A duality theorem for an overdetermined eigenvalue problem
type: Item
instance of: scholarly article
title: A duality theorem for an overdetermined eigenvalue problem
zbMATH Open document ID: 0842.35066
DOI: 10.1007/BF00917447
author: N. B. Willms
author: Marc Chamberland
author: Graham M. L. Gladwell
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1995-09-20T00:00:00Z
review text: The overdetermined eigenvalue problem for the Laplacian \[ \Delta u + \lambda u = 0 \quad \text{in } \Omega, \qquad u = b, \;{\partial u \over \partial n} = c \quad \text{on } \partial \Omega \tag{P} \] is studied in a bounded domain \(\Omega\), for \(\lambda\), \(b > 0\). Call \(w\) a Helmholtz function of wave number \(\lambda\) in \(\Omega\), if \(w\) satisfies the equation \(\Delta w + \lambda w = 0\). The main result of this paper is: There exists a solution \(u \in C^2 (\Omega) \cap C^1 (\overline \Omega)\) of (P) if and only if \(\int_\Omega wdx = d \int_{\partial \Omega} wd \sigma\) for all Helmholtz functions \(w\) of wave number \(\lambda\) \((d\) is an arbitrary fixed constant).
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 798850
zbMATH Keywords: overdetermined boundary condition
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00917447
OpenAlex ID: W2069212903
description: scientific article; zbMATH DE number 798850