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Regularity of the free boundary in parabolic phase-transition problems. (English) Zbl 0891.35164
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 \(F\).

35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
35K55 Nonlinear parabolic equations
Full Text: DOI
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