Regularity in degenerate elliptic and parabolic free boundary problems

In this thesis, the main topic is the study of some free boundary problems, more precisely the investigation of regularity issues in degenerate elliptic and parabolic ones. Specifically, three different problems are treated. The first one is the one-phase Stefan problem, for which the regularity of flat free boundaries is dealt with by relying on perturbation arguments leading to a linearization of the problem. This approach is inspired by the elliptic counterpart. The second problem concerns the question of the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group. Following the ideas exploited in the Euclidean setting, a necessary condition about the existence of such tool in that noncommutative setting is found. The last problem faced is related to almost minimizers of the p-Laplacian. In particular, the optimal Lipschitz continuity of almost minimizers, for p greater or equal than 2, is proved as well as the regularity of the free boundary is studied.