Calculus of Variations and Geometric Measure Theory

A. Iannizzotto - S. J. N. Mosconi - M. Squassina

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

created by squassina on 25 Jul 2018



Inserted: 25 jul 2018
Last Updated: 25 jul 2018

Year: 2018


We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $s\in(0,1)$ and $p\ge 2$ (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 H\"older 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.

Keywords: boundary regularity, Fractional $p$-Laplacian, Weighted H\"older regularity