Boundary regularity for a parabolic obstacle type problem

Mathematical background: The regularity of free boundaries has been extensively studied over the last thirty years and the literature on the Stefan problem is vast. This problem however (that is, the Stefan problem without sign restriction) was, to the author’s knowledge, first studied by L. A. Caffarelli, A. Petrosyan and H. Shahgholian in [CPS]. The authors of [CPS] showed that a solution is C1,1 in space and Lipschitz in time, and that the free boundary is locally analytic under an assumption on the density of {u = 0} backward in time (see Definition 4 and Theorem 4). The regularity of the free boundary near contact points with the fixed boundary was investigated by D. E. Apushkinskaya, N. N. Uraltseva and H. Shahgholian in [ASU1]. The authors of [ASU1] consider the free boundary close to a fixed boundary with zero Dirichlet condition. They also assume that u > 0. Under these conditions they prove that the free boundary is uniformly C1,α away from the fixed boundary and Lipschitz as a graph over the fixed boundary near a contact point. Recently D. E. Apushkinskaya, N. Matevosyan and N. N. Uraltseva [AMU] extended the results of [ASU1] to solutions without sign restriction.