IJPAM: Volume 23, No. 1 (2005)

CONVEXITY AND UNIQUENESS OF THE SOLUTION
TO THE LIOUVILLE EQUATION

Tomohiko Sato$^1$, Takashi Suzuki$^2$
$^{1,2}$Division of Mathematical Science
Department of System Innovation
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonakashi, Osaka, 560-8531, JAPAN
$^1$e-mail: t-sato@sigmath.es.osaka-u.ac.jp
$^2$e-mail: suzuki@sigmath.es.osaka-u.ac.jp


Abstract.In this paper we study uniqueness and convexity of the level sets of the solution to the Liouville equation

\begin{displaymath}
- \Delta v = \lambda V(x) e^v \quad \mbox{in $\Omega$}, \qquad v=0 \quad \mbox{on $\partial \Omega$},
\end{displaymath}

where $\Omega \subset \mathbb{R}^2$ is a bounded domain with smooth boundary, $V(x)>0$ is a positive-valued $C^1(\overline{\Omega})$ function, and $\lambda >0$ is a constant.

Received: May 10, 2005

AMS Subject Classification: 35J60

Key Words and Phrases: Liouville equation, blow-up analysis, Green's function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 23
Issue: 1