A Minimization Problem Associated With The Chern-Simons Model With Double Vortex Points On A Torus
by Chang-Shou Lin   Shusen Yan  

Vol. 8 No. 4 (2013) P.491~P.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.

KEYWORDS
concentration, mean field equation, Chern-Simons model,

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35J60, 58E11

MILESTONES

Received: 2013-06-06
Revised : 2013-08-29
Accepted: 2013-08-29

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