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

Abstract

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 $(-Δ)^s u = g$ in $Ω$, $u \equiv 0$ in $\R^n\setminusΩ$, for some $s\in(0,1)$ and $g \in L^\infty(Ω)$, then $u$ is $C^s(\R^n)$ and $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$ for some $α\in(0,1)$, where $δ(x)={\rm dist}(x,\partialΩ)$. 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 $u/δ^s$. Namely, the $C^β$ norms of $u$ and $u/δ^s$ in the sets $\{x\inΩ: δ(x)\geqρ\}$ are controlled by $Cρ^{s-β}$ and $Cρ^{α-β}$, respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian \cite{RS-CRAS,RS}.