scholarly journals The structure of the free boundary in the fully nonlinear thin obstacle problem

2017 ◽  
Vol 316 ◽  
pp. 710-747 ◽  
Author(s):  
Xavier Ros-Oton ◽  
Joaquim Serra
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Fully Nonlinear ◽  
Thin Obstacle

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 The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals On the fine structure of the free boundary for the classical obstacle problem

2018 ◽  
Vol 215 (1) ◽  
pp. 311-366 ◽  
Author(s):  
Alessio Figalli ◽  
Joaquim Serra
Keyword(s):  
Fine Structure ◽  
Free Boundary ◽  
Obstacle Problem

On the Curvature of the Free Boundary for the Obstacle Problem in Two Dimensions

2004 ◽  
Vol 142 (1-2) ◽  
pp. 1-5 ◽  
Author(s):  
Bj�rn Gustafsson ◽  
Makoto Sakai
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Two Dimensions
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