The singular set in the Stefan problem

Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim

doi:10.1090/jams/1026

The singular set in the Stefan problem
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by Alessio Figalli, Xavier Ros-Oton and Joaquim Serra;

J. Amer. Math. Soc. 37 (2024), 305-389

DOI: https://doi.org/10.1090/jams/1026

Published electronically: July 3, 2023

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Abstract:

In this paper we analyze the singular set in the Stefan problem and prove the following results:

The singular set has parabolic Hausdorff dimension at most $n-1$.

The solution admits a $C^\infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$.

In $\mathbb {R}^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \mathbb {R}$ has Hausdorff dimension at most $1/2$.

These results provide us with a refined understanding of the Stefan problem’s singularities and answer some long-standing open questions in the field.

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