The Stefan problem, dating back to the XIXth century, is probably the most classical and well-known free boundary problem. The regularity of free boundaries in such problem was developed in the groundbreaking paper (Caffarelli, Acta Math. 1977). The main result therein establishes that the free boundary is $C^\infty$ in space and time, outside a certain set of singular points.

The fine understanding of singularities is of central importance in a number of areas related to nonlinear PDEs and Geometric Analysis. In particular, a major question in such context is to establish estimates for the size and structure of the singular set. The goal of this talk is to present some new results in this direction for the Stefan problem. This is a joint work with A. Figalli and J. Serra.

Short bio: Xavier Ros-Oton is an ICREA Research Professor at the University of Barcelona since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He works on elliptic and parabolic PDEs, and has established important results on free boundary problems and integro-differential equations. He is the PI of an ERC Starting Grant, and has received several awards for young mathematicians in Spain.