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 smooth domain $\Omega$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case $p\in(1,2)$ and with $s\in(0,1)$. We prove that $u/{\rm d}_\Omega^s$ has a $\alpha$-H\"older continuous extension to the closure of $\Omega$, ${\rm d}_\Omega(x)$ meaning the distance of $x$ from the complement of $\Omega$. This result corresponds to that of ref. [28] for the degenerate case $p\ge 2$.