Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions

TL;DR

This work analyzes the Stefan problem in four spatial dimensions and shows that, for almost every time, the free boundary is a smooth (in space) 3D manifold. The authors refine the parabolic obstacle framework by proving a frequency gap that excludes intermediate homogeneities, derive a pointwise $C^{3+\\beta}$ expansion at the maximal singular stratum via an epiperimetric inequality for a parabolic Weiss energy, and then obtain a smooth regular covering of the singular set. They further prove a quadratic cleaning mechanism and a corresponding GMT-based rectifiability decomposition, which together yield almost-everywhere smoothness in the 4+1-dimensional setting. The results advance understanding of parabolic free boundaries, offering sharp structural description and a robust multi-scale approach applicable to higher dimensions.

Abstract

We show that the free boundary of a solution of the Stefan problem in $\mathbb R^{4+1}$ is a $3$-dimensional manifold of class $C^\infty$ in $\mathbb R^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the singular set $Σ\subset \mathbb R^{n+1}$ can be decomposed in two parts $Σ=Σ^\infty\cup Σ^*$, where $Σ^\infty$ is covered by one $(n-1)$-dimensional manifold of class $C^\infty$ in $\mathbb R^{n+1}$ and its projection onto the time axis has Hausdorff dimension 0, while $Σ^*$ is parabolically countably $(n-2)$-rectifiable.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4: Frequency gap
• Lemma 1
• proof : Proof of Theorem \ref{['teo:frequency gap']}
• Lemma 2
• proof
• proof : Proof of Lemma \ref{['lem:test function']}