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

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}.

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

scholarly journals Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

2017 ◽  
Vol 34 (3) ◽  
pp. 533-570 ◽  
Author(s):  
Nicola Garofalo ◽  
Arshak Petrosyan ◽  
Camelia A. Pop ◽  
Mariana Smit Vega Garcia
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Fractional Laplacian

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 of the free boundary in the biharmonic obstacle problem

2019 ◽  
Vol 58 (6) ◽  
Author(s):  
Gohar Aleksanyan
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Small Ball ◽  
One Dimensional ◽  
Coincidence Set ◽  
Accessible Domain

Abstract In this article we use a flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible domain, we derive the $$C^{1,\alpha }$$C1,α-regularity of the free boundary in a small ball centred at the origin. From the $$C^{1,\alpha }$$C1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $$ C^{3,\alpha }$$C3,α up to the free boundary, and therefore $$C^{2,1}$$C2,1. In the end we study an example, showing that in general $$ C^{2,\frac{1}{2}}$$C2,12 is the best regularity that a solution may achieve in dimension $$n \ge 2$$n≥2.

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