Well-posedness for a cauchy problem associated to time dependent free boundaries with nonlocal leading terms

Abstract

We study a class of free boundary problems, where the normal velocity of the interface is proportional to the derivative of the solution of an elliptic PDE; we give a simple, explicit criterion for the well-posedness of the linearized Cauchy problem. The method is then applied to two classical problems; the stefan problem and the Muskat problem.