On the regularity in obstacle problems

This thesis we deal with classical and thin problems. In the first chapter we establish Weiss’ and Monneau’s type quasi-monotonicity formulas are for quadratic energies having matrix of coefficients which are Dini, double-Dini continuous, respectively. Free boundary regularity for the corresponding classical obstacle problems under Hölder continuity assumptions is then deduced. In the third chapter of this thesis we consider the thin boundary obstacle problem for a 1 general class of non-linearities and we prove the optimal C 1, 2 -regularity of the solutions in any space dimension. In the last chapter we establish a quasi-monotonicity formula for an intrinsic frequency function related to solutions to thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev W 1,p coefficients, with p bigger than the space dimension. From this we deduce several regularity and structural properties of the corresponding free boundaries at those distinguished points with finite order of contact with the obstacle. In particular, we prove the rectifiability and the local finiteness of the Minkowski content of the whole free boundary in the case of Lipschitz coefficients.