The Variational Inequality Approach to the One-Phase Stefan Problem

Considering the one-phase Stefan problem, we present an account of some recent mathematical results within the framework of variational inequalities. We discuss several situations corresponding to different boundary conditions and different geometries, like the exterior problem, the continuous casting model, and the degenerate case of the quasi-steady model. We develop a few continuous-dependence results explaining their relevance to the stability properties of the solution and of the free boundary, including the asymptotic behaviour for large time, the stability for homogenization, and the perturbation of the Dirichlet boundary conditions. AMS (MOS) subject classifications (1980). Primary 35R35; secondary 35K60, 35K99.