H\"older estimates for viscosity solutions of equations of fractional $p$-Laplace type

Abstract

We prove H\"older estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional $p$-Laplace equation $$ \text{PV} \int_{\mathbb{R}^n}\frac{|u(x)-u(x+y)|^{p-2}(u(x)-u(x+y))}{|y|^{n+sp}}\, dy =0, $$ where $s\in (0,1)$ and $p>2$ or $1/(1-s)<p<2$. Our results also apply for inhomogeneous equations with more general kernels, when $p$ and $s$ are allowed to vary with $x$, without any regularity assumption on $p$ and $s$. This complements and extends some of the recently obtained H\"older estimates for weak solutions.