scholarly journals An epiperimetric inequality for the lower dimensional obstacle problem

2019 ◽  
Vol 25 ◽  
pp. 39
Author(s):  
Francesco Geraci
Keyword(s):  
Free Boundary ◽  
Convergence Analysis ◽  
Obstacle Problem ◽  
Invent Math ◽  
Lower Dimensional

In this paper we give a proof of an epiperimetric inequality in the setting of the lower dimensional obstacle problem. The inequality was introduced by Weiss [Invent. Math.138(1999) 23–50) for the classical obstacle problem and has striking consequences concerning the regularity of the free-boundary. Our proof follows the approach of Focardi and Spadaro [Adv. Differ. Equ.21(2015) 153–200] which uses an homogeneity approach and aΓ-convergence analysis.

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

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
Sign in / Sign up

Export Citation Format

Share Document