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

摘要

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 (-?)~su = g in ?,u 三 0 in ?~n?, for some s ? (0，1) and g ? L~∞(?), then u is Cs(?~n) u/δ~S|? is C~α up to the boundary ?? for some α ? (0,1),where δ(x) = dist(x, ??). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Holder estimates for u and u/δ~s. Namely, the C~β norms of u and u/8s in the sets {x ??:≥ ρ} are controlled by Cp~(s-β) and Cp(a~β), 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]).