Regularity of Free Boundaries in Obstacle Problems

Abstract

Free boundary problems are those described by PDE that exhibit a priori unknown (free) interfaces or boundaries. Such type of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, 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. This is one of the main results for which he got the Wolf Prize in 2012 and the Shaw Prize in 2018. The goal of these notes is to introduce the obstacle problem, prove some of the main known results in this context, and give an overview of more recent research on this topic.