scholarly journals Regularity of the singular set in the fully nonlinear obstacle problem

2021 ◽  
Author(s):  
Ovidiu Savin ◽  
Hui Yu
Keyword(s):  
Obstacle Problem ◽  
Singular Set ◽  
Fully Nonlinear

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.

Removable singularities of fully nonlinear elliptic equations

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Friedmar Schulz
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Singular Set ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Removable Singularities ◽  
Nonlinear Elliptic ◽  
Monge Ampere

In this paper we consider fully nonlinear elliptic equations of the formincluding the Monge–Ampère, the Hessian and the Weingarten equations and give conditions which ensure that a singular set

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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