Regularity for the fractional $p$-Laplace equation

Abstract

Higher Sobolev and Hölder regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le \frac{p-2}{p}\text{or}\frac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W_{\mathrm{loc}}^{1+s,2}$-regularity for the fractional Laplace equation of any order $s\in (0,1)$.