Hölder estimates for viscosity solutions of equations of fractional $p$-Laplace type

Abstract

We prove Hölder 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 Hölder estimates for weak solutions.

References in corpus (2)

• Local behavior of fractional $p$-minimizers
• $C^{1,\alpha }$ regularity of solutions of degenerate fully non-linear elliptic equations

Cited by in corpus (5)

• Local behavior of fractional $p$-minimizers
• Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes
• The Brezis-Nirenberg problem for the fractional $p$-Laplacian involving critical Hardy-Sobolev exponents
• Asymptotics of Dirichlet Problems to Fractional p-Laplacian Functionals-Approach in De Giorgi Sense
• The obstacle problem for nonlinear integro-differential operators