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 {(-Delta)(s)u = f in Omega u = g in R-n\Omega are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s is an element of (0, 1) is a fixed parameter, Omega is a bounded, open subset of R-n (n >= 1) with C-2-boundary, and (-Delta)(s) is the fractional Laplacian operator, that may be defined as (-Delta)(s)u(x) := c(n, s) integral(Rn) 2u(x) - u(x + y) - u(x - y)/vertical bar y vertical bar(n+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 Omega.