Uniqueness and existence results on viscosity solutions of some free-boundary problems

This work studies several examples of nonlocal and non-geometric free boundary problems: the Hele-Shaw problem, the Stefan problem and a flame propagation model. There are two main parts in our investigation. First, we introduce a notion of viscosity solutions for the one phase Hele-Shaw and Stefan problems when there is no surface tension. We prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of the Porous Medium Equation to those of the Hele-Shaw problem. We also generalize our method to free boundary problems with convex operators. Second, we study a free boundary problem describing the propagation of laminar flames. The problem arises as the limit of a singular perturbation problem. We introduce the notion of viscosity solutions to show a maximal principle property of the solutions. Using this property, we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity solution for two classes of initial data.