Regularity of interfaces in phase transitions via obstacle problems

TL;DR

This paper reviews recent advances in the regularity theory of obstacle problems, focusing on the structure of singular free boundary points and applications to phase transition models like Stefan problems.

Contribution

It summarizes new results on free boundary regularity, including progress on Schaeffer's conjecture and smoothness in phase transition models.

Findings

Structure of singular free boundary points characterized

Almost everywhere smoothness of free boundary in Stefan problem

Progress on Schaeffer's conjecture regarding free boundary regularity

Abstract

The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some recent results on the structure of singular free boundary points. Then, we show some selected applications to the generic smoothness of the free boundary in the stationary obstacle problem (Schaeffer's conjecture), and to the smoothness of the free boundary in the one-phase Stefan problem for almost every time.