Published Paper
Inserted: 9 apr 2021
Last Updated: 19 aug 2024
Journal: J. Amer. Math. Soc
Year: 2024
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 \( S \subset \mathbb{R} \) has Hausdorff dimension at most \( \frac{1}{2} \).
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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