Xavier-Ros Oton: Regularity of free boundaries in obstacle problems Lecture I

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. The goal of this course is to introduce these free boundary problems, prove some of the main known results in this context, and give an overview of the current research and open problems.