Regularity of the free boundary in parabolic phase-transition problems
type: Item
instance of: scholarly article
title: Regularity of the free boundary in parabolic phase-transition problems
zbMATH Open document ID: 0891.35164
DOI: 10.1007/BF02551583
published in: Acta Mathematica
publication date: 1997-11-05T00:00:00Z
review text: The comprehensive paper is devoted to the study of regularity properties of the free boundary for general parabolic two-phase free boundary problems in several space variables. An important application of this type of phase-transition problems is the two-phase Stefan problem describing the melting/solidification of a material with a solid-liquid interphase. The authors consider (so-called) viscosity solutions whose free boundary is given (locally) by a Lipschitz graph. In general, one cannot expect the smoothing of the free boundary \(F\). This is shown by means of a counter example for the Stefan problem at the end of the paper. Therefore, a class of (in a sense non-degenerate) problems is studied, for which the authors prove that the regularity of the free boundary can be pushed to \(C^1\) and the assumed viscosity solution is also a classical solution. The non-degeneracy condition states, roughly speaking, that the two heat fluxes are not vanishing simultaneously on \(F\).
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1083664
zbMATH Keywords: parabolic free boundary problem
zbMATH Keywords: two-phase Stefan problem
zbMATH Keywords: regularity of the free boundary
author: I. Athanasopoulos
author: Luis Caffarelli
author: Sandro Salsa
MaRDI profile type: Publication
cites work: A theorem of real analysis and its application to free boundary problems
cites work: Variational problems with two phases and their free boundaries
cites work: Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems
cites work: A Harnack inequality approach to the regularity of free boundaries. I: Lipschitz free boundaries are \(C^{1,\alpha}\)
cites work: A Harnack inequality approach to the regularity of free boundaries part II: Flat free boundaries are Lipschitz
cites work: Continuity of the temperature in the two-phase Stefan problem
cites work: A class of non-degenerate two-phase stefan problems in several space variables
cites work: Inequalities for the Green Function and Boundary Continuity of the Gradient of Solutions of Elliptic Differential Equations.
description: scientific article; zbMATH DE number 1083664