We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight \(|y|^a\) for \(a \in (-1,1)\). Such problem arises as the local extension of the obstacle problem for the fractional heat operator \((\partial _t - \Delta _x)^s\) for \(s \in (0,1)\). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation (\(a=0\)).

中文翻译：

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退化抛物方程的薄障碍问题中奇异集的结构

我们针对稀薄抛物方程的权重为\（| y | ^ a \）对于\（a \ in（-1,1）\）的奇异集研究奇异集。这样的问题是由于分数热算子\（（\ partial_t-\ Delta _x）^ s \）对于\（s \ in（0,1）\）的障碍问题的局部扩展而产生的。我们的主要结果建立了自由边界奇异集的完整结构和规则性。为此，我们证明了Almgren-Poon，Weiss和Monneau型单调性公式，这些公式将热方程（\（a = 0 \））的情况进行了推广。