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 (−∆)u = g in Ω, u ≡ 0 in R\Ω, for some s ∈ (0, 1) and g ∈ L∞(Ω), then u is C(R) and u/δ|Ω is C up to the boundary ∂Ω for some α ∈ (0, 1), where δ(x) = dist(x, ∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and u/δ. Namely, the C norms of u and u/δ in the sets {x ∈ Ω : δ(x) ≥ ρ} are controlled by Cρs−β and Cρα−β , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian [19, 20].