The singular set in the Stefan problem
Alessio Figalli, Xavier Ros‐Oton, Joaquim Serra · Journal of the American Mathematical Society · 2023
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 n-1 . The solution admits a C ∞ C^\infty -expansion at all singular points, up to a set of parabolic Hausdorff dimension at most n − 2 n-2 . In R 3 \mathbb {R}^3 , the free boundary is smooth for almost every time t t , and the set of singular times S ⊂ R \mathcal S\subset \mathbb {R} has Hausdorff dimension at most 1 / 2 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.