2013 / December Volume 8 No.4
A Minimization Problem Associated With The Chern-Simons Model With Double Vortex Points On A Torus
| Published Date |
2013 / December
|
|---|---|
| Title | A Minimization Problem Associated With The Chern-Simons Model With Double Vortex Points On A Torus |
| Author | |
| Keyword | |
| Download | |
| Pagination | 491-503 |
| Abstract | In this paper, we will study the following minimization problem
\[
\inf\bigl\{\frac12 \int_\Omega |Du|^2-8\pi \ln \int_\Omega e^{u+u_0} : \; u\in H\bigr\},
\]
where $ \bigl\{ u\in H_{loc}^1(\mathbb R^2):\; u\; \text{is doubly periodic
in}\; \Omega, \text{and}\; \int_\Omega u=0
\bigr\}$, $ u_0(x)=-4\pi G(x,p_1)-4\pi G(x,p_2)$ and $G(x, p)$ is
the Green function of $-\Delta $ in $\Omega$
with singularity at $p$ subject to the periodic boundary condition.
We will introduce a quantity $D(p)$ for $p\in\Omega$ and prove that if
$D(p)>0$ at a maximum point of $u_0$, then the above problem has a minimizer.
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| AMS Subject Classification |
35J60, 58E11
|
| Received |
2013-08-29
|
| Accepted |
2013-08-29
|