\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 28, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/28\hfil
Compactness for Ginzburg-Landau functionals]
{Compactness results for Ginzburg-Landau type functionals with
general potentials}

\author[M. Kurzke\hfil EJDE-2010/28\hfilneg]
{Matthias Kurzke} 

\address{Matthias Kurzke \newline
Institut f\"ur Angewandte Mathematik, 
Universit\"at Bonn,
Endenicher Allee 60, 53115 Bonn, Germany}
\email{kurzke@iam.uni-bonn.de}

\thanks{Submitted September 22, 2009. Published February 18, 2010.}
\thanks{Supported by SFB 611}
\subjclass[2000]{35J50, 35B25}
\keywords{Gamma-convergence; compactness for Jacobians; \hfill\break\indent
Ginzburg-Landau functional}

\begin{abstract}
 We study compactness and $\Gamma$-convergence  for Ginzburg-Landau
 type functionals. We only assume that the
 potential is continuous and positive definite close to one
 circular well, but allow large zero sets inside the well.
 We show that the relaxation of the assumptions does not change
 the results to leading order unless the energy is very large.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}
\label{sec:int}
We study the family of functionals
\begin{equation}
 \label{eq:thefn}
 E_\varepsilon(u)=\frac12\int_U |\nabla u|^2 + \frac{1}{\varepsilon^2} P(u)
\end{equation}
for a smooth bounded domain $U\subset \mathbb{R}^2$ and 
$u\in H^1(U;\mathbb{C})$,
where $P$ is a nonnegative function with $P=0$ on 
$S^1=\{z\in \mathbb{C} : |z|=1\}$. 

In the case where 
$P(u)=P_{gl}(u)=\frac12(1-|u|^2)^2$,
this functional is the 
Ginzburg-Landau functional
\begin{equation}
\label{eq:theglf}
G_\varepsilon(u)=\int_U \frac12|\nabla u|^2 +\frac{1}{4\varepsilon^2}(1-|u|^2)^2 ,
\end{equation}
 which has been widely studied
since the groundbreaking work of Bethuel, Brezis and H\'elein \cite{BBH_GL}.
They considered minimizers and solutions of the Euler-Lagrange
equations under Dirichlet boundary condition
and were able to show convergence results
and detailed energy asymptotics. Their methods relied on 
Rellich-Poho\u{z}aev type identities and thus on the PDE.

Later, Sandier \cite{San98} and Jerrard \cite{Jer99} were
able to show the essential lower bounds using more direct
comparison arguments and without PDE arguments.
This approach was refined and 
later used by Jerrard and Soner \cite{JerSon02a,
JerSon02} to show compactness for the Jacobians and a 
$\Gamma$-convergence result 
for the energies. Independently, Alberti-Baldo-Orlandi \cite{Alb,ABO}
obtained a proof in the general case of maps from $\mathbb{R}^{n+k}$ 
to $\mathbb{R}^n$ and an energy related to the $k$-Dirichlet energy.
In the context of the magnetic Ginzburg-Landau functional,
a detailed presentation can be found
in the monograph by Sandier-Serfaty \cite{SanSer_GL}.

The PDE approach of Bethuel-Brezis-H\'elein \cite{BBH_GL}
has been generalized to potentials $P$ that vanish near $S^1$ 
to infinite order by Hadiji and Shafrir \cite{HadSha06}. 
The main new feature is that certain potentials with
sufficiently slow growth allow for a vortex energy that is 
not $\pi|\log\varepsilon|+O(1)$, but instead $\pi|\log\varepsilon|-o(|\log\varepsilon|)$.

Potentials given by distance functions to homeomorphic images
of $S^1$ instead of $S^1$ itself were also studied,
see the work of Shafrir \cite{Sha04} and Andr\'e and Shafrir
\cite{AndSha07}.

In this article, we investigate a different type of generalization:
we allow the potential to vanish on a larger set, and
in particular to have zeroes inside the unit ball.
One simple such potential is
\begin{equation}
 \label{eq:cshp}
 P_{csh}(u)=|u|^2(1-|u|^2)^2.
\end{equation}
The study of this potential has applications in the
theory of Chern-Simons-Higgs vortices. In Kurzke-Spirn \cite{KS_JFA},
the analysis of \eqref{eq:thefn} with this potential is the basis
for a $\Gamma$-convergence analysis of the static Chern-Simons-Higgs
functional,
\begin{equation}
 \label{eq:cshfn}
 G^{\varepsilon,\mu}_{csh}(u,A)=\frac12\int_U |(\nabla-iA)u|^2 + \frac{\mu^2}{4}
\frac{|\mathop{\rm curl}-h_{ex}|^2}{|u|^2}+\frac{1}{\varepsilon^2} |u|^2(1-|u|^2)^2,
\end{equation}
which contains an interaction with a magnetic vector potential $A:U\to\mathbb{R}^2$ and 
an external magnetic field $h_{ex}$.
The particular potential \eqref{eq:cshp} has also been studied using the methods
devised by Bethuel-Brezis-H\'elein \cite{BBH_GL} for $P_{gl}$. Lassoued-Lefter
\cite{LaLe98} showed a convergence result for minimizers in bounded
domains, and Ma \cite{Ma08} studied quantization properties 
for solutions of the corresponding Euler-Lagrange equations in $\mathbb{R}^2$.
Both of these works rely heavily on the Poho\u{z}aev identity
and on the equations, requiring some differentiability of the potential.

In contrast to these works, in the present article
we only require the potential $P$ to be continuous and nonnegative in $\{|z|\le 1\}$ 
and comparable to the Ginzburg-Landau potential near $S^1$.
We do not require any further smoothness or symmetry assumptions,
and allow the potential to vanish on large subsets of the unit ball. 
To ensure the presence of vortices,
we study situations where $|u|=1$ on the boundary (for example,
this can be ensured by a Dirichlet boundary condition), although this 
assumption can be weakened. 

Our method is based on a few elementary Modica-Mortola type arguments
and the co-area formula. By an observation of G. Orlandi, these
arguments lead to a bound on the standard Ginzburg-Landau functional,
and it is then possible to apply the standard compactness results of 
\cite{ABO,JerSon02a,JerSon02} also to the generalized functional. 
The method is applicable to the functional studied in \cite{KS_JFA}, 
and yields a new and much shorter and clearer proof 
of the central compactness statement there.

The $\Gamma$-convergence result turns out to be essentially the same as for the 
Ginzburg-Landau energy, in contrast to
the weighted Ginzburg-Landau energy with vanishing weight
as studied by Lefter-Radulescu \cite{LeRa97}, 
where $P(u)$ is replaced by $w(x)P(u)$ with a nonnegative function 
$w: U\to [0,\infty)$,
and the behaviour depends on the growth of the weight function near its zeroes.

A difference between our result and Ginzburg-Landau theory can be seen 
at very high energies: our compactness theorem only holds for a smaller
range of energies, and we even have an explicit example
of failure for very large energies, see Remark \ref{rem:counterex}.

\subsection*{Notation and assumptions}
We will deal with sequences $(u_\varepsilon)$ of functions
$u_\varepsilon\in C^1(\overline U;\mathbb{C})$, labeled by 
an arbitrary (but assumed fixed) sequence $\varepsilon=\varepsilon_k\to 0$. Subsequences 
will be taken from this fixed sequence. As we are dealing mostly 
with compactness issues, the regularity requirement 
is mostly technical. 

We will generally assume 
$|u_\varepsilon|\le 1$, $|u_\varepsilon|=1$ on $\partial U$, where
$U$ is a Lipschitz domain in $\mathbb{R}^2$. The assumption
$|u_\varepsilon|\le 1$ is technical and could be replaced by 
suitable growth conditions on $P(z)$ for $|z|\ge 1$.

The assumption $|u_\varepsilon|=1$ on $\partial U$ is slightly more 
restrictive: it completely rules out vortices on the boundary
and at first glance
seems to make it impossible to use the results 
presented here for dynamical situations where
vortices enter or leave through the boundary.
However, under reasonable assumptions on $P$ and $U$, 
it is possible
to use the extension technique of Kurzke-Spirn \cite{KS_JFA}
to show that any $u_\varepsilon$ on $U$ with $|u_\varepsilon|\to 1$ in $L^2(U)$
can be extended to 
$\tilde u_\varepsilon$ on $\tilde U\supset U$ with $|\tilde u_\varepsilon|=1$ on 
$\partial \tilde U$, and then the compactness results of 
this article can be applied in $\tilde U$.

Our assumptions on the potential $P$ are as follows.
\begin{itemize}
\item[(A1)] $P\in C^0(\{|z|\le 1\})$, $P\ge 0$, $P(S^1)=\{0\}$
\item[(A2)] There is a $\kappa\in(0,1)$ and an $a>0$ such that
$P(z)\ge a(1-|z|^2)^2$ for $|z|\in [1-\kappa,1]$.
\end{itemize}
We note that both $P_{csh}$ and $P_{gl}$ satisfy these assumptions.

For $a,b\in\mathbb{C}$ we write $(a,b):=\text{Re}(\overline{a}b)$ for the
scalar product. Identifying $\mathbb{R}^2$ and $\mathbb{C}$, we can write the 
Jacobian of $u$ as $\frac12\mathop{\rm curl}(iu,\nabla u)$.

Our main results are a compactness result for the Jacobians,
Theorem \ref{thm:jaccomp}, holding for energies $E_\varepsilon(u_\varepsilon)\ll \frac1\varepsilon$,
and $\Gamma$-convergence results
for the energy scalings $E_\varepsilon(u_\varepsilon)\approx |\log\varepsilon|$
and $E_\varepsilon(u_\varepsilon)\approx |\log\varepsilon|^2$,
Theorems \ref{thm:cgcg1} and \ref{thm:gammac}. Unlike in the Ginzburg-Landau
case, the Jacobian bounds of Theorem \ref{thm:jaccomp}
cannot be extended to energies in the range $\frac1\varepsilon\ll E_\varepsilon(u_\varepsilon)
\ll \frac{1}{\varepsilon^2}$, see Remark \ref{rem:counterex}.

\begin{remark}\label{rem:radial}
 For the questions studied in this article, we may
assume without loss of generality that $P$ is radial,
more precisely, we can assume $P(u)=V(|u|)$ 
for some $V$ with $V\in C^0([0,1])$ with $V\ge 0$, $V(1)=0$ and
$V(t)\ge a(1-t^2)^2$ for $t\in[1-\kappa,1]$. To see this, set
\begin{equation}
 \label{eq:vfromw}
 V^-(\rho)=\inf_{|u|=\rho} P(u) \quad\text{and}\quad V^+(\rho)=
\sup_{|u|=\rho} P(u).
\end{equation}
It is clear that 
$\int_U V^-(|u|)\le \int_U P(u)\le \int_U V^+(|u|)
$ 
and so compactness and lower bound results for functionals involving
$V^-$ carry over to $P$, while upper bound results for $V^+$ 
imply upper bound results for $P$. 
We note that by compactness,
$V^-\le V^+\le C(P) V^-$ and so any $\Gamma$-convergence results
for radial functionals that are invariant under rescaling 
$V\mapsto \sigma V$ for $\sigma>0$ automatically hold for non-radial $P$.
\end{remark}

\subsection*{Acknowledgments}
The author is grateful to Giandomenico Orlandi who suggested 
a significant improvement of a first draft of
 the article using Propositions
\ref{prop:glbd} and \ref{prop:GL1}. This greatly simplified the proofs. 

The author enjoyed the support of DFG SFB 611.
\section{Energy bounds and Jacobian compactness}
\begin{definition} \rm
For any compact set $K\subset\mathbb{R}^2$, we define its \emph{radius}
$\mathbf{r}(K)$ as 
\begin{equation}
 \label{eq:radius}
 \mathbf{r}(K)=\inf \Big\{ \sum_{i=1}^N r_i : K \subset \bigcup B_i, B_i 
\text{ a closed ball of radius $r_i$}
\Big\}
\end{equation}
\end{definition}
Radius of a set and $1$-dimensional Hausdorff measure $\mathcal{H}^1$ of its perimeter
are related by the following  inequality:
\begin{equation}
 \label{eq:radiusineq}
 \mathbf{r}(K)\le \frac12 \mathcal{H}^1(\partial K).
\end{equation}
For a proof, see \cite[p. 71]{SanSer_GL}.

The radius also has the monotonicity property
$\mathbf{r}(A)\le \mathbf{r}(B)$ for $A\subset B$.

\begin{proposition}\label{prop:startcov0}
There exists a constant $C=C(a)>0$ such that
for $u\in C^1(\overline U,\mathbb{C})$ with $|u|=1$ on $\partial U$, $E_\varepsilon(|u|)\le M$
and for any $\delta<\kappa$ and any $\varepsilon$, we have the following covering 
estimate for $K_\delta:=\{ |u|\le 1-\delta \}$:
\begin{equation}
 \label{eq:radd}
 \mathbf{r}(K_\delta) \le C\frac{\varepsilon M}{\delta^2}.
\end{equation}
In particular, the measure of $\{ |u|\le 1-\delta \}$ satisfies
\begin{equation}\label{eq:measfromr}
|K_\delta| \le \tilde C \varepsilon^2 M^2
\end{equation}
for $\tilde C=\frac{4\pi C(a)}{\kappa^2}$.
\end{proposition}
\begin{proof}
Setting $\rho=|u|$, the energy bound 
$$\frac12\int_U |\nabla \rho|^2 + \frac1{\varepsilon^2} V(\rho) \le M
$$  
implies by Cauchy's inequality that 
$$\int_U |\nabla\rho|\sqrt{V(\rho)} \le \varepsilon M.
$$
Using the co-area formula, this shows 
$$\int_{1-\kappa}^1 \sqrt{V(t)} \mathcal{H}^1( \{\rho=t\}) dt \le \varepsilon M.
$$
Using the assumption (A2) on the potential, we see that
in particular
$$\int_{1-\delta}^{1-\frac\delta 2} a(1-t^2) \mathcal{H}^1( \{ \rho=t \}) \le \varepsilon M.
$$
Using that $(1-t^2)\ge (1-(1-\frac\delta 2)^2) \ge \frac\delta 2$ in $(1-\delta,1-\frac\delta 2)$ and the
mean value theorem, we obtain the existence of $t_0\in(1-\delta,1-\frac\delta 2)$ such that
\begin{equation}
\label{eq:hm1est}
\mathcal{H}^1(\{x\in U : \rho(x) = t_0 \}) \le \frac{4\varepsilon M}{a\delta^2}  
\end{equation}
 Using that $\{\rho\le t_0\}$ is compactly contained in $U$ by the boundary condition $|u|=1$, it follows that
$\partial\{x\in U : \rho(x)\le t_0\}=\{x\in U : \rho(x) = t_0 \})$. Hence we can use \eqref{eq:radiusineq} and 
see that $\mathbf{r}(\{\rho\le t_0\})\le \frac{4\varepsilon M}{a\delta^2}$. The monotonicity of $\mathbf{r}$ now yields \eqref{eq:radd}.

To prove \eqref{eq:measfromr}, we note that $K_\delta \subset \bigcup B_{r_i}(a_i)$ 
with $\sum r_i \le \mathbf{r}(K_\delta)$. Hence $|K_\delta|\le \pi \sum r_i^2 
\le \pi (\sum r_i)^2 \le \pi (\mathbf{r}(K_\delta))^2$, and \eqref{eq:measfromr} follows.
\end{proof}

The following observation was communicated to the author by Giandomenico Orlandi:
\begin{proposition}\label{prop:glbd}
For $\alpha\in(0,1)$ and $\varepsilon>0$ sufficiently small there exists a 
$C=C(a)>0$ such that the following holds:
For every  $u \in C^1(\overline U, \mathbb{C})$ with $|u|\le 1$ in $U$, $|u|=1$ on $\partial U$ 
that satisfies the energy bound $E_\varepsilon(|u|)=M\le \varepsilon^{-\alpha}$,
there holds the bound
\begin{equation}
\label{eq:oest}
\int_{U} (1-|u|^2)^2 \le C\varepsilon^{2-\alpha} M.
\end{equation}
\end{proposition}
\begin{proof}
We split $U=\{|u|\le 1-\frac\kappa2\}
\cup \{|u|>1-\frac\kappa2\}$.
Using \eqref{eq:hm1est} applied to $\delta=\frac\kappa2$ 
and \eqref{eq:radiusineq},
\begin{align*}
% \label{eq:nmodco1}
 \int_{\{|u|\le 1-\frac\kappa2\}} (1-|u|^2)^2
& \le \pi \mathbf{r}(\{|u|\le 1 -\frac\kappa2 \})^2 
\\  &\le C(a)(\varepsilon M)^2 
\\ &\le \varepsilon^2 \varepsilon^{-\alpha} M \\ 
&=C\varepsilon^{2-\alpha} M
\end{align*}
since $M\le \varepsilon^{-\alpha}$.
On the other hand, using (A2) we estimate
\begin{align*}
 %\label{eq:nmodco2}
 \int_{\{|u|>1-\frac\kappa2\}} (1-|u|^2)^2 
&\le \frac{1}{a} \int_{\{|u|>1-\frac\kappa2\}} V(|u|)
\\ &\le C(a)\varepsilon^2 E_\varepsilon(|u|) \\ &\le C\varepsilon^2 M
\\ &\le C\varepsilon^{2-\alpha} M.
\end{align*}
Combining these estimates we obtain \eqref{eq:oest}.
\end{proof}
\begin{remark}
From the proof of Proposition \ref{prop:glbd} we see that the 
assumption $|u|\le 1 $ in $U$ could be replaced by 
assuming that  (A2) also holds for $|z|>1$.
\end{remark}
\begin{proposition}\label{prop:GL1}
Let $u\in C^1(\overline U,\mathbb{C})$ with $E_\varepsilon(u)\le M$, $|u|\le 1$ in $U$, 
$|u|=1$ on $\partial U$, where 
$M\le \varepsilon^{-\alpha}$ for some $\alpha\in (0,1)$. Then for $\eta=\varepsilon^{1-\frac{\alpha}{2}}$, 
the Ginzburg-Landau functional $G_\eta(u)$ as defined in \eqref{eq:theglf}
satisfies the bound
\begin{equation}
\label{eq:GL1bd}
G_\eta(u)\le CM
\end{equation}
If $(u_\varepsilon)$ is a sequence satisfying the assumptions above,
 then the 
following bounds hold for the sequence $\eta_\varepsilon=\varepsilon^{1-\frac{\alpha}{2}}\to 0$:
\begin{itemize}
\item[(i)] If $E_\varepsilon(u_\varepsilon)\le \varepsilon^{-\alpha}$ then 
$G_{\eta_\varepsilon}(u_\varepsilon)\le C\eta_\varepsilon^{-\frac{2\alpha}{2-\alpha}}$.
\item[(ii)] If $E_\varepsilon(u_\varepsilon)\le K|\log\varepsilon|$ then 
$G_{\eta_\varepsilon}(u_\varepsilon)\le CK\frac{2}{2-\alpha} |\log\eta_\varepsilon|$.
\item[(iii)] If $E_\varepsilon(u_\varepsilon)\le K|\log\varepsilon|^2$ then 
$G_{\eta_\varepsilon}(u_\varepsilon)\le CK\frac{4}{(2-\alpha)^2} |\log\eta_\varepsilon|^2$.
\end{itemize}
\end{proposition}
\begin{proof}
From \eqref{eq:oest} we obtain 
\[
\eta^{-2} \int_U (1-|u|^2)^2 \le C \eta^{-2} \eta^2 M=CM,
\]
and trivially 
\[
\frac12\int_U |\nabla u|^2 \le E_\varepsilon(u)\le M.
\]
The bounds (i)-(iii) follow from \eqref{eq:GL1bd} by simply
expressing the energy bounds in terms of $\eta_\varepsilon$.
\end{proof}
\begin{theorem}
\label{thm:jaccomp}
Let $(u_\varepsilon)$ be a sequence in $C^1(\overline U, \mathbb{C})$ with $|u_\varepsilon|=1$ on $\partial U$,
$|u_\varepsilon|\le 1 $ in $U$ and 
$E_\varepsilon(u_\varepsilon)\le M_\varepsilon$, where 
$$|\log\varepsilon| \le M_\varepsilon \le \varepsilon^{-\alpha}$$ for some $\alpha\in(0,1)$.
Set
$$\mu_\varepsilon=\frac{|\log\varepsilon|}{M_\varepsilon} \mathop{\rm curl}(iu_\varepsilon,\nabla u_\varepsilon).
$$ 
Then the sequence of measures $(\mu_\varepsilon)$   
is precompact in $(C_0^{0,\beta})^*$ for any $\beta\in(0,1)$.
\end{theorem}
\begin{proof}
By Proposition \ref{prop:GL1}, $G_{\eta_\varepsilon}(u_\varepsilon)\ll \eta^{-2}$. 
Hence we can apply Theorem 1.1 of \cite{JerSon02}, which yields 
the claim. 
\end{proof}
\begin{remark}
 \label{rem:counterex}
 For the Ginzburg-Landau energy, the analog of the 
 previous compactness result 
holds for energies up to $G_\varepsilon(u)\ll\varepsilon^{-2}$. 
In fact, we used that
result in its full strength, as
$E_\varepsilon(u_\varepsilon)\ll \varepsilon^{-1}$ just implies 
$G_\eta(u_{\varepsilon})\ll \eta^{-2}$.
That we cannot relax the assumption $E_\varepsilon(u_\varepsilon)\ll \varepsilon^{-1}$
 is not just an artifact of the proof: for our more general potentials,
the following example shows that compactness may fail for 
higher energies.
\end{remark}
\begin{example}
Set $V(\rho)=(\rho^2-1)^2(\rho^2-\frac14)^2\chi_{[\frac12,1]}(\rho)$.
Let $\frac1\varepsilon\ll d_\varepsilon$.
Then there exist constants $C_1, C_2$ and a sequence of functions 
$(u_\varepsilon)$, 
$u_\varepsilon\in C^1(\overline{B_1(0)},\mathbb{C})$
with $u_\varepsilon=1$ on $\partial B_1(0)$, 
$E_\varepsilon(u_\varepsilon)\le C_1 d_\varepsilon$ and
\begin{equation*}
  \|\mathop{\rm curl}(iu_\varepsilon,\nabla u_\varepsilon)\|_{(C_0^{0,1})^*} 
  \ge C_2 E_\varepsilon(u_\varepsilon).
\end{equation*}
In particular, the measures $(\mu_\varepsilon)$ as in Theorem \ref{thm:jaccomp}
will be unbounded. 
\end{example}
\begin{proof}
 Setting $d=d_\varepsilon$, we choose 
 \begin{equation*}
   u_\varepsilon(r,\theta)=
   \begin{cases}
     \frac12 (4r)^d e^{id\theta} & 0\le r<\frac14\\
     \frac12(2-4r)^d e^{id\theta} & \frac14\le r< \frac12\\
     0 & \frac12\le r<1-\varepsilon \\
     1-\frac{1-r}{\varepsilon} & 1-\varepsilon\le r\le 1.
   \end{cases}
 \end{equation*}
Then it is not hard to check that $|u_\varepsilon|\le \frac12$ on $B_{1/2}$ and
$E_\varepsilon(u_\varepsilon)=Cd_\varepsilon+\frac{c}{\varepsilon}\le Cd_\varepsilon$.
Testing with the $C^{0,1}$ function
\begin{equation*}
 \zeta(r,\theta)=
 \begin{cases}
   \frac18 & r<\frac18 \\
   \frac14-r & \frac18\le r<\frac14 \\
   0 & r\ge \frac14
 \end{cases}
\end{equation*}
shows the claim for the Jacobians.
\end{proof}
\section{Gamma limits}
In this section, we use the results of 
the previous section to obtain $\Gamma$-limit results in two scaling regimes:
One is the smallest energy regime where vortices appear, 
namely $M\approx |\log\varepsilon|$. The second is the
``natural'' energy scaling $M\approx |\log\varepsilon|^2$,
the only scaling where both
the Jacobian and the current $(iu,\nabla u)$  make a significant contribution.
A full discussion of possible regimes can be found in \cite{JerSon02}.
We note that results for Chern-Simons-Higgs type
functionals \eqref{eq:cshfn} can be deduced as in \cite{KS_JFA},
at least for certain parameter regimes.

\begin{theorem}[Compactness and $\Gamma$-convergence in the $|\log\varepsilon|$ 
scaling] \label{thm:cgcg1}
 If $(u_\varepsilon)$, \\ $u_\varepsilon\in C^1(\overline U; \mathbb{C})$ 
satisfies $|u_\varepsilon|\le 1$, $|u_\varepsilon|=1$ on $\partial U$ and 
$E_\varepsilon(u_\varepsilon)\le K|\log\varepsilon|$, then the sequence 
$\mu_\varepsilon=\frac12\mathop{\rm curl}(iu_\varepsilon,\nabla u_\varepsilon)
$
is precompact in $(C_0^{0,\beta})^*$, and any subsequential weak limit $\mu$
satisfies 
\begin{equation}
 \label{eq:gamma1str}
\mu=\pi \sum_{j=1}^n d_j \delta_{a_j}
\end{equation}
for some $n\in \mathbb{N}$, $a_j\in U$ and $d_j \in \mathbb{Z}$. Moreover,
\begin{equation}
 \label{eq:gamma1}
 \|\mu\|_{\mathscr{M}}=\pi \sum_{j=1}^n |d_j|
\le \liminf_{\varepsilon\to 0}\frac{1}{2|\log\varepsilon|} \int_U |\nabla u_\varepsilon|^2
\le \liminf_{\varepsilon\to 0}\frac{1}{|\log\varepsilon|} E_\varepsilon(u_\varepsilon).
\end{equation}
Conversely, for any $\mu$ of the form \eqref{eq:gamma1str} there
exists a sequence $(u_\varepsilon)$, $u_\varepsilon\in C^1(\overline U;\mathbb{C})$ such that
\begin{equation}
 \label{eq:gamma1ub}
 \|\mu\|_{\mathscr{M}} = \lim_{\varepsilon\to 0}\frac{1}{|\log\varepsilon|} E_\varepsilon(u_\varepsilon)
\end{equation}
\end{theorem}
\begin{proof} 
Let $\alpha\in(0,1)$.
From Proposition \ref{prop:GL1}, $G_\eta(u_\varepsilon)\le C|\log\varepsilon|$, where 
$\eta=\varepsilon^{1-\frac{\alpha}{2}}$. We have
$|\log\eta|=(1-\frac\alpha 2)|\log\varepsilon|$ and so 
$$G_\eta(u_\varepsilon)\le \frac{C}{1-\frac\alpha 2} |\log\eta|.
$$
In particular, we can apply the standard compactness theory for
the Ginzburg-Landau functional in the form of Theorem 3.1 of \cite{JerSon02a},
which yields the compactness and the 
structure statement on $\mu$. Furthermore, it yields the bound
 as well as the following
\begin{equation}
\label{eq:etalobd}
\|\mu\|_{\mathscr{M}} \le \frac{1}{|\log\eta|} G_{\eta_\varepsilon}(u_\varepsilon).
\end{equation}
By the observation made in the Remark following Theorem 1.1 of \cite{ABO}
(the bound in \eqref{eq:etalobd} does not depend on the shape of the potential), 
we can replace $(1-|u|^2)^2$ by $\sigma (1-|u|^2)^2$ and then let $\sigma\to 0$
and arrive at
\begin{equation}
\|\mu\|_{\mathscr{M}} \le \frac{1}{2|\log\eta_\varepsilon|} \int_U |\nabla u_\varepsilon|^2.
\end{equation}
Finally, we let $\alpha\to 0$ so $\frac{|\log\eta_\varepsilon|}{|\log\varepsilon|}=
1-\frac\alpha 2\to 1$, and we obtain \eqref{eq:gamma1}.
The upper bound construction of \cite{JerSon02a} provides \eqref{eq:gamma1ub}.
Here our different potential does not substantially change the proof, cf. Remark \ref{rem:radial}:
Since
$P(u)\le K(P)(1-|u|^2)^2$ and so $\varepsilon^{-2} P(u)\le 
\frac12{\tilde \varepsilon}^{-2}(1-|u|^2)^2$ for $\tilde \varepsilon=\sqrt{2K(P)} \varepsilon$
and $\frac{|\log\varepsilon|}{|\log\tilde\varepsilon|}\to 1$ for $\varepsilon\to 0$, the precise form 
of the potential is irrelevant.
\end{proof}

\begin{theorem}[Compactness and $\Gamma$-convergence in the $|\log\varepsilon|^2$ scaling]
\label{thm:gammac}
Assume \\ that $(u_\varepsilon)$, 
 $u_\varepsilon\in C^1(\overline U; \mathbb{C})$ 
satisfies $|u_\varepsilon|\le 1$, $|u_\varepsilon|=1$ on $\partial U$ and 
$E_\varepsilon(u_\varepsilon)\le K|\log\varepsilon|^2$. Then a subsequence of 
$v_\varepsilon=\frac1{|\log\varepsilon|}(iu_\varepsilon,\nabla u_\varepsilon)$ converges weakly in $L^2$ 
to $v\in L^2(U)$. The measures $w_\varepsilon=\mathop{\rm curl}(iu_\varepsilon,\nabla u_\varepsilon)$ 
converge subsequentially in $(C_0^{0,\beta})^*$ to 
$w=\mathop{\rm curl} v\in H^{-1}(U)$. 

The energy satisfies 
\begin{equation}
 \label{eq:gammaliminf}
 \liminf_{\varepsilon\to 0}\frac{1}{|\log\varepsilon|^2} E_\varepsilon(u_\varepsilon)
 \ge \liminf_{\varepsilon\to 0}\frac{1}{2 |\log\varepsilon|^2} \int_U |\nabla u_\varepsilon|^2
 \ge \frac12\int_U |v|^2 + 
\frac12\|\mathop{\rm curl} v\|_{\mathscr{M}}.
\end{equation}

Furthermore, for any $v\in L^2(U)$ such that $\mathop{\rm curl} v$ is a Radon measure,
there exists a sequence $(u_\varepsilon)$ such that $v_\varepsilon$ and $w_\varepsilon$ defined
as above converge weakly in $L^2$ and in $(C^{0,\beta})^*$, respectively,
and such that equality holds in \eqref{eq:gammaliminf}.
\end{theorem}
\begin{proof}
 We note that $\|v_\varepsilon\|^2_{L^2}
\le\left\|\frac{v_\varepsilon}{|u_\varepsilon|}\right\|^2_{L^2}\le K$. By Theorem
\ref{thm:jaccomp}, the $w_\varepsilon$ are compact.

We again choose $\alpha\in (0,1)$
and set $\eta_\varepsilon=\varepsilon^{1-\frac\alpha2}$.
By Proposition
\ref{prop:GL1}, the Ginzburg-Landau energy satisfies the bound
\begin{equation}
G_{\eta_\varepsilon}(u_\varepsilon) \le C|\log \eta_\varepsilon|^2.
\end{equation} 
Hence we can use Theorem 1.2 of \cite{JerSon02} and 
obtain the compactness and compactness and structure results of the theorem.
To prove \eqref{eq:gammaliminf}, we note that the cited theorem implies
\begin{equation}
 \label{eq:gammaliminf2}
 \liminf_{\varepsilon\to 0}\frac{1}{|\log\eta_\varepsilon|^2} G_{\eta_\varepsilon}(u_\varepsilon)
 \ge \liminf_{\varepsilon\to 0}\frac{1}{2 |\log\eta_\varepsilon|^2} \int_U |\nabla u_\varepsilon|^2
 \ge \frac12\int_U |v|^2 + 
\frac12\|\mathop{\rm curl} v\|_{\mathscr{M}},
\end{equation}
where we have again used the observation of \cite{ABO} to drop the 
potential term in the lower bound. Just as in the proof of the 
previous theorem, we may now let $\alpha\to 0$, which 
implies $\frac{|\log\eta_\varepsilon|^2}{|\log\varepsilon|^2}\to 1$, hence 
\eqref{eq:gammaliminf}.

For the construction of a recovery sequence that shows equality,
we note that the sequence constructed in Proposition 7.1 of 
\cite{JerSon02} can be used without changes, keeping in mind the 
remarks made at the end of the proof of Theorem \ref{thm:cgcg1}.
\end{proof}
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\end{document}