The structure of the singular set in the thin obstacle problem for degenerate parabolic equations
2021 ◽
Vol 60
(3)
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Keyword(s):
AbstractWe study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 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 ).
2016 ◽
Vol 32
(2)
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pp. 327-332
2000 ◽
Vol 25
(1-2)
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pp. 73-99
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Keyword(s):
2001 ◽
Vol 8
(3)
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pp. 343-361
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2005 ◽
Vol 2005
(6)
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pp. 607-617
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