Weak and viscosity solutions of the fractional laplace equation

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation (Equation) are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s G (0, 1) is a fixed parameter, Ω is a bounded, open subset of ℝⁿ (n ≥ 1) with C²-boundary, and (-Δ)s is the fractional Laplacian operator, that may be defined as (Equation) for a suitable positive normalizing constant c(n,s), depending only on n and s. In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem. As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of (-Δ)^s is strictly positive in Ω.