Fine boundary regularity for the degenerate fractional $p$-Laplacian

Abstract

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Hölder regularity up to the boundary, that is, $u/d^s\in C^{\alpha }(\overline{\Omega })$ for some $\alpha \in (0,1)$, $d$ being the distance from the boundary.

arXiv comment38 pages, 3 figures