Probabilistic Solutions to 3D Stefan-Gibbs-Thomson Problems under Radial Symmetry: Regularity and Uniqueness
Yucheng Guo, Princeton University
We study physical solutions to the three-dimensional Stefan-Gibbs-Thomson Problems under radial symmetry, of which the existence has been established in [Stefan problem with surface tension: global existence of physical solutions under radial symmetry, Nadtochiy and Shkolnikov 2023]. In this talk, I will focus on the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. We also establish a variety of regularity estimates for the free boundary and for the temperature function in the course of the proof. This talk is based on joint work with Sergey Nadtochiy and Mykhaylo Shkolnikov.