A two phase elliptic singular perturbation problem with a forcing term
type: Item
instance of: scholarly article
title: A two phase elliptic singular perturbation problem with a forcing term
zbMATH Open document ID: 1111.35136
author: Claudia Lederman
author: Noemí Wolanski
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2007-02-13T00:00:00Z
review text: The authors study the following two phase elliptic singular perturbation problem \[ \Delta u^\varepsilon =\beta_\varepsilon(u^\varepsilon) +f^\varepsilon\quad \text{in}\;\Omega\subset \mathbb R^N, \] where \(\varepsilon>0,\) \(\beta_\varepsilon(s)={1\over \varepsilon} \beta\left({s\over \varepsilon}\right)\) with a Lipschitz continuous function \(\beta,\) satisfying \(\beta>0\) in \((0,1),\) \(\beta\equiv0\) outside \((0,1)\) and such that \(\int \beta(s)\,ds=M,\) and the functions \(u^\varepsilon\) and \(f^\varepsilon\) are supposed to be uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit in the presence of sources. Uniform estimates and passing to the limit when \(\varepsilon\to0\) enable the authors to show that the limit functions are solutions of the two phase free boundary problem \[ \begin{aligned} &\Delta u= f\chi_{\{u\not\equiv0\}}\quad \text{in}\;\Omega\setminus \partial\{u>0\},\cr &| \nabla u^+| ^2-| \nabla u^-| ^2=2M\quad \text{on}\;\Omega\cap \partial\{u>0\}, \end{aligned} \] where \(f=\lim f^\varepsilon\) in a viscosity sense, and in pointwise sense at the regular free boundary points. It is shown in addition that the free boundary is smooth whence, under appropriate assumptions, the limit functions are classical solutions to the considered free boundary problem. Some of the results are new even in the case \(f^\varepsilon\equiv0.\) The results at the paper apply also to other combustion models, such as ones with nonlocal diffusion and/or transport. Several of these applications are discussed and, in some cases full regularity of the free boundary is obtained.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 80A25
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5125018
zbMATH Keywords: two phase free boundary problem
zbMATH Keywords: viscosity solutions
zbMATH Keywords: regularity
zbMATH Keywords: combustion
MaRDI profile type: Publication
full work available at URL: j.matpur.2006.10.008
OpenAlex ID: W2064092942
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DOI: 10.1016/J.MATPUR.2006.10.008
description: scientific article; zbMATH DE number 5125018