Solution of the Robin problem for the Laplace equation
type: Item
instance of: scholarly article
title: Solution of the Robin problem for the Laplace equation
zbMATH Open document ID: 0938.31005
DOI: 10.1023/A:1023267018214
author: Dagmar Medková
published in: Applications of Mathematics
publication date: 2000-05-22T00:00:00Z
full work available at URL: https://eudml.org/doc/33003
review text: Let \(G\subset \mathbb {R}^m\) (\(m>2\)) be an open set with compact boundary \(\partial G\neq \emptyset \). Let \(\lambda \) be a finite Borel measure supported by \(\partial G\) and suppose that the single layer potential \(U\lambda \) of the measure \(\lambda \) is continuous and bounded on \(\partial G\). Further let \(\mu \) be a finite signed Borel measure supported by \(\partial G\). A generalized Robin problem for the Laplace equation on \(G\) is formulated in the following way: Find a function \(u\in L^1(\lambda)\) on \(\operatorname {cl}G\) such that (a) \(u\) is harmonic on \(G\), (b) \(u\) is finely continuous in \(\lambda \)-a.a. points of \(\partial G\) (c) \(|\nabla u|\) is integrable over all bounded open subsets of \(G\), (d) \(N^Gu+u\lambda =\mu \), where \(N^Gu\) denotes the weak characterization of the normal derivative of \(u\). The solution of that problem is represented by the single layer potential \(U\nu \) of a signed measure \(\nu \) on \(\partial G\). It is proved that if \(G\) has a smooth boundary or \(m=3\) and \(G\) has a piecewise-smooth boundary then there is a solution of that problem with given \(\mu \) if and only if \(\mu (\partial H)=0\) for all bounded components \(H\) of \(\operatorname {cl}G\) for which \(\lambda (\partial H)=0\).
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1450147
zbMATH Keywords: Laplace equation
zbMATH Keywords: Robin problem
zbMATH Keywords: single layer potential
reviewed by: Miroslav Dont
MaRDI profile type: Publication
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OpenAlex ID: W56584169
description: scientific article; zbMATH DE number 1450147