Regularity of the free boundary in parabolic phase-transition problems

Abstract

In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic two-phase free boundary problem is the Stefan problem, a simplified model describing the melting (or solidification) of a material with a solid-liquid interphase. The concept of solution can be stated in several ways (classical solution, weak so- lution on divergence form, or viscosity solution) and as usual, one would like to prove that the (weak) solutions that may be constructed, are in fact as smooth and classical as possible. Locally, a classical solution of the Stefan problem may be described as following: On the unit cylinder Q1 =B1 “ (-1, 1) we have two complementary domains, ~ and QI\~, separated by a smooth surface S=(OI2)NQ1. In fl and QI\~ we have two smooth solutions, Ul and u2, of the heat equations