%ABanerjee, Agnid%ABanerjee, Agnid%ADanielli, Donatella%ADanielli, Donatella%AGarofalo, Nicola%AGarofalo, Nicola%APetrosyan, Arshak%APetrosyan, Arshak%BJournal Name: Calculus of Variations and Partial Differential Equations; Journal Volume: 60; Journal Issue: 3; Related Information: CHORUS Timestamp: 2022-12-25 09:36:26
%D2021%ISpringer Science + Business Media
%JJournal Name: Calculus of Variations and Partial Differential Equations; Journal Volume: 60; Journal Issue: 3; Related Information: CHORUS Timestamp: 2022-12-25 09:36:26
%K
%MOSTI ID: 10223849
%PMedium: X
%TThe structure of the singular set in the thin obstacle problem for degenerate parabolic equations
%XAbstract
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$${\left|y\right|}^{a}$for$$a \in (-1,1)$$$a\in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$${({\partial}_{t}-{\Delta}_{x})}^{s}$for$$s \in (0,1)$$$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 ($$a=0$$$a=0$).

%0Journal Article