Least supersolution approach to regularizing free boundary problems
type: Item
instance of: scholarly article
title: Least supersolution approach to regularizing free boundary problems
zbMATH Open document ID: 1170.35101
author: Diego R. Moreira
published in: Archive for Rational Mechanics and Analysis
publication date: 2009-02-09T00:00:00Z
review text: In the interesting paper under review, the author studies a free boundary problem obtained as a limit for \(\varepsilon \to 0\) of the regularizing family of semilinear elliptic equations \(\Delta u = \beta_\varepsilon(u) F(\nabla u)\), where \(\beta _\varepsilon\) approximates the Dirac delta function at the origin and \(F\) is a Lipschitz continuous function bounded away from \(0\) and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces uniformly in \(\varepsilon.\) This allows to prove that the free boundary of a limit has the ``right'' (in the measure theoretical sense) weak geometry. A classification of the global profiles of the blow-up analysis is carried out by means of suitable barriers with curvature, and the limit functions are proven to be viscosity and pointwise solution (\({\mathcal H}^{n-1}\) almost everywhere) to the free boundary problem. Finally, it is proved that the free boundary is a \(C^{1,\alpha}\)-surface around \({\mathcal H}^{n-1}\)-almost everywhere point.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 5503635
zbMATH Keywords: free boundary problems
zbMATH Keywords: least supersolution
zbMATH Keywords: viscosity solution
zbMATH Keywords: level surface
zbMATH Keywords: barrier
MaRDI profile type: Publication
full work available at URL: s00205-008-0113-9
OpenAlex ID: W2052124652
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DOI: 10.1007/S00205-008-0113-9
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description: scientific article; zbMATH DE number 5503635