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arXiv · 1902.07457

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

Agnid Banerjee·Donatella Danielli·Nicola Garofalo·Arshak Petrosyan
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Abstract

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 ($a=0$).

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Agnid Banerjee, Donatella Danielli, Nicola Garofalo, Arshak Petrosyan. 2019-02-20. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations. https://arxiv.org/abs/1902.07457

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