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\markboth{ Maxwell-Chern-Simons-Higgs system }{ Dongho Chae }
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electronin Journal of Differential Equations, Conf. 04, 2000, pp. 11--15\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Remarks on the relativistic self-dual Maxwell-Chern-Simons-Higgs system
\thanks{ {\em Mathematics Subject Classifications:} 35Q80.
\hfil\break\indent {\em Key words:} Maxwell-Chern-Simons-Higgs
system, admissible solutions, Chern-Simons limit problem
\hfil\break\indent \copyright 2000 Southwest Texas State
University  and University of North Texas. \hfil\break\indent
Published July 12, 2000. } }

\date{}
\author{Dongho Chae}
\maketitle
\begin{abstract}
In this note we present some recent developments on the
topological multi-vortex solutions of the  self-dual
Maxwell-Chern-Simons-Higgs system in ${\mathbb R}^2$.
 We find that all the topological solutions are admissible in the sense defined in \cite{CK}.
We also discover that the convergence of the topological solution to the
solutions of the self-dual Chern-Simons equations can be improved to be
strong.
\end{abstract}

\newtheorem{Thm}{Theorem}


\section{Introduction}
We are concerned with the semilinear elliptic system in ${\mathbb R}^2$:
\begin{eqnarray}
\Delta u &=& 2q^2 (e^u - 1) - 2q\kappa A_0 + 4\pi \sum_{j=1}^m
\delta(z-z_j) \label{ueq}  \\
\Delta A_0 &=& \kappa q (1 - e^u) + (\kappa^2 + 2q^2 e^u) A_0 \label{rawAeq},
\end{eqnarray}
where $q>0$ is the charge of electron, and $\kappa >0$ is the Chern-Simons
coupling constant.
Each point of the prescribed set ${\bf Z}=\{ z_1, \cdots, z_m \}$ is called a
vortex point.
We consider two different physically-meaningful sets of boundary conditions related to finiteness of total energy (\cite{Lee}, \cite{CK}).
One is the topological boundary condition,
\begin{equation}
\lim_{|z|\to \infty} u= 0, \quad \label{tb}
\lim_{|z|\to \infty} A_0 =0, \label{tbc}
\end{equation}
and the other is the nontopological boundary condition,
\begin{equation}
\lim_{|z|\to \infty} u= -\infty, \quad \label{nb}
\lim_{|z|\to \infty} A_0 =-\frac{q}{\kappa}, \label{nbc}
\end{equation}

The system (1)-(2) arises from a Jaffe-Taubes\cite{Taubes} reduction of the
Bogomolnyi equations of the self-dual
Maxwell-Chern-Simons-Higgs system\cite{Lee}.
One major motivation for introducing the system was unification of two
apparently separate systems,
namely, the Abelian Higgs system and the Chern-Simons system.
The Abelian Higgs system reduces to
\begin{equation}
\Delta u = 2q^2 (e^{u} - 1) + 4\pi \sum_{j=1}^m \delta(z-z_j) , \label{ah}
\end{equation}
while the Chern-Simons system reduces to
\begin{equation}
\Delta u= 4l^2  e^u (e^{u} - 1) + 4\pi \sum_{j=1}^m \delta(z-z_j) ,\label{cs}
\end{equation}
where $l$ is a physical parameter related to the Chern-Simons
constant.
The Abelian Higgs system (\ref{ah}) has finite-energy solutions under only the topological boundary conditions (\ref{tb}), while the Chern-Simons system has finite-energy solutions under either boundary conditions (\ref{tb}) or (\ref{nb}).
For the precise meaning of the finite
energy of these systems, we refer to \cite{Taubes}.

The mathematical analysis  of the topological multi-vortex solutions for the
Abelian Higgs system and the
Chern-Simons system is studied in \cite{Taubes} and \cite{Wang}, \cite{Spruck}.
We note that recently existence of non-topological multi-vortex solutions of
(\ref{cs}) is established in \cite{CI}.
The existence of topological multi-vortex solutions of (\ref{ueq})-(\ref{rawAeq})
was established by a simple variational argument in \cite{CK}.
Moreover, the existence of so-called admissible solutions was established by an
iteration scheme in \cite{CK}.
We recall the definition of  admissible topological solution of (\ref{ueq})-
(\ref{tbc}).
\paragraph{Definition}
A topological solution  pair $(u, A_0)$ of (\ref{ueq})-(\ref{rawAeq}) is called
admissible if
it satisfies one of the following inequalities.
\begin{itemize}
\item[(i)] $A_0 \leq 0$
\item[(ii)] $u \leq 0$
\item[(iii)] $A_0\geq \frac{q}{\kappa} (e^{u} -1 )$
\item[(iv)] $v\leq u_a ^q$, where $u_a ^q$ is the solution the (topological)
Abelian Higgs equation (\ref{ah}).
\end{itemize}

We remark that the conditions (i)-(iv) are shown to be equivalent to each
other\cite{CK}.
In \cite{CK} we also considered the two convergence problems of the {\em
admissible} topological solutions of (\ref{ueq})-(\ref{rawAeq}) .
One is the Abelian Higgs limit, namely the problem of identifying the behavior of
the solution $u^{\kappa , q}$ of (\ref{ueq})-(\ref{rawAeq}) as $\kappa \to 0$ with $q$ kept
fixed.
The other is the Chern-Simons limit, the similar problem as both $\kappa$ and
$q$ go to
infinity with the ratio $l=q^2/\kappa$ kept fixed.
In  the Abelian Higgs limit we proved in \cite{CK} that admissible topological
solution of  (\ref{ueq})-(\ref{rawAeq}) converges strongly to the solution of
the Abelian Higgs solution, while in the Chern-Simons limit we could just show
that our solution
is weakly consistent to the Chern-Simons equation.
For the precise statements of these results see \cite{CK}.
The natural open questions raised were
\begin{enumerate}
\item Is any topological solution of (\ref{ueq})-(\ref{rawAeq}) , which is
smooth except at the points $z_1, \cdots, z_m$, admissible?
\item Can we strengthen the sense of convergence in the Chern-Simons limit
problem?
\end{enumerate}
Question 1 is concerned with the physical validity of the model system,
(\ref{ueq})-(\ref{rawAeq}), since
$u\leq 0$ is equivalent to the condition $|\phi|^2 =e^u \leq 1$ for the Higgs field $\phi$.
Question 2, combined with the already established strong convergence in the
Abelian Higgs limit,
 is concerned with the rigorous verification of the physical argument that
the Maxwell-Chern-Simons-Higgs model is a unification of the Abelian Higgs model
and
the Chern-Simons model\cite{Lee}.
In \cite{CK1} we answer these two questions in the affirmative.
In the next section we state our results and their implications.

For further discussion, we introduce the background function $u_0$
defined by
\[
  u_0= \sum_{j=1}^m  \ln\left(
              \frac{|z - z_j|^2}{1 + |z - z_j|^2}\right),
\]
and we set $u = v + u_0$ to remove the singular inhomogeneous term
in (\ref{ueq}).  Then (\ref{ueq}) and (\ref{rawAeq}) become
\begin{eqnarray}
\Delta v &=& 2q^2 (e^{v+u_0} - 1) - 2q\kappa A_0 + g, \label{veq}\\
\Delta A_0 &=& \kappa q (1 - e^{v+u_0}) + (\kappa^2 + 2q^2 e^{v+u_0}) A_0 \label{Aeq}
\end{eqnarray}
with the topological boundary condition
\begin{equation}
\lim_{|z|\to \infty} v= 0, \quad
\lim_{|z|\to \infty} A_0 =0, \label{bc}
\end{equation}
where
\[
g  = \sum_{j=1}^m \frac{4}{(1 + |z - z_j|^2)^2}.
\]
In this setting the Chern-Simons equation (\ref{cs}) becomes
\begin{equation}
\Delta v =4l^2 e^{v+u_0} (e^{v+u_0} -1 )+g. \label{cs1}
\end{equation}

\section{Main Results}
The following result is proved in \cite{CK1}.
\begin{Thm}
Suppose ${\bf Z} =\{ z_1, \cdots, z_m\}\subset {\mathbb R}^2$ is given as before. Then any
topological solution $(u, A_0 )$ in $C^2 ({\mathbb R}^2 \setminus {\bf Z} )$ is admissible.
\end{Thm}
One immediate consequence of Theorem 1 and the argument of the construction
is that the solution constructed in Section 3 is maximal. On the other
hand, due to the monotonicity of the minimizing functional $\cal{F}$,
\begin{eqnarray} \label{func}
{\cal F}(v) &=& \int \biggl[ \frac{1}{2}|\Delta v|^2 - (\Delta g -\kappa^2g )v
             + 2q^4(e^{v+u_0}-1)^2  \nonumber \\
            && + \frac{1}{2}\kappa^2 |\nabla v|^2
             + 2q^2e^{v+u_0}|\nabla (v+u_0)|^2 \biggr] dx,
\end{eqnarray}
as established in Section 4 of \cite{CK}, the solution constructed in
Section 2 of \cite{CK} by the variational  method is minimal.
Thus we have, as a result,  constructed the maximal and the minimal
solutions of the system.
As remarked in \cite{CK1}, the strong convergence in the Chern-Simons limit for
admissible solutions in the periodic
boundary condition can be extended to the case of our solutions of the system
(\ref{veq})-(\ref{Aeq}). In particular, due to
Theorem 1 we can remove the condition of admissibility, and obtain:
\begin{Thm}
Let $(v^{\kappa,q}, A_0^{\kappa,q})$ be any  topological solution of  (\ref{veq})-(\ref{Aeq}). Then
$v^{\kappa,q} \to v_{cs}^l$, and $\frac{\kappa}{q}A_0^{\kappa,q} \to e^{v_{cs}^l +u_0} -1$,
both in $H^1 ({\mathbb R}^2)$
 as $\kappa \to \infty$ with $\frac{q^2}{\kappa} = l$ kept fixed, where
$v_{cs}^l$ denotes a topological solution
of (\ref{cs1}).
\end{Thm}
We now have further open problems to consider for the system (\ref{ueq})-
(\ref{rawAeq}):
\begin{enumerate}
\item Prove uniqueness, or  multiplicity of topological solution.
\item Prove existence of non-topological multi-vortex solutions.
\end{enumerate}
\paragraph{Acknowledgements.}
The author would like to thank Prof. J. Jang for helpful
discussions on topics issued in this note. This research is
supported partially by KOSEF(K97070202013), BSRI-MOE, GARC-KOSEF
and SNU Research Fund.

\begin{thebibliography}{0}
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topological multi-vortex solutions in the
relativistic self-dual Chern-Simons theory}, IMA Preprint Series {\bf 1629},
(1999).
\bibitem{CK} D. Chae and N. Kim, {\it Topological multi-vortex solutions of the
self-dual
Maxwell-Chern-Simons-Higgs system },  J. Differential Equations {\bf 134 },
no. 1, pp. 154--182(1997).
\bibitem{CK1} D. Chae and N. Kim, {\it Vortex condensates in the relativistic
self-dual
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(1997).
\bibitem{Taubes} A. Jaffe and C. Taubes, Vortices and Monopoles,
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\bibitem{Wang} R. Wang, {\it The Existence of Chern-Simons Vortices},
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\end{thebibliography}

\noindent{\sc Dongho Chae}\\
        Department of Mathematics\\
        Seoul National University\\
        Seoul 151-742, Korea \\
        e-mail: dhchae@math.snu.ac.kr
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