scholarly journals The regular free boundary in the thin obstacle problem for degenerate parabolic equations

2021 ◽  
pp. 1
Author(s):  
A. Banerjee ◽  
D. Danielli ◽  
N. Garofalo ◽  
A. Petrosyan
Keyword(s):  
Free Boundary ◽  
Parabolic Equations ◽  
Obstacle Problem ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Thin Obstacle

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

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Agnid Banerjee ◽  
Donatella Danielli ◽  
Nicola Garofalo ◽  
Arshak Petrosyan
Keyword(s):  
Parabolic Equations ◽  
Obstacle Problem ◽  
Degenerate Parabolic Equations ◽  
Singular Set ◽  
Heat Operator ◽  
Complete Structure ◽  
Local Extension ◽  
Monotonicity Formulas ◽  
Degenerate Parabolic ◽  
Thin Obstacle

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 ).

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