Maximal solution of the Liouville equation in doubly connected domains

TL;DR

This paper investigates the asymptotic behavior of solutions to the Liouville equation in doubly connected domains, revealing the existence of a special curve and describing the limiting harmonic function and integral behavior as the parameter tends to zero.

Contribution

It establishes the existence of a curve and describes the asymptotic limit of solutions to the Liouville equation in doubly connected domains, linking the behavior to the domain's conformal class.

Findings

Existence of a simple closed curve in the domain for solution sequences.

Asymptotic convergence of scaled solutions to a harmonic function outside the curve.

Limit of the scaled integral of the exponential of solutions relates to the domain's conformal class.

Abstract

In this paper we consider the Liouville equation $\Delta u +\lambda^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $\Omega$. We show that there exists a simple, closed curve $\gamma\subset \Omega$ such that for a sequence $\lambda_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{\lambda_n}}\to H$, where $H$ is a harmonic function in $\Omega\setminus\gamma$ and $\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega e^{\,u_n}\,dx\to 8\pi c_\Omega$, where $c_\Omega$ is a constant depending on the conformal class of $\Omega$ only.