The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (-Delta)(s)u = g in Omega, u equivalent to 0 in R-n\Omega, for some s is an element of (0, 1) and g is an element of L-infinity (Omega), then u is C-s (R-n) and u/delta(s)/Omega is C-alpha up to the boundary partial derivative Omega for some a is an element of (0, 1), where delta(x) = dist(x, partial derivative Omega). For this, we develop a fractional analog of the Krylov boundary Hamack method. Moreover, under further regularity assumptions on g we obtain higher order Holder estimates for u and u/delta(s). Namely, the C-beta norms of u and u/delta(s) in the sets {x is an element of Omega: delta(x) >= rho} are controlled by C rho(s-beta) and C rho(alpha-beta), respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19,20]). (C) 2013 Elsevier Masson SAS. All rights reserved.