Regularity properties of a free boundary near contact points with the fixed boundary.
type: Item
instance of: scholarly article
title: Regularity properties of a free boundary near contact points with the fixed boundary.
zbMATH Open document ID: 1050.35157
DOI: 10.1215/S0012-7094-03-11611-7
author: Henrik Shahgholian
author: Nina Uraltseva
published in: Duke Mathematical Journal
publication date: 2003-01-01T00:00:00Z
review text: This paper concerns the regularity properties of solutions of a free boundary problem. Let \(\Omega\) be a domain in the half space \(\mathbb{R}^n_+= \{x\in\mathbb{R}^n: x_1> 0\}\), \(n\geq 2\), and suppose that a function \(u\) satisfies \[ \Delta u=\chi_\Omega\quad\text{in}\quad B^+,\quad u=|\nabla u|=0\quad\text{in}\quad B^+\setminus\Omega,\quad\text{and}\quad u= 0\quad\text{on}\quad \Pi\cap B, \] where \(B\) is the unit ball with center at the origin, \(B^+= \{x\in B: x_1> 0\}\), \(\Pi= \{x_1= 0\}\), and \(\chi_\Omega\) denotes the characteristic function of \(\Omega\). The free boundary \(\Gamma(u)\) is defined by \[ \Gamma(u)= \{x: u(x)=|\nabla u(x)|= 0\}\cap \partial\Omega. \] The main purpose of this paper is to study the behavior of the free boundary near the contact points \(\Gamma_0(u)= \Gamma(u)\cap\Pi\) with neither \(u\geq 0\) nor Lipschitz regularity assumption on \(\partial\Omega\). The authors show, among other things, that the free boundary is the graph of a \(C^1\) function near the contact points. The interior points of the free boundary have been well studied by \textit{L. A Caffarelli}, \textit{L. Karp} and \textit{H. Shahgholian} [Ann. Math. (2) 151, 269--292 (2000; Zbl 0960.35112)].
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 1911421
zbMATH Keywords: free boundary
zbMATH Keywords: regularity
zbMATH Keywords: overdetermined problem
zbMATH Keywords: contact points
zbMATH Keywords: interior points
Wikidata QID: Q56967409
MaRDI profile type: Publication
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full work available at URL: s0012-7094-03-11611-7
OpenAlex ID: W1991192948
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description: scientific article; zbMATH DE number 1911421