scholarly journals On the singular set in the thin obstacle problem: higher-order blow-ups and the very thin obstacle problem

2021 ◽  
Vol 14 (5) ◽  
pp. 1599-1669
Author(s):  
Xavier Fernández-Real ◽  
Yash Jhaveri
Keyword(s):  
Obstacle Problem ◽  
Higher Order ◽  
Singular Set ◽  
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 ).

scholarly journals A duality-type method for the obstacle problem

2013 ◽  
Vol 21 (3) ◽  
pp. 181-196 ◽  
Author(s):  
Diana Rodica Merlusca
Keyword(s):  
Sobolev Spaces ◽  
Obstacle Problem ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
Higher Order ◽  
Type Method ◽  
Quadratic Optimization Problem ◽  
Approximate Problem ◽  
Duality Property ◽  
New Type

Abstract Based on a duality property, we solve the obstacle problem on Sobolev spaces of higher order. We have considered a new type of approximate problem and with the help of the duality we reduce it to a quadratic optimization problem, which can be solved much easier.

scholarly journals Regularity for the fully nonlinear parabolic thin obstacle problem

2021 ◽  
pp. 2150011
Author(s):  
Georgiana Chatzigeorgiou
Keyword(s):  
Parabolic Equation ◽  
Viscosity Solution ◽  
Nonlinear Parabolic Equation ◽  
Obstacle Problem ◽  
Elliptic Case ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Nonlinear Parabolic ◽  
Parabolic Case ◽  
Thin Obstacle

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

scholarly journals Regularity results of the thin obstacle problem for the p(x)-Laplacian

2019 ◽  
Vol 276 (2) ◽  
pp. 496-519 ◽  
Author(s):  
Sun-Sig Byun ◽  
Ki-Ahm Lee ◽  
Jehan Oh ◽  
Jinwan Park
Keyword(s):  
Obstacle Problem ◽  
Regularity Results ◽  
Thin Obstacle
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