The singular set in the Stefan problem

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 $S \subset R$ 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.

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 $S \subset R$ 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.

文献摘要