# The singular set in the Stefan problem

Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

**24/3/21**Published in : arXiv:2103.13379

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 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.