On prescribing geodesic curvature on \(D^2\)
type: Item
instance of: scholarly article
title: On prescribing geodesic curvature on \(D^2\)
zbMATH Open document ID: 1086.53054
author: Pan Liu
author: Wei Huang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-09T00:00:00Z
review text: Let \((D, g_0)\) be the unit disc with the Euclidean metric \(g_0\). Given a continuous function \(k\) on \(S^1=\partial D\), it is an interesting problem whether \(k\) can be the geodesic curvature of some flat metric \(g\) which is pointwise conformal to the metric \(g_0\). If we set \(g=e^{2u} g_0\), where \(u\) is a function defined on \(\overline{D}\), the problem is equivalent to solve the following equation: \[ -\Delta u=0 \quad\text{in}\quad D,\qquad \frac{\partial u}{\partial n}+1=k e^{u} \quad\text{on}\quad \partial D, \tag{1} \] where \(\partial u/\partial n\) is the outer normal derivative of \(u\). The problem (1) may happen to be insolvable for general \(k\). In the paper under review the authors obtain some conditions for solvability of the problem (1) in the case when \(k\) possesses some kinds of symmetries.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 2132716
zbMATH Keywords: geodesic curvature
zbMATH Keywords: minimax principle
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
full work available at URL: j.na.2004.09.013
OpenAlex ID: W2090952331
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DOI: 10.1016/J.NA.2004.09.013
description: scientific article; zbMATH DE number 2132716