Vortrag
What is... a free boundary problem?
Vortrag von Prof. Dr. Xavier Ros-Oton
Datum: 27.02.18 Zeit: 17.15 - 18.30 Raum:
Free boundary problems are those described by PDE's that exhibit a priori unknown (free) interfaces or boundaries. In other words, there are two unknowns in these problems: the solution of a PDE, and a free boundary which determines the domain in which the PDE is satisfied. Such type of problems appear in Probability, Physics, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDE's, Calculus of Variations, and Geometric Measure Theory. The main mathematical challenge is understanding the regularity of free boundaries. The obstacle problem is the most classical and motivating example in the study of free boundary problems. A milestone in this context is a classical paper of L. Caffarelli (Acta Math. 1977), in which he established for the first time the regularity of free boundaries in the obstacle problem. This is one of the main results for which he got the Wolf Prize in 2012. The goal of this talk is to present and motivate different free boundary problems, explain the main known results in this context, and give an overview of the current research and open problems.