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We study the Stefan problem with surface tension and radially symmetricinitial data. In this context, the notion of a so-called physical solution,which exists globally despite the inherent blow-ups of the melting rate, hasbeen recently introduced in [NS23]. The paper at hand is devoted to the proofthat the physical solution is unique, the first such result when the freeboundary is not flat, or when two phases are present. The main argument relieson a detailed analysis of the hitting probabilities for a three-dimensionalBrownian motion, as well as on a novel convexity property of the free boundaryobtained by comparison techniques. In the course of the proof, we establish awide variety of regularity estimates for the free boundary and for thetemperature function, of interest in their own right.

Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry