Regularity of the free boundary in the obstacle problem.

Free boundary problems are those described by PDEs that exhibit a-priori unknown (free) interfaces or boundaries. Such type of problems appears in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, and Geometric Measure Theory. The main mathematical challenge is to understand the regularity of free boundaries. The Stefan problem and the obstacle problem are the most classical and motivating examples in the study of free boundary problems. A milestone in this context is the classical work of Caffarelli, in which he established for the first time the regularity of free boundaries in the obstacle problem, outside a certain set of singular points. The goal of this thesis is to study the regularity of solutions and free boundary in the obstacle problem.