Two-Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front

We continue to develop the regularity theory of general two-phase free boundary problems for parabolic operators. In \cite{FS} we establish the optimal (lipschitz) regularity of a viscosity solution under the assumptions that the free boundary is locally a flat Lipschitz graph and a nondegeneracy condition holds. Here, on one side we improve this result by removing the nondegeneracy assumption, on the other side we prove the smoothness of the front. The proof relies in a crucial way on a local stability result stating that, for a certain class of operators, under small perturbations of the coefficients flat free boundaries remain close and flat.