We investigate the homogenization of Stefan-type problems with oscillating diffusion coefficients. Both cases of periodic and random (stationary ergodic) medium are considered. The proof relies on the coincidence of viscosity solutions and weak solutions (which are the time derivatives of the solutions of an obstacle problem) for the Stefan problem. This coincidence result is of independent int...

We study the 1D contracting Stefan problem with two moving boundaries that describes the freezing of a supercooled liquid. The problem is borderline ill–posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, ρ0(x) does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times...

The Stefan problem with small surface tension e is considered. Assuming that the classical Stefan problem (with s = 0) has a smooth free boundary T, we denote the temperature of the solution by 60 and consider an approximate solution 60 + su for the case where e ^ 0, e small. We first establish the existence and uniqueness of u , and then investigate the effect of u on the free boundary T. It i...