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

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight | y | a for a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t - Δ x ) s for s ∈ ( 0 , 1 ) . Our main result establishes the...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 60; no. 3
Main Authors Banerjee, Agnid, Danielli, Donatella, Garofalo, Nicola, Petrosyan, Arshak
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021
Springer Nature B.V
Subjects
Analysis
Barriers
Calculus of Variations and Optimal Control; Optimization
Control
Free boundaries
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
Thermodynamics
35R11
35K85
35R35
35K65
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Summary:We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight | y | a for a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t - Δ x ) s for 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 ).
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-021-01938-2