Harmonic functions which vanish on a cylindrical surface
type: Item
instance of: scholarly article
title: Harmonic functions which vanish on a cylindrical surface
zbMATH Open document ID: 1327.31012
author: Stephen J. Gardiner
author: Hermann Render
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-09-30T00:00:00Z
review text: Let \(a>0\), \(N\geq 3\) and \(\Omega_a\) be the finite cylinder \(B'\times (-a,a)\), where \(B'\) is the open unit ball in \({\mathbb R}^{N-1}\). The main result of this paper establishes that any harmonic function \(h\) on \(\Omega_a\) that vanishes continuously on \(\partial B'\times (-a,a)\) has a harmonic extension \(\tilde h\) to \({\mathbb R}^{N-1}\times (-a,a)\). The authors also establish the following estimate: for any \(b\in (0,a)\) there is a constant \(c=c(N,a,b,h)\) such that \(|\tilde h(x)|\leq c\|x'\|^{1-N/2}\) for all \(x=(x',x_N)\in ({\mathbb R}^{N_1}\setminus B')\times (-b,b)\). The approach relies on the analysis of the convergence properties of double series expansion of the Green function for an infinite cylinder.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 6487342
zbMATH Keywords: harmonic continuation
zbMATH Keywords: Green function
zbMATH Keywords: cylindrical domains
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.08.077
OpenAlex ID: W1875618824
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DOI: 10.1016/J.JMAA.2015.08.077
description: scientific article; zbMATH DE number 6487342