Regularity of a free boundary problem
type: Item
instance of: scholarly article
title: Regularity of a free boundary problem
zbMATH Open document ID: 0965.35194
DOI: 10.1007/BF02921977
author: Lavi Karp
author: Henrik Shahgholian
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: Define \(v\) to be a Newtonian potential: \[ v(x)=\int \limits_\Omega E_n(x-y) f(y)dy, \] \noindent where \[ E_n=\begin{cases} \frac 1 {2 \pi} log |x |\quad \quad \quad n=2, \\ \frac 1 {(2-n)\omega _n}|x |^{2-n} \quad n \geq 3, \end{cases} \] \(\omega _n\) is the area of the unit sphere in \(\mathbb{R}^n\) and \(\Omega\) is a bounded open set in \(\mathbb{R}^n\). The authors assume that \(f\) is a real analytic function in some neighbourhood of \(x_0 \in \partial \Omega\) and \(v\) is analytically continuable from the complement of \(\Omega\) in some neighbourhood of \(x_0\). The main problem is to study under which conditions one can conclude that \(\partial \omega \cap B_\rho (x_0) (0<\rho \leq r)\) is a real analytic surface.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 31A25
zbMATH DE Number: 1542107
zbMATH Keywords: Newtonian potential
zbMATH Keywords: analytic (harmonic) continuation
zbMATH Keywords: module of continuity
zbMATH Keywords: blow up
reviewed by: Mykola Krasnoschok
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921977
OpenAlex ID: W2044390060
description: scientific article; zbMATH DE number 1542107