Serrin's result for hyperbolic space and sphere
type: Item
instance of: scholarly article
title: Serrin's result for hyperbolic space and sphere
zbMATH Open document ID: 0941.35029
DOI: 10.1215/S0012-7094-98-09102-5
author: S. Kumaresan
author: J. Prajapat
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The following result is proved: Theorem. Let \(\Omega\subset H^n\), where \(H^n\) is the \(n\)-dimensional hyperbolic space, be a bounded domain. Let \(u\in C^2(\overline\Omega)\) be a positive solution of \[ \Delta u+f(u)=0,\quad \text{in }\Omega, \] with boundary conditions \[ u=0,\quad {\partial u\over\partial\nu}=c\quad \text{on }\partial\Omega, \] where \(f\) is a \(C^1\) function, \(c\) is a constant and \(\partial u\over\partial\nu\) denotes the normal derivative along the inward normal field on \(\partial\Omega\). Then (i) \(\Omega\) is a geodesic ball and (ii) \(u\) is radially symmetric. This result generalizes a result of \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304-318 (1971; Zbl 0222.31007)] concerning a similar problem in \(\mathbb{R}^n\).
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1425071
zbMATH Keywords: Laplace-Beltrami operator in hyperbolic spaces
zbMATH Keywords: radially symmetric solutions
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09102-5
OpenAlex ID: W1965529785
description: scientific article; zbMATH DE number 1425071