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 (-Δ su = f inΩ u = g in Rn ω are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s (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 (-Δ )su(x) := c(n, s) Z ∫ Rn 2u(x) - u(x + y) - u(x - y) yn+2s dy. 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 (-Delta;)s is strictly positive in ω.