Regularity of the free boundary in parabolic phase-transition problems.

*(English)*Zbl 0891.35164The 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 \(F\).

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 \(F\).

Reviewer: J.Steinbach (München)

##### MSC:

35R35 | Free boundary problems for PDEs |

80A22 | Stefan problems, phase changes, etc. |

35K55 | Nonlinear parabolic equations |

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\textit{I. Athanasopoulos} et al., Acta Math. 176, No. 2, 245--282 (1996; Zbl 0891.35164)

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