Weak and viscosity solutions of the fractional Laplace equation

Abstract

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s2 (0; 1) is a xed parameter, is a bounded, open subset of R n (n > 1) with C 2 -boundary, and ( ) s is the fractional Laplacian operator, that may be dened as