Liouville's formula in arbitrary planar domains
type: Item
instance of: scholarly article
title: Liouville's formula in arbitrary planar domains
zbMATH Open document ID: 1122.35029
author: Maria Luiza Leite
author: Francisco A. Brito
author: Jorge Hounie
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: This paper deals with the Liouville equation \[ \Delta_x u(x,y)= Ke^{-2u(x,y)},\quad (x,y)\in \Omega,\tag{1} \] where \(K\) is a real constant, \(u\) is a real \(C^2\) function and \(\Omega\subset\mathbb R^2\) is a plane domain. The aim of this work is to extend Liouville's formula to an arbitrary planar domain. When \(\Omega\) is not simply connected, the authors show that each solution of (1) in an arbitrary domain \(\Omega\subset \mathbb R^2\simeq\mathbb C\) is naturally associated to a Riemannian surface over \(\Omega\) by means of analytic continuation; the representation formula will hold in terms of a meromorphic function defined on a Riemannian surface defined above applying results of \textit{K. S. Chou} and \textit{T. Y. H. Wan} [Pac. J. Math. 163, No. 2, 269--276 (1994; Zbl 0794.35049)]. The authors present an integral formula describing all solutions of Liouville's equation (1) in the punctured disk. Moreover, they also discuss solutions on the plane with one and two punctures.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 2136806
zbMATH Keywords: Liouville's equation
zbMATH Keywords: Liouville's formula
zbMATH Keywords: Covering space
MaRDI profile type: Publication
full work available at URL: j.na.2004.10.018
OpenAlex ID: W1998185099
cites work: Q4177796
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DOI: 10.1016/J.NA.2004.10.018
description: scientific article; zbMATH DE number 2136806