A symmetry result for the torsion equation in an unbounded domain in 3D
type: Item
instance of: scholarly article
title: A symmetry result for the torsion equation in an unbounded domain in 3D
zbMATH Open document ID: 1379.35201
author: Lizhou Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-08-03T00:00:00Z
review text: An overdetermined problem is obtained by associating to a partial differential equation so many conditions that no solution exists, in general. The situation may be compared, roughly speaking, to a linear algebraic system with more (independent) equations than unknowns. Nevertheless, it may well happen that an overdetermined problem does possess a solution in some particular domain \(\Omega\). For instance, a celebrated result by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304--318 (1971; Zbl 0222.31007)] and \textit{H. F. Weinberger} [Arch. Ration. Mech. Anal. 43, 319--320 (1971; Zbl 0222.31008)] asserts that the following overdetermined problem: \[ \begin{cases} \Delta u = -1 &\text{in \(\Omega\),} \cr u = 0 &\text{on \(\partial \Omega\),} \cr |\nabla u| = c &\text{on \(\partial \Omega\),} \end{cases} \] where \(\Omega\) is a bounded, smooth domain in \(\mathbb R^N\), \(N \geq 2\), and \(c\) is a positive constant, is solvable if and only if \(\Omega\) is a ball. The case when \(\Omega\) is an \textit{exterior domain}, i.e., a domain whose complement is compact, was investigated by \textit{A. Aftalion} and \textit{J. Busca} [C. R. Acad. Sci., Paris, Sér. I 324, 633--638 (1997; Zbl 0876.35032)] and by \textit{W. Reichel} [Arch. Ration. Mech. Anal. 137, 381--394 (1997; Zbl 0891.35006)]. The reviewed paper is part of a branch of current research on similar problems in unbounded domains: see, for instance, [\textit{F. Hélein} et al., Pacific J. Math. 250, 319--334 (2011; Zbl 1211.35207); \textit{A. Ros} et al., Commun. Pure Appl. Math. 70, 1223--1252 (2017; Zbl 1373.35207); J. Differ. Equations 255, 951--977 (2013; Zbl 1284.35297); \textit{F. Schlenk} and \textit{P. Sicbaldi}, Adv. Math. 229 (2012; Zbl 1233.35147); \textit{M. Traizet}, Geom. Funct. Anal. 24, 690--720 (2014; Zbl 1295.35344)]. To be more precise, let \(u\) be a continuous function over \(\mathbb R^3\) satisfying \(0 \leq u \leq a\) in \(\mathbb R^3\) and such that the set \(\{\, 0 < u < a \,\}\) i…
reviewed by: Antonio Greco
Mathematics Subject Classification ID: 35N25
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B06
zbMATH DE Number: 6756247
zbMATH Keywords: distance function to the catenoid
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.05.065
OpenAlex ID: W1977742058
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DOI: 10.1016/J.JMAA.2017.05.065
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description: scientific article; zbMATH DE number 6756247