Paris/Berkeley/Bonn/Zürich Analysis Seminar: On the singular set in the Stefan problem and a conjecture of Schaeffer

Seminar | October 25 | 9:10-10 a.m. | 238 Sutardja Dai Hall

 Xavier Ros-Oton, University of Zürich

 Department of Mathematics

Free boundary problems are those described by PDE's that exhibit a priori unknown (free) interfaces or boundaries. The Stefan problem is the most classical and motivating example in the study of free boundary problems. It describes the evolution of a medium undergoing a phase transition, such as ice passing to water. A milestone in this context is the classical work of Caffarelli (Acta Math. 1977), in which he established for the first time the regularity of free boundaries in the Stefan problem, outside a certain set of singular points. The goal of this talk is to present some new results concerning the size of the singular set in the Stefan problem, proving in particular that, in $\mathbb R^3$, for almost every time the free boundary is smooth, with no singularities. This is a joint work with A. Figalli and J. Serra.