Omega}_H }
a^{-1} \left|{\nabla \eta_j }\right|^2 d x +
\int_{\Omega}\left|{\eta_j}\right|^2 d x> 0.
$$
On the other hand, Using $ f = (\eta_j - \eta_j^j)_{+} $ as
a test function for \eqref{Etan}, then
$$
\int_{\Omega\setminus{\overline\Omega}_H }
a^{-1} \left|{\nabla (\eta_j - \eta_j^j)_{+} }\right|^2 d x +
\int_{\Omega}\eta_j (\eta_j - \eta_j^j)_{+} d x =0,
$$
so that
$(\eta_j - \eta_j^j)_{+}  = 0 $ a.e., i.e.
$ \eta_j^j =\mathop{\rm ess\, sup}{}_{\Omega}  \eta_j$. By using test
functions from $C_0^1(\Omega\setminus{\overline\Omega}_H)$ in
\eqref{Etan}, we see that
\begin{equation} \label{Eq:LOCAL}
\begin{gathered}
 - \nabla \cdot a^{-1} \nabla \eta_j + \eta_j = 0 \quad\mbox{in }
  \Omega\setminus{\overline\Omega}_H, \\
\eta_j = 0 \quad \mbox{on } \partial \Omega, \mbox{ for }
1 \leq j \leq n.
\end{gathered}
\end{equation}
Since $\eta_j$ is nonconstant, by the maximum principle  in
\eqref{Eq:LOCAL}, there is no local  maxima for $\eta_j$ in
$\Omega\setminus{\overline\Omega}_H$.  If $\eta_j^k = \eta_j^j$
for some $k \neq j$, fix $ \sigma $ such that $\Omega_k^{\sigma}
\cap \Omega_H = \Omega_k$.  Let $ c_k = \mathop{\rm ess\,sup}{}_
{\Omega_k^{\sigma}\setminus \Omega_k^{\sigma/2}}\eta_j$,  then
 $c_k <  \eta_j^j =  \mathop{\rm ess, sup}{}_{\Omega} \eta_j$.

 Using $ f = \chi_k^{\sigma} (\eta_j - c_k)_{+}
\not\equiv 0$ as a test function in \eqref{Etan}, we have
$$
0 = \int_{\Omega_k^{\sigma}\setminus{\overline\Omega}_k  }
a^{-1} \left|{\nabla f }\right|^2 d x + \int_{\Omega_k^{\sigma}} f
\eta_j d x> 0,
$$
a contradiction, so that
 $\eta_j^j > \eta_j^k$ for $k \neq j$, $1 \leq k, j \leq n$.
Using $\eta_j$ as a test function in \eqref{Etan}, we have
$(\eta_j^j)^2  \mathop{\rm meas}\{\Omega_j\} < 2\pi\eta_j^j, $ so that $
\eta_j^j < 2\pi/ \mathop{\rm meas}\{\Omega_j\}$. Therefore,
(iii) is proved.
\end{proof}

\noindent{\bf Property (iv)} $\eta_j\in C^0({\overline\Omega})\cap V$,
$j =1,2,\dots, n$. %\label{Property:iv}

\begin{proof}
 By \eqref{Eq:LOCAL} and
$ a^{-1} \in  C^1({\overline\Omega} \setminus \Omega_H)$,
we apply the standard estimate for the weak solution of an
elliptic equation (say \cite{GT} theorem 8.8 at page 183 and
theorem 8.12 at page 186), $ \eta_j(x)$ in $H^2(\Omega^\prime)$,
for any  $\Omega^{\prime} \subset\subset
{\overline\Omega}\setminus{\overline\Omega}_H$, by Sobolev
embedding, $\eta_j(x) \in
C^0({\overline\Omega}\setminus{\overline\Omega}_H)$. Since $
\eta_j(x) $ is a bounded constant in ${\overline\Omega}_k, 1\leq k
\leq n$, with $\eta_j \in H^1({\overline\Omega})$, i.e.,
\begin{equation}\label{CONTINUITY}
 \eta_j(x) \in
H^1({\overline\Omega})\cap
C^0({\overline\Omega}\setminus{\overline\Omega}_H).
\end{equation}
We show   $\eta_j(x)$ is $C^0$ on the boundary of $\Omega_H$.
Since $\Omega_k$ is Lipschitz, $k=1,2,\dots, n$, there is
a constant $\sigma_k < r_1$, for any $x_0 \in \Omega\setminus\Omega_H$
 with $d =\mathop{\rm dist}\{{x_0}, \, {{\overline\Omega}_k}\} < \sigma_k$,
 we can find a rectangle with sides parallel to the local coordinate
 axes, its top $\overline{x_0 x_1}$, above the boundary graph and its
bottom $\overline{y_0 y_1}$, below the graph, $ \mathop{\rm dist}\{
{x_0}, \, {x_1} \} = d^{1+\alpha_k/3}$, $\mathop{\rm dist}\{ {x_0}, \,
{y_1} \} = c_k d$, where $c_k$ is a constant depending on
$\Omega_k$, $\overline{x_0 x_1}$ represents the line segment
starting at $x_0$ and ending at $x_1$, $\overline{y_0 y_1}$ is the
line segment starting at $y_0$ and ending at $y_1$.  Note the
rectangle is not unique, but it does not matter.

 \noindent {\bf Claim:} For all $f\in
C^{0}({\overline\Omega}\setminus{\overline\Omega}_H)\cap V$,
\[
d^{-1-\alpha_k/3}
 \int_{x_0}^{x_1} \left|{f(x_s) - f_k}\right| d H^1
 \leq b_k d^{\alpha_k/3}(x) \|f\|_V ,
 \]
where the integral is in ${\overline{x_0 x_1}}$, and $ x_s = (1-s)
x_0 + s x_1$, and $ s\in [0,1]$.

\noindent{\bf Proof of the Claim:}
Assume $f\in C^1(\Omega)\cap V$, then $ f(y) = f_k \mbox{ in }
\overline{y_0 y_1} $  and
$$
\int_{x_0}^{x_1} \left|{f(x_s) - f_k}\right| d H^1
=\int_{x_0}^{x_1} \left|{f(x_s) - f(y_s)}\right| d H^1
= \int_{x_0}^{x_1}\Big|{ \int_{x_s}^ {y_s}
  \nabla f \cdot {\rm\bf n}_{x_s y_s} d H^1}\Big| d H^1 \,,
$$
where ${\rm\bf n}_{x_s y_s}$ is the unit vector from $x_s$ to
$y_s$. So that
\begin{align*}
  \int_{x_0}^{x_1} \left|{f(x_s) - f_k}\right| d H^1
&\leq \int_{x_0}^{x_1}\int_{x_s}^ {y_s}
 \left|{\nabla f}\right|  d x \\
 & \leq \left(
 \int_{x_0}^{x_1} \int_{x_s}^ {y_s}
 a^{-1} \left|{\nabla f}\right|^2  d x \right)^{1/2} \cdot
 \left( \int_{x_0}^{x_1} \int_{x_s}^ {y_s}
 a(x)  d x \right)^{1/2} \\
&\leq \|f\|_V \left( \int_{x_0}^{x_1} \int_{x_s}^ {y_s}
 C_1 d^{\alpha_k}  d x  \right)^{1/2} \\
&\leq b_k d^{1+2\alpha_k/3} \|f\|_V \,,
\end{align*}
here $b_k= C_1 c_k$. From the above inequality,
$$
d^{-1-\alpha_k/3} \int_{x_0}^{x_1} \left|{f(x_s) - f_k}\right| d H^1
 \leq b_k d^{\alpha_k/3} \|f\|_V \,.
$$
Because  $C^1(\Omega)\cap V$ is dense in $C^0(\Omega)\cap V$, we
have proved the claim. \smallskip

Now we continue the proof of Property (iv).
Since $ \eta_j(x) \in
C^{0}(\Omega_k^{\sigma_k} \setminus {\overline\Omega}_k)$, for
every $x_0 \in \Omega_k^{\sigma_k} \setminus {\overline\Omega}_k$,
we have
\begin{align*}
&\left|{\eta_j(x_0) - \eta_j^k}\right|\\
&= \leq d^{-1-\alpha_k/3}
\int_{x_0}^{x_1} \left|{\eta_j(x_s) - \eta_j^k}\right| d H^1
  +  d^{-1-\alpha_k/3} \int_{x_0}^{x_1}
\left|{\eta_j(x_s) - \eta_j(x_0)}\right| d H^1\\
&\leq  b_k d^{\alpha_k/3} \|\eta_j \|_V +
 d^{-1-\alpha_k/3} \int_{x_0}^{x_1}
\left|{\eta_j(x_s) - \eta_j(x_0)}\right| d H^1  ,
\end{align*}
To estimate $ d^{-1-\alpha_k/3} \int_{x_0}^{x_1}
\left|{\eta_j(x_s) - \eta_j(x_0)}\right| d H^1$, consider
\eqref{Eq:LOCAL} in the ball $B_{d/2}(x_0)$. \\ Scaling
$B_{d/2}(x_0)$ to a unit ball, \eqref{Eq:LOCAL} becomes
$$
- 4d^{-2} \nabla_{y} \cdot a^{-1}(y d/2 + x_0) \nabla_y \eta_j = -
\eta_j \quad \mbox{in } B_1(0)\,.
$$
Apply H{\"o}lder estimate  \cite[theorem 8.22 page 200]{GT}
in the  ball $B_{2 d^{\alpha_k/3}}(0)$ for the dilated equation,
we have $\mbox{ osc } \eta_j \leq C  d^{\beta \alpha_k/3}
\|\eta_j\|_{L^{\infty}}$, where $C$ depends on $\beta, \Omega_k$
and $\alpha_k$, $\beta\in (0,1)$.  By  Property (iii),
$\|\eta_j\|_{L^{\infty}} \leq 1/ \mathop{\rm meas}\{\Omega_j\}$, so that
$\left|{\eta_j(x_s) - \eta_j(x_0)}\right| \leq C(\Omega_k,
\alpha_k) d^{\beta \alpha_k/3 }$, i.e.,
\begin{equation} \label{eq:calphak}
\left|{\eta_j(x_0) - \eta_j^k}\right| \leq b_k d^{\alpha_k/3}
\|\eta_j \|_V + C(\Omega_k,\alpha_k)  d^{ \beta \alpha_k/3}
 \leq \tilde b_k d^{ \beta \alpha_k/3 } (1 +
\|\eta_j \|_V),
\end{equation}
 for all $x_0 \in \Omega_k^{\sigma_k} \setminus {\overline\Omega}_k $.
Since $d(x)\to 0$ as $x\to \partial  \Omega_k$, $\eta_j(x) \to
\eta_j^k $  at $\partial  \Omega_k$, i.e., $\eta_j(x)$  is
continuous at $\partial  \Omega_k$.  Combining this with
(\ref{CONTINUITY}), we have
 $\eta_j \in C^0({\overline\Omega})$, and
Property (iv) is proved.
 \end{proof}

{\bf Property (v)} %\label{Property:v}
$\eta_j > 0 $ a.e.
in $\Omega$, $j=1,2,\dots,n$. Hence $\eta_j^k > 0$,
  $1 \leq k, j \leq n$.

\begin{proof} We prove that
 $ \mathop{\rm meas}\{x \in {\overline\Omega}| \eta_j(x) = 0\} = 0$
for $ j = 1,2,\dots, n$.  By (iv), it makes sense to talk
about the level set of $ \eta_j(x)$.  Let $\Gamma^j_0 = \{x \in
{\overline\Omega}| \eta_j(x) = 0\}$, $ \Gamma^j_{\delta} = \{x \in
{\overline\Omega}| \eta_j(x) \leq \delta\}$, then  $\Gamma^j_0
\subset \Gamma^j_{\delta}$, $ \mathop{\rm meas}\{ \Gamma^j_{\delta} \setminus
\Gamma^j_0 \} \to 0 $ as $\delta \to 0$. By $\eta_j^j > 0$,
${\overline\Omega}_j \cap \Gamma^j_{\delta} = \emptyset$
 for $\delta < \eta_j^j$. In the proof, we  always
assume  $\delta < \eta_j^j$, i.e., $ {\overline\Omega}_j \cap
\Gamma^j_{\delta} = \emptyset$.

Denote $\chi_{\Gamma^j_{\delta}}$ as the characteristic function of the
set $ \Gamma^j_{\delta}$, $\delta \geq 0$.  Set
\[
g_{\delta}(v) = \begin{cases} v, & v \leq \delta/2 \\
(\delta - v)_{+}, & v >  \delta/2
\end{cases}
\quad \mbox{and}\quad
h_{\delta}(v) =  \begin{cases} v/2, & v \leq 2\delta/3 \\
(\delta - v)_{+}, & v > 2\delta/3\,.
\end{cases}
\]
Using $f_{\delta} =  \chi_{\Gamma^j_{\delta}} \ h_{\delta}(\eta_j)
\in H_0^1(\Omega)\cap V $
 as a test function in \eqref{Etan}, we have
$$
\int_{\Gamma^j_{\delta}\setminus {\overline\Omega}_H}
 a^{-1} h'_{\delta}(\eta_j) \left|{ \nabla \eta_j  }\right|^2 d x +
\int_{\Gamma^j_{\delta}} \eta_j f_{\delta} d x = 0.
$$
 By the sign of $h'_{\delta}$ in $ \Gamma^j_{\delta} $,
\begin{align*}
\int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup
\Gamma^j_{2\delta/3})}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
&= 1/2 \int_{\Gamma^j_{2\delta/3}\setminus {\overline\Omega}_H}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x +
 \int_{\Gamma^j_{\delta}} \eta_j f_{\delta} d x \\
&\leq  1/2 \int_{\Gamma^j_{2\delta/3}\setminus
{\overline\Omega}_H}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x +
 \int_{\Gamma^j_{\delta}} \left|{\eta_j}\right|^2 d x\,.
\end{align*}
 Do the same thing in  $\Gamma^j_{2/3 \delta},
\Gamma^j_{2^2/3^2 \delta}, \Gamma^j_{2^3/3^3 \delta}, \dots$, to get
\begin{align*}
\int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup
\Gamma^j_{2\delta/3})}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
&\leq \sum_{k=0}^{\infty} 2^{-k} \int_{\Gamma^j_{ 2^k\delta / 3^k
}} \left|{\eta_j}\right|^2 d x \\
& < \sum_{k=0}^{\infty} 2^{-k} \int_{\Gamma^j_{\delta}}
\left|{\eta_j}\right|^2 d x
 = 2 \int_{\Gamma^j_{\delta}} \left|{\eta_j}\right|^2 d x \,.
 \end{align*}
Similarly,
\[
\int_{\Gamma^j_{2\delta/3}
\setminus({\overline\Omega}_H\cup \Gamma^j_{4\delta/9})}
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
< 2 \int_{\Gamma^j_{2\delta/3}}\left| {\eta_j}\right|^2 d x \,.
\]
Summing the above two equations, we have
$$
\int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup
\Gamma^j_{4\delta/9})}
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
< 4 \int_{\Gamma^j_{\delta}} \left| {\eta_j}\right|^2  d x \,.
$$
Use the test function $\chi_{\Gamma^j_{\delta}} \
g_{\delta}(\eta_j)$ in \eqref{Etan}, get
$$
\int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup \Gamma^j_{\delta/2})}
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
=   \int_{\Gamma^j_{\delta/2}\setminus {\overline\Omega}_H}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x +
\int_{\Gamma^j_{\delta}} \eta_j g_{\delta}(\eta_j) d x\,,
$$
i.e.,
\[
\int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup
\Gamma^j_{\delta/2})}
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
> \int_{\Gamma^j_{\delta/2}\setminus {\overline\Omega}_H}
  a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x \,.
\]
So that
\begin{align*}
\int_{\Gamma^j_{\delta}\setminus{\overline\Omega}_H }
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
&\leq
 2 \int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup \Gamma^j_{\delta/2})}
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x \\
&\leq 2 \int_{\Gamma^j_{\delta}\setminus({\overline\Omega}_H\cup
\Gamma^j_{4\delta/9})} a^{-1} \left|{ \nabla \eta_j  }\right|^2 d
x \leq 8 \int_{\Gamma^j_{\delta}} \left|{\eta_j}\right|^2 d x \,.
\end{align*}
 Consider the function $g(x)= \chi_{\Gamma^j_{\delta}}
(\delta - \eta_j)_{+}\in H^1(\Omega)$, then $\nabla g = \nabla
\eta_j$ a.e. in $ \Gamma^j_{\delta}$. By $g \equiv 0 $ on
$\Omega_j$, and $ \mathop{\rm meas}\{\Omega_j\}> 0$, we can apply Sobolev
inequality to $g(x)$, then
\begin{align*}
\int_{\Gamma^j_{\delta}} \left|{g}\right|^2 d x&\leq c(\Omega)
\int_{\Gamma^j_{\delta}} \left|{\nabla g}\right|^2 d x = c(\Omega)
\int_{\Gamma^j_{\delta}} \left|{\nabla \eta_j }\right|^2 d x \\ &
\leq c(a, \Omega)
\int_{\Gamma^j_{\delta}\setminus{\overline\Omega}_H }
 a^{-1} \left|{ \nabla \eta_j  }\right|^2 d x
\leq c(a, \Omega) \int_{\Gamma^j_{\delta}} \left|{\eta_j}\right|^2
d x\,.
\end{align*}
Therefore,
$$
\delta^2  \mathop{\rm meas}\{\Gamma^j_0\} \leq
\int_{\Gamma^j_{\delta}} \left|{g}\right|^2 d x \leq c(a, \Omega)
\int_{\Gamma^j_{\delta}} \left|{\eta_j}\right|^2 d x \leq  c(a,
\Omega) \delta^2  \mathop{\rm meas}\{\Gamma^j_{\delta}\setminus \Gamma^j_0
\}\,,
$$
i.e., $ \mathop{\rm meas}\{\Gamma^j_0\} \leq c(a, \Omega)
 \mathop{\rm meas}\{\Gamma^j_{\delta}\setminus \Gamma^j_0 \} \to 0$,
 as $\delta \to 0$,
thus $ \mathop{\rm meas}\{\Gamma^j_0\} = 0$, $ \eta_j > 0 $ a.e..
Since $ \mathop{\rm meas}\{\Omega_k\}> 0$, for $ 1\leq k \leq n$,
and $ \eta_j>0 $ a.e.,
 $\eta_j^k = \frac {\int_{\Omega_k} \eta_j(x) d x} { \mathop{\rm meas}\{\Omega_k\}}
  > 0$.
So that Property (v) is proved.
\end{proof}

\noindent{\bf Property (vi)} % \label{Property:vi}
For every domain $G_k$ with $G_k\cap
{\overline\Omega}_H={\overline\Omega}_k$,
$\eta_j^k < \sup_{G_k} \eta_j$, where $k\neq j$,  $k, j = 1,2, \dots, n$.

\begin{proof} If $\eta_j$ is constant on any subdomain of
 $\Omega\setminus{\overline\Omega}_H$, then $\nabla\eta_j$
 is zero, from \eqref{Eq:LOCAL},
$\eta_j$ is also zero, which contradict with $(v)$. So that,
$\eta_j$ is not a constant on any subdomain of
 $\Omega\setminus{\overline\Omega}_H$.
We use contradiction to prove Property (vi). Assume $\eta_j^k =
\sup_{G_k} \eta_j$, for some $G_k$ as in (vi).
 Set
\[
\sigma < \mathop{\rm dist}\{ {\partial G_k },  {{\overline\Omega}_k} \},
\quad c_k =\mathop{\rm ess\,sup}_{\Omega_k^{\sigma} \setminus
 {\overline\Omega}_k^{\sigma/2}} \eta_j,
\]
 use $f = \chi_{\Omega_k^{\sigma}} (\eta_j - c_k)_{+}$
 as a test function in \eqref{Etan}, for $k\neq j$,
\[
\int_{\Omega_k^{\sigma} \setminus {\overline\Omega}_k} a^{-1}
\left|{\nabla f }\right|^2 d x + \int_{ \Omega_k^{\sigma} } f
\eta_j d x = 0
\]
 i.e., $ f\equiv 0,  \eta_j^k = c_k >0$. So that
$\eta_j$ achieves its nonzero local maximum in $\Omega_k^{\sigma}
\setminus {\overline\Omega}_k^{\sigma/2}$, which contradicts with
the maximum principle applicable to \eqref{Eq:LOCAL}, hence
Property (vi) holds.
\end{proof}

\noindent{\bf Property (vii)}
If $\alpha_k>1$, $k=1,2,\dots,n$, then $\eta_j\in C^{1}({\overline\Omega})$,
$j=1,2,\dots,n$, where $\alpha_k$ is from \eqref{Eq:ApproxAx}.

\begin{proof}
Without loss of generality, we write
$\eta_j$ as $\eta$ in the proof. The $C^1$ continuity of $\eta$ in
${\overline\Omega}\setminus{\overline\Omega}_H$ is from the
standard elliptic argument (See \cite{GT}, theorem 8.33 on page
210). We focus on the proof of the $C^1$ continuity of $\eta$
close to $\Omega_H$, and show that $\left|{\nabla \eta}\right|$ is
forced to 0 as $x$ close $\partial\Omega_H$.  For $T\leq
\eta_j^j$, denote $\Sigma_T=\{x|\eta(x)\geq T\}$, use the test
function $(\eta-T)_{+}$ in (\ref{Etan}), then
$$
\int_{\Sigma_T\setminus{\overline\Omega}_H }
a^{-1} \left|{\nabla \eta}\right|^2 d x +  \int_{\Sigma_T}\eta
(\eta-T) d x = 2\pi(\eta_j^j-T).
$$
 Apply the co-area formula (see \cite{EG}),
\begin{equation}\label{Eq:coarea}
\int_{\{\eta=T\}\setminus{\overline\Omega}_H } a^{-1}
\left|{\nabla \eta}\right| d H^1(x) +  \int_{\Sigma_T}\eta d x =
2\pi .
\end{equation}
For any point $x_0$ close to $\Omega_k$ with $d=\mathop{\rm dist}\{
{x_0}, \, {\Omega_k} \} \leq r_1$,   (\ref{Etan}) becomes
\eqref{Eq:LOCAL} in $B_{d/2}(x_0)$, with $a(x)$ of order
$d_k^{\alpha_k}(x)$, where $d_k(x)=\mathop{\rm dist}\{ {x}, \,
{\Omega_k} \}$. Scaling $B_{d/2}(x_0)$ to the unit ball $B_1(0)$,
write $\tilde {\eta}(y) =\eta(dy/2 + x_0)$, then \eqref{Eq:LOCAL}
can be simplified as
$$
-\triangle _y (\tilde \eta-\eta(x_0)) + \frac d 2 \frac {\nabla_x
a} {a} \cdot \nabla_y (\tilde \eta -\eta(x_0)) =- a \big(\frac d
2 \big) ^2 \tilde \eta, \qquad \mbox{ in } B_1(0).
$$
Apply H{\"o}lder estimate  \cite[Theorem 8.32 page 210]{GT}
in the  ball $B_{1/2}(0)$ for the above dilated equation, then
$\tilde \eta \in C^{1, \beta}(B_{1/2}(0))$, $\forall\beta \in
(0,1)$, and  $\exists\tilde C $ depending only on $\alpha_k, C_1$,
such that
$$
\left|{\nabla_y \tilde \eta }\right|_{ C^{0, \beta}(B_{1/2}(0))}
\leq \tilde C \Big( \left|{\tilde \eta - \eta(x_0)
}\right|_{C^0(B_{1}(0))} + \Big|{a(x_0) (\frac d 2)
^2 \tilde \eta}\Big|_{C^0(B_{1}(0))} \Big).
$$
Fix $\beta$. From \eqref{Eq:ApproxAx}, $a(x_0)$ is bounded by
 $d^{\alpha_k}$. Pull back to $B_{d/2}(x_0)$, then
\begin{equation}\label{eq:bddetak}
 \Big|{ \big(\frac d 2\big) ^{1+\beta} \nabla_x \eta }\Big|_{
 C^{0,\beta}(B_{d/4}(x_0))}  \leq \tilde C \left( \left|{\eta - \eta(x_0)
}\right|_{C^0(B_{d/2}(x_0))} +  d^{\alpha_k+2}  \right)\,.
\end{equation}
Let $T=\eta(x_0), M=\left|{ \nabla \eta (x_0)}\right|$,we iterate
to obtain the bound on $\nabla \eta (x_0)$.
 First by the uniform bound of $\eta$ and \eqref{eq:bddetak},
 $\left|{ \nabla \eta }\right|_{ C^{0,\beta}(B_{d/4}(x_0))}  \leq \tilde C_1
 d^{-1-\beta}$,
 so that $\left|{ \nabla \eta (x)}\right|>\frac {M} {2}$ for $x\in
B_{\tilde r_1}(x_0)$, where $\tilde r_1= \big(\frac {M} {2 \tilde
C_1 }\big)^{1/\beta} d^{1+1/\beta}$.

 By \eqref{Eq:coarea},
$\int_{\{\eta=T\}\cap B_{\tilde r_1}(x_0)} a^{-1} \left|{\nabla
\eta}\right| d H^1(x) \leq 2\pi$, then
  $\frac M {2d^{\alpha_k}}  \mathop{\rm meas}\{\{\eta=T\}\cap B_{\tilde r_1}(x_0)\}
  \leq 2\pi$.
 If $ \mathop{\rm meas}\{\{\eta=T\}\cap B_{\tilde r_1}(x_0)\} \geq \tilde r_1$, then
 $\frac {M^{1+1/\beta} d^{1+1/\beta}}{2d^{\alpha_k}(2\tilde C_1)^{1/\beta}} \leq 2 \pi$,
  i.e. $M \leq \tilde C_2 d^{\beta\alpha_k/(1+\beta)-1}$.

If $ \mathop{\rm meas}\{\{\eta=T\}\cap B_{\tilde r_1}(x_0)\} <
 \tilde r_1$, by the continuity of $\eta$ and the intermediate value theorem,
 $\{\eta=T\}\cap B_{\tilde r_1}(x_0)$ will be a closed curve inside
 $B_{\tilde r_1}(x_0)$, we use $(T-\eta)_{+}\chi_{B_{\tilde r_1}(x_0)}$
 as the test function in (\ref{Etan}),  then
 $$\int_{\{\eta\leq T \}\cap B_{\tilde r_1}(x_0) }
\left( - a^{-1} \left|{\nabla \eta }\right|^2 + \eta (T-\eta)
\right) d x = 0\,,$$ so that \begin{equation*}\begin{split} \frac
{M} {2C_1d^{\alpha_k}}  \mathop{\rm meas}\{\{\eta\leq T \}\cap B_{\tilde
r_1}(x_0)\}&\leq \int_{\{\eta\leq T \}\cap B_{\tilde r_1}(x_0) }
\eta (T-\eta) d x
\\ & \leq C  \mathop{\rm meas}\{\{\eta\leq T \}\cap B_{\tilde r_1}(x_0)\},
\end{split}
\end{equation*}
 we have  $M\leq \tilde C_2 d^{\alpha_k}$.
Hence $\left|{ \nabla \eta (x)}\right| \leq \tilde C_2
d^{-1+\beta\alpha_k/(1+\beta)}$ for all
 $x \in \Omega_k^{r_1}$, which yields
 $$ \left|{\tilde \eta - \eta(x_0)}\right|_{C^0(B_{d/2}(x_0))}
  \leq \tilde C_2 d^{\beta \alpha_k/(1+\beta)}\,.$$
Back to \eqref{eq:bddetak},  then
 $\left|{ \nabla \eta }\right|_{ C^{0,\beta}(B_{d/4}(x_0))}  \leq \tilde C_3
 d^{-1-\beta+\beta\alpha_k/(1+\beta)}$.
  Consider in $B_{\tilde r_2}(x_0)$, where
 $\tilde r_2=\big(\frac {M} {2 \tilde
C_3 }\big)^{1/\beta} d^{1+1/\beta-\alpha_k/(1+\beta)}$.
 Using the same way as above, we obtain
\[
\frac {M^{1+1/\beta} d^{1+1/\beta -\alpha_k/(1+\beta)}}
{2d^{\alpha_k} (2\tilde C_3)^{1/\beta} } \leq 2 \pi,
\]
  i.e., $M \leq \tilde C_4 d^{-1+ \beta\alpha_k/(1+\beta)+
  \beta\alpha_k/(1+\beta)^2}$.
Iterate $N$ times,
$$
M \leq \tilde C_{2N} d^{-1+ \beta\alpha_k
((1+\beta)^{-1}+
  (1+\beta)^{-2}+\dots+(1+\beta)^{-N}) } =
   \tilde C_{2N}d^{-1+\alpha_k (1- (1+\beta)^{-N-1})}.
$$
Take $N=1+ \lfloor \log_{1+\beta} \frac {\alpha_k}
{\alpha_k-1}\rfloor$, then $\gamma_k=(1-(1+\beta)^{-N-1})\alpha_k
-1>0$, and $\left|{ \nabla \eta (x)}\right| \leq \tilde C_{2N}
d^{\gamma_k}$ for any $x$ with $\mathop{\rm dist}\{ {x}, \, {\Omega_k}
\} \leq r_1/2$.
  Thus as $x$ approaches $\Omega_k$, $\left|{ \nabla \eta (x)}\right|$
  approaches 0 with the order of $d_k^{\gamma_k}(x)$.
 The above argument is held for all $x$ close to
$\Omega_k, k=1,2, \dots,n$,  hence
  as $x$ approaches $\Omega_H$, $\left|{ \nabla \eta (x)}\right|$
  approaches 0.
 \end{proof}

 Similar to  Properties (i)--(vii) of $\eta_k$, $k=1,2,\dots, n$, we have
the following results.

\begin{lem} \label{Lemma:ETA0}
$\eta_0$ has the following properties:
\begin{itemize}
\item[(i)] $0 \leq \eta_0 \leq 1 $ in $\Omega$
\item[(ii)] $\eta_0\in   C^0({\overline\Omega})\cap V$
\item[(iii)]  Assume $\eta_0^k $ is the value of $\eta_0$ on
$\Omega_k$, then $ \eta_0^j \neq 1, $ for $1 \leq k \leq n$.
\item[(iv)] $\eta_0 \neq 0$ a.e..
 $\eta_0 \neq 1$  a.e.; i.e.
$\eta_0^j \neq 0 $  for $1 \leq k \leq n$.
\item[(v)] For any subdomain $G_k$ with $G_k\cap
{\overline\Omega}_H={\overline\Omega}_k$,
$\eta_0^k < \sup_{G_k} \eta_0$, for $ 1 \leq k \leq n$.
\item[(vi)] If $\alpha_k>1$, $k=1,2,\dots,n$, then $\eta_0\in
C^{1}({\overline\Omega})$.
\end{itemize}
\end{lem}

\begin{proof} (i) can be proved by using test
functions $ (\eta_0 - 1)_{+} = \max\{ \eta_0 - 1, 0 \}$ and
$ f = \min\{ \eta_0 , 0 \}$ for  \eqref{Eta0} respectively.

The proof of (ii) is the same as the proof of  Property (iv) above.

Using test functions from
$C_0^1(\Omega\setminus{\overline\Omega}_H)$ in \eqref{Eta0}, we get
\begin{equation} \label{Eq:ETA0}
\begin{gathered}
 - \nabla \cdot a^{-1} \nabla \eta_0 + \eta_0 = 0 \quad\mbox{ in }
  \Omega\setminus{\overline\Omega}_H,\\
\eta_0 = 1 \quad \mbox{on } \partial \Omega.
\end{gathered}
\end{equation}
Fix $ \sigma $ such that $\Omega_k^{\sigma} \cap \Omega_H =
\Omega_k$.  If $\eta_0^k  = 1 $ for some $ k$, let $ c_k =
\mathop{\rm ess\,sup}_ {\Omega_k^{\sigma}\setminus \Omega_k^{\sigma/2}} \eta_0$,
 then $c_k < 1 =  \mathop{\rm ess\, sup}_{\Omega} \eta_0 $
by the maximum principle for \eqref{Eq:ETA0}.
Use $ f = \chi_k^{\sigma} (\eta_0 - c_k)_{+} \not\equiv 0$ as a
test function in \eqref{Eta0},
$$
0 = \int_{\Omega_k^{\sigma}\setminus{\overline\Omega}_k  }
a^{-1} \left|{\nabla f }\right|^2 d x + \int_{\Omega_k^{\sigma}} f
\eta_0 d x
> 0,$$ a contradiction, so that  $\eta_0^k < 1 $ for $ 1 \leq k \leq n$.
 $(iii)$ is showed.

Applying the maximum principle for \eqref{Eq:ETA0}, $\eta_0$ can
not achieve the maximum value in
$\Omega\setminus{\overline\Omega}_H$, combine with $(iii)$, then
$\eta_0 \neq 1 $ a.e..

 To show $\eta_0 \neq 0 $ a.e., we use the same idea as in the proof of
Property (v) to show  $\mathop{\rm meas}\{\Gamma^j_0\} = 0$, where
$\Gamma^j_0 = \{x \in {\overline\Omega}|\eta_0(x) = 0 \}$.
Let $ \Gamma^j_{\delta}\{x \in {\overline\Omega}| \eta_0(x) \leq \delta \}$,
then $\Gamma^j_0 \subset \Gamma^j_{\delta}$,  and
 $ \mathop{\rm meas}\{\Gamma^j_{\delta} \setminus \Gamma^j_0 \} \to 0 $ as
$\delta \to 0$. By $\eta_0 = 1 $ on  $ \partial \Omega$,
 $\Gamma^j_{\delta} \cap  \partial \Omega = \emptyset $
for all $0 \leq \delta < 1 $.
 Use $f_{\delta} = - \chi_{\Gamma^j_{\delta}} ( \delta - \eta_0 )_{+} \in
H_0^1(\Omega)\cap V $
 as a test function in \eqref{Eta0}, then
$$
\int_{\Gamma^j_{\delta}\setminus {\overline\Omega}_H}
 a^{-1} \left|{ \nabla f_{\delta}  }\right|^2 d x +
\int_{\Gamma^j_{\delta}} \eta_0 f_{\delta} d x = 0, \mbox{ i.e., }
\int_{\Gamma^j_{\delta}\setminus {\overline\Omega}_H}
  a^{-1} \left|{ \nabla f_{\delta}  }\right|^2 d x  =
  \int_{\Gamma^j_{\delta}} \eta_0 \left|{f_{\delta}}\right|  d x \,.$$
By the Sobolev embedding,
\begin{align*}
\int_{\Gamma^j_{\delta}} \left|{f_{\delta}}\right|^2 d x
&\leq c(\Omega) \int_{\Gamma^j_{\delta}} \left|{\nabla f_{\delta}
}\right|^2 d x \\
&\leq c(a, \Omega) \int_{\Gamma^j_{\delta}\setminus{\overline\Omega}_H }
 a^{-1} \left|{ \nabla f_{\delta} }\right|^2 d x \\
&\leq c(a, \Omega) \int_{\Gamma^j_{\delta}} \eta_0
\left|{f_{\delta}}\right| d x \,.
\end{align*}
Therefore,
\[\delta^2  \mathop{\rm meas}\{\Gamma^j_0\} \leq
\int_{\Gamma^j_{\delta}\setminus\Gamma^j_0}
\left|{f_{\delta}}\right|^2 d x \leq c(a, \Omega) \delta^2
\mathop{\rm meas}\{\Gamma^j_{\delta}\setminus \Gamma^j_0 \};
\]
i.e., $\mathop{\rm meas}\{\Gamma^j_0\} \leq c(a,\Omega)
 \mathop{\rm meas}\{\Gamma^j_{\delta}\setminus \Gamma^j_0 \} \to 0$,
as $\delta \to 0$. Therefore,  $ \mathop{\rm meas}\{\Gamma^j_0\} = 0$ and
(iv) is proved.

The proof of (v) is the same as the proof
of  Property (vi) above.
To prove (vi), use $(T-\eta_0)_{+}$ as the test function in
\eqref{Eta0}. The rest is similar to the proof of  Property
(vii) above, use the coarea formula,
elliptic estimates and iteration to obtain the desired result.
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:regularity}]
The representation of the $h_D$ follows from the linearity of
(\ref{Heq}) and its uniqueness of solution. The regularity of
$h_D$ follows from the regularity of $\eta_0$ and $\eta_j$,
$j=1,2, \dots,n$. \end{proof}

\section{Consequence of the Limit Problem}

In this section, we follow  \cite{ABP} closely to give a few
applications of the limit problem. Set
\begin{gather*}
a_{ij} = \int_{\Omega\setminus{\overline\Omega}_H }
 a^{-1} \nabla \eta_i \cdot \nabla \eta_j d x
+  \int_{\Omega} \eta_i \eta_j  d x, 1\leq i,j \leq n ,\\
 b_j =
\int_{\Omega\setminus{\overline\Omega}_H } a^{-1} \nabla \eta_0
\cdot \nabla \eta_j d x +  \int_{\Omega} (\eta_0 - 1) \eta_j  d x,
\quad 1 \leq j \leq n ,
\\
b_0 = \int_{\Omega\setminus{\overline\Omega}_H } a^{-1} \nabla
\eta_0 \cdot \nabla \eta_0 d x +  \int_{\Omega} (\eta_0 - 1)^2 d
x,
\end{gather*}
and apply the same argument as  \cite[Theorem 3.4]{ABP}
(also see  \cite[Lemma 2.2]{AB}). We can represent the local
 minimum energy of $J_0$ as the follows.

\begin{lem} \label{Thm:Energy} Fix $h_e$.
 If $(\psi_D,A_D)$  minimizes
 $J_0$ in $H_{a,D}^1\times H^1(\Omega;{\mathbb R}^2)$,  then
\begin{equation}\label{Eq:J0Val}
J_0(\psi_D,A_D)=\int_\Omega |\nabla \sqrt a |^2 d x
+\sum_{i=1}^{n}\sum_{j=1}^{n} a_{ij} d_i d_j+ 2 \sum^n_{j=1} b_j
d_j h_e+b_0 h_e^2 \,.
\end{equation}
\end{lem}

For a minimizing sequence of $J_{\epsilon}$ in ${\mathcal{M}}$, we
also prove the following result.

\begin{thm}\label{thm:4.2}
Fix $h_e$ and a sequence $\epsilon_k\to 0^+$ as $k\to\infty $.
  Let $(\psi_{\epsilon_k}, A_{\epsilon_k}) $
 minimize $J_{\epsilon_k}$
in $H^1(\Omega;\mathbb{C})\times H^1 (\Omega;\mathbb{R}^2)$ under
gauge (\ref{Gauge0}), then $|\psi_{\epsilon_k}|\to \sqrt a$  in
$C(\overline\Omega)$, and there is a subsequence
$(\psi_{\epsilon_{k_\ell}},A_{\epsilon_{k_\ell}})\to (\psi_D,A_D)$
in ${\mathcal{M}}$ as $\ell \to \infty$, where $(\psi_D,A_D)$ is a
minimizer of $J_0$ in ${\mathcal{M}}_0$, and  $(\psi_D,A_D)\in
H^1_{a,D}\times H^1(\Omega;{\mathbb R}^2) $ for some $D \in
{\mathbb Z}^n$. Consequently,
$J_{\epsilon_{k_\ell}}(\psi_{\epsilon_{k_\ell}},A_{\epsilon_{k_\ell}})\to
J_0(\psi_D,A_D)$ as $\ell \to \infty$, moreover, for any  $ 0 <
\sigma < r_1 $ and $\ell$ sufficiently large,
$|\psi_{\epsilon_{k_\ell}}|$ is uniformly positive outside
$\bigcup_{j=1}^{n} {\overline\Omega}^{\sigma}_j$, and the degree
of $\psi_{\epsilon_{k_\ell}}$  around $\overline{\Omega^{\sigma}_j
}$ is $d_j, j =1,2, \dots, n$.
\end{thm}

\begin{proof}
Use the same argument as \cite[Theorem 4.2]{ABP}, we can prove the first
part of the theorem; the second
part follows from the definition of the degree in \eqref{DegU} and
the fact that $(\psi_{\epsilon_{k_\ell}},A_{\epsilon_{k_\ell}})\to
(\psi_D,A_D)$ in ${\mathcal{M}}$  as $k_\ell \to \infty$.
\end{proof}

 From Theorem~\ref{thm:4.2}, for sufficiently small $\epsilon$, the
vortex set of the minimizers of $J_{\epsilon}$ in ${\mathcal{M}}$
is forced to close the zero set of $a(x)$, by zero set of $a(x)$
corresponds to the normal impurities in the inhomogeneous
superconductor,  the vortices of the minimizers of $J_{\epsilon}$
is pinned near the normal impurities, which verifies the
effectiveness of the pinning mechanism by adding normal impurities
to a superconductor.

\begin{proof}[Proof of Theorem~\ref{thm:4.6}] %\label{proofthm:4.6}
By Theorem~\ref{Thm:1.5}, $H^1_{a,D}$ is both open
and closed in $H_a^1$, we can always find $r>0$ sufficiently
small, such that
${\mathcal{B}}_r\cap{\mathcal{M}}_0={\mathcal{B}}_r\cap
[H^1_{a,D}\times H^1(\Omega;{\mathbb R}^2)]$.  Apply the same
argument as in  \cite[Theorem~4.6 ]{ABP}, we  derive
Theorem~\ref{thm:4.6}.
\end{proof}

\subsection*{Acknowledgements}
The main results of the paper were done when the author was a student in the
Math department at Purdue University. The author thanks his
advisor, Professor Bauman,  for her endless assistance and helpful
suggestions for this paper and other results.


\section{Appendix}
In this part, we list some lengthy proofs omitted in Section II.

\begin{proof}[Proof of Lemma~\ref{Lem:1.4}] %}\label{Proof:1.4}
 First we can find a sequence of nested $C^{\infty} $ domains to
approximate $\Omega_H$, say they are
$$
\Omega_H^1 \supset \supset \Omega_H^2 \supset \supset \dots
 \supset \supset \Omega_H \quad
\mbox{with } \mathop{\rm dist}\{ {\partial \Omega_H^m} , {\partial
\Omega_H}\} \to 0, \mbox{ as } m \to \infty.
$$
Then by the proposition of Schoen and Uhlenbeck \cite[page~267]{SU}
 (also see \cite[Lemma A.11 page~244]{BN0} ), there
exists a sequence $\{v^m\}_{m=1}^{\infty}$,  such that
\begin{equation}\label{Eq:SU}
v^m\in C^2({\overline\Omega}\setminus\Omega_H^m; S^1)\cap
 H_{\rm loc}^1(\Omega \setminus{\overline\Omega}_H ),
 v^m\to v \mbox{ in } H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H),
 \mbox{ as } m\to\infty.
\end{equation}
 From \eqref{DegU}, for any $G_j$ satisfying (\ref{Eq:Gj}),
$f_{G_j}$ satisfying  (\ref{Eq:FGj}), the degree of $v^m$ on $\partial G_j$
will converge to the degree of $v=u/\sqrt{a}$ on $\partial G_j$,
i.e. for all $m$ sufficiently large, we have
\begin{equation}\label{DegV}
d_j= \deg( v^m, \partial G_j) = \frac 1 {\pi} \int_{G_j\setminus
{\overline\Omega}_j}{\mathbb\bf J}(v^m f_{G_j}) dx = \frac 1 { 2
\pi i} \int_{\partial G_j} (v^m)^*(v^m)_\tau \, d s\,.
\end{equation}
Here $\tau$ is the counterclockwise tangent vector field of
$\partial G_j$, and the most right-hand side is derived by using
integral by part, it is  the standard definition of the degree
(winding number) of $C^1$ function on $\partial G_j$.  We define
a sequence of  real two-dimensional vector fields by
\begin{equation}\label{Angle}
F^m(x)=- \sum_{j=1}^n d_j \nabla\theta_j + i (v^m)^*\nabla v^m,
\quad m = 1, 2, \dots, \; x \in \Omega
\setminus{\overline\Omega}_H^m.
\end{equation}
Note that $\nabla\theta_j$ is a single-valued smooth vector field
on $\Omega \setminus{\overline\Omega}_H$,
 and $\int_{\partial G_j} \nabla\theta_j \cdot \tau \, d s = 2 \pi$,
$1\leq j \leq n$,  for any $G_j$ as in (\ref{Eq:Gj}). From
\eqref{DegV},
 $\oint_C F^m\cdot \tau  d s =0$ for any closed curve
$C \subset\subset \Omega \setminus{\overline\Omega}_H$,  and all
$m$ sufficiently large with $ C \subset \Omega
\setminus{\overline\Omega}_H^m$. Hence there exists a $\phi^m\in
H^1( \Omega \setminus{\overline\Omega}_H^m)$,
 such that $\nabla\phi^m=F^m$ in $ \Omega \setminus{\overline\Omega}_H^m $
 for all $m$ sufficiently large.
Use (\ref{Angle}), then $v^m\nabla\phi^m = -v^m \sum_{j=1}^n
d_j\nabla\theta_j + i\nabla v^m$.  As a result,
$ v^m(x) =e^{i\phi^m(x)}\cdot e^{i \sum_{j=1}^n d_j\theta_j(x)}
=e^{i[\phi^m(x) + \sum_{j=1}^n d_j \theta_j(x)] }$.

Since $v^m\to v = u / \sqrt{a} $ in
$H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$,  as $ m \to
\infty$,  $ \nabla \theta_j \in  C^{\infty}(
{\overline\Omega}\setminus\Omega_H)$, it follows that
$ e^{i\phi^m}\equiv v^m\cdot e^{-i \sum_{j=1}^n d_j
\theta_j(x)} \to  v\cdot  e^{-i  \sum_{j=1}^n d_j \theta_j(x)}$
 in  $H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$,
as $  m \to \infty$,
 and  $\nabla\phi^m\to -  \sum_{j=1}^n d_j \nabla\theta_j +
i v^*\nabla v $ in $L_{\rm loc}^2(\Omega\setminus{\overline\Omega}_H)$,
 as $ m \to \infty$.
 It follows that, after possibly subtracting
 constants $2\pi k_m$,    $k_m\in {\mathbb Z}, m=1, 2, \dots$,
$\phi^m\to \phi$ in $H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$, as
 $ m \to \infty$, where
 $\phi \in H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$, and
$v = e^{i(\phi+ \sum_{j=1}^n d_j \theta_j)}$
 in $ H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$.
Setting $ \Theta (x) = \phi (x)+ \underset{j=1}{\overset n\sum}
 d_j\theta_j$,  then $u = \sqrt{a} v = \sqrt{a} e^{i\Theta (x)}$.
 By $\left|{\nabla \theta_j}\right| \leq C(\Omega_H), \quad 1\leq j \leq n$,
\begin{align*}
\int_{ \Omega\setminus{\overline\Omega}_H } a|\nabla\phi|^2
&= \le \int_{\Omega\setminus{\overline\Omega}_H }
 |\nabla u|^2+ C \int_{\Omega\setminus{\overline\Omega}_H}
 a |\sum_{j=1}^n d_j\nabla \theta_j |^2 \\
&\le \int_ {\Omega\setminus{\overline\Omega}_H } |\nabla
u|^2+C(\Omega_H, D, a ) \,.
\end{align*}
To show $\phi\in H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H  )$
is unique (up to an additive constant $2\pi k$, $k \in {\mathbb Z}$).
 Assume   $\tilde\phi \in H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H  )$,
satisfying $ u= \sqrt{a} e^{i(\tilde\phi+ \sum_{j=1}^n d_j\theta_j)}$,
then $ e^{i(\phi-\tilde\phi)}=1$ in
 $ H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H  )$,  with
$\phi-\tilde\phi \in H_{\rm loc}^1(\Omega\setminus{\overline\Omega}_H)$, so
$\phi-\tilde\phi = 2\pi k$,  for some $k \in {\mathbb Z}$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{Lemma:2.2}] %\label{Proof:Lemma:2.2}
 Our proof is standard. We first construct
a family of functions in $V$ to approximate a given $f \in V$,
then use a mollifier to smooth them, and apply the diagonal rule
to finish the proof.
Take $\sigma < r_1$, so that
$\Omega_j^{\sigma} \cap \Omega_k^{\sigma} = \emptyset, k \neq j,
 1 \leq k, j \leq n$.
Let
\[
 \alpha(r) =  \begin{cases}
  1,   & 0\leq r \leq \frac 1 2 \\
 2 - 2 r, & \frac 1 2 < r \leq 1 \\
 0,    & r > 1\,.
\end{cases}
\]
 Then $| \frac d {d r} \alpha | \leq 2$
If $x=(\tilde x, x_d) \in {\mathcal{U}}_k^j \setminus {\overline\Omega}_j$
represented in the local coordinate system of
${\mathcal{U}}_k^j$, define the shift of $x$ away from ${\overline\Omega}_j$
in sense of the local coordinate as
$$
m_{\sigma}^{k j}(x) = x +
 \alpha( \frac { \phi_k^j(\tilde x) - x_d } {\sigma})
((\tilde x, \phi_k^j(\tilde x)) - x ),
$$
it can be verified that
$\left|{\nabla m_{\sigma}^{k j}(x)}\right| \leq c(\Omega_H  )$
a.e. in ${\mathcal{U}}_k^j \setminus {\overline\Omega}_j$.

 For any $f \in V, \sigma $ small enough, define
$$
f_{\sigma}(x) =  \begin{cases}
\sum_{k=1}^{n_j} \beta_k^j(x) f(m_{\sigma}^{k j}(x))
& \mbox{for } x \in \Omega_j^{ \sigma } \subset \Omega_j^{r_1},
 j=1,2, \dots, n ,\\
f(x) & \mbox{for other } x,
\end{cases}
$$
where $\{ \beta_k^j \}_{k=1} ^{n_j}$ is the partition  of unity
from \eqref{Eq:unity}.
Clearly $f_{\sigma} \in H^1(\Omega)$, we verify  $f_{\sigma} \in V$.
For  $x \in \Omega_j^{\sigma/2g}$,
 by \eqref{Eq:ApproxDx}, $\left|{\phi_k^j(\tilde x) - x_d}\right|
\leq g d(x) \leq \sigma/2$,  $ m_{\sigma}^{k j}(x)=(\tilde x,
\phi_k^j(\tilde x) ) \in \partial \Omega$, so that  $
f_{\sigma}(x) = f_j =$ constant, and  $\nabla  f_{\sigma} = 0$;
from \eqref{Eq:ApproxAx}, $a(x) > C_0 (\sigma/2g)^{\alpha_j}$ for
$ x\in \Omega\setminus \Omega_j^{\sigma/2g}$, then $ f_{\sigma}
\in V$.
Moreover, calculate in the local coordinates, and sum together the
difference of $f$ and $f_{\sigma}$ in $V-$norm, we obtain,
\begin{equation*}
 \| f_{\sigma} - f \|_V^2
 \leq  C(\Omega_H) \int_{\Omega^{ \sigma} \setminus
{\overline\Omega}_H }
  (a^{-1} \left|{\nabla f}\right|^2 + f^2) d x \,.
\end{equation*}
Since   $ \mathop{\rm meas}\{\Omega^{\sigma} \setminus {\overline\Omega}_H\}\to
0$, as $ {\sigma} \to 0$,  we have $\| f_{\sigma} - f \|_V \to 0$,
as $ {\sigma} \to 0$.

Choose a  mollifier
\begin{equation}\label{Eq:MOLLIFIER}
\rho  \in C_0^{\infty}(B_1(0))\cap C_0^{\infty}({\mathbb R}^d),
\quad \rho \geq 0, \quad \mbox{and} \int _{B_1(0)} \rho(x) d x = 1,
\end{equation}
 and set $\rho_m (x) = m^d \rho ( m x)$, where $B_1(0)$ is the
unit ball in ${\mathbb R}^d$.
Then for $m > 6g/{\sigma}$, $\rho_m   * f_{\sigma} \in
C^{\infty}(\Omega)\cap V$, and $ \rho_m
* f_{\sigma} = f_j   $ in $ {\overline\Omega}_j^{\sigma/3g}$.

 From (\ref{Eq:ApproxAx}), $\left|{a(x)}\right|\geq C_0
(\sigma/3g)^{\alpha_j} \geq C({\overline\Omega}_H,\sigma) > 0 $ in
$\Omega \setminus {\overline\Omega}^{{\sigma}/3g}$,  we obtain
\begin{equation*}
\| f_{\sigma} - \rho_m   * f_{\sigma}\|_V^2 \leq
C({\overline\Omega}_H,\sigma)
 \| f_{\sigma} - \rho_m   * f_ {\sigma})\|_{H^1}^2  \,.
  \end{equation*}
Hence by $\rho_m   * f_ {\sigma} \to f_ {\sigma}, m \to \infty $
in $H^1$,  we have $\rho_m   * f_ {\sigma} \to f_ {\sigma}, m \to
\infty $ in $V$.
Apply the diagonal rule,  pick up the $\rho_{m_{\sigma}}  *
f_{\sigma} \in C^{\infty}(\Omega)\cap V$, such that \\
 $\rho_{m_{\sigma}}  * f_{\sigma} \to f $ in $V$,
as $ {\sigma} \to 0$.
\end{proof}

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\end{document}