Fine boundary regularity for the singular fractional p-Laplacian

Abstract

We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded $C^{1,1}$ domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with $s\in (0,1)$. We prove optimal up-to-the-boundary regularity of u, which is $C^s\left(\bar{\Omega }\right)$ for any $p>1$ and, in the singular case $p\in (1,2)$, that $u/d_{\Omega }^s$ has a Hölder continuous extension to the closure of Ω, $d_{\Omega }\left(x\right)$ meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case $p⩾2$ is considered.