Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian.

Abstract

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian (−Δ)s in a Lipschitz bounded domain Ω⊂Rn satisfying the exterior ball condition. The weight is a power of the distance to the boundary ∂Ω of Ω that accounts for the singular boundary behavior of the solution for any 0<s<1. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension n, which is optimal for n=2.