发表机构
Scuola Normale Superiore; EPFL SB MATH Institute of Mathematics; Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas（高等师范学院; 洛桑联邦理工学院数学研究所; 西班牙国家研究委员会数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对模拟薄冰层融化的薄斯蒂芬问题，通过变换转化为抛物型西诺里尼问题，证明其奇异集的维数上界，建立低维下的正则性，并结合经典方法与薄情形新要素完成分析。
AI 中文摘要
我们研究$\boldsymbol{\text{R}}^{n+1}\times\boldsymbol{\text{R}}$中薄斯蒂芬问题解的自由边界的奇异集，该问题用于模拟水面上薄冰层的融化过程。经过适当变换，我们将其简化为研究具有时间单调性条件的抛物型西诺里尼（薄障碍）问题。首先，我们证明自由边界局部为定义在$n$维薄空间上的可微融化函数$t=\tau(x)$的图像，且奇异集与$\tau$的临界集重合。接下来，我们证明奇异集的抛物型豪斯多夫维数至多为$n$。随后，我们建立了直至维数$n+1=4$的一般性正则性，证明自由边界在几乎所有时间都是正则的。最后，在障碍满足光滑性假设的条件下，我们证明奇异集包含在一个$C^\text{\text{infty}}$超曲面中，该超曲面位于抛物型维数至多为$n-1$的集合之外。我们的分析结合了经典斯蒂芬问题的研究思路与若干针对薄情形的新要素，包括二阶爆破的频率公式、 epi-度量不等式、高阶展开式以及奇异集的精细分层。
英文摘要
We study the singular set of the free boundary for solutions of the thin Stefan problem in $\mathbb{R}^{n+1}\times \mathbb{R}$, which models the melting of a thin sheet of ice over water. After a suitable transformation, we reduce to studying the parabolic Signorini (thin obstacle) problem with a time-monotonicity condition. First, we prove that the free boundary is locally the graph of a differentiable melting function $t=τ(x)$ defined on the $n$-dimensional thin space, and that the singular set coincides with the critical set of $τ$. Next, we show that the singular set has parabolic Hausdorff dimension at most $n$. We then establish generic regularity up to dimension $n+1=4$, proving that the free boundary is regular for almost every time. Finally, under smoothness assumptions on the obstacle, we show that the singular set is contained in a $C^\infty$-hypersurface outside a set of parabolic dimension at most $n-1$. Our analysis combines ideas developed for the classical Stefan problem with several new ingredients specific to the thin setting, including frequency formulas for second blow-ups, epiperimetric inequalities, higher order expansions, and a fine stratification of the singular set.
