Regularity for $(s,p)$-harmonic functions

Verena Bögelein (Salzburg)

Abstract: We report on higher Sobolev and Hölder regularity results for local weak solutions of the fractional $p$-Laplace equation of order $s\in(0,1)$ with $1<p<\infty$. The relevant estimates are stable when the fractional order $s$ reaches 1, and the known Sobolev regularity estimates for weak solutions of the local $p$-Laplace equation are recovered. As an application we establish Calderòn-Zygmund type estimates at the gradient level for the associated fractional $p$-Poisson equation.
The talk is based on joint work with Frank Duzaar (Salzburg), Kristian Moring (Salzburg), Naian Liao (Salzburg), Giovanni Molica Bisci (Urbino), and Raffaella Servadei (Urbino).