# The singular set in the Stefan problem

created by figalli on 09 Apr 2021

[BibTeX]

Preprint

Inserted: 9 apr 2021
Last Updated: 9 apr 2021

Year: 2021

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 $\R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \R$ 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