Regularity theory for singular nonlocal diffusion equations

TL;DR

This paper establishes continuity and H"older regularity for solutions to a class of singular, nonlocal diffusion equations with rough kernels, extending known results even for fractional Laplacian cases.

Contribution

It proves regularity results for nonlinear, singular nonlocal equations with rough kernels, including cases where solutions change sign and the nonlinearity is mildly oscillatory.

Findings

Solutions are continuous under broad conditions.

Solutions are H"older continuous if the nonlinearity is controlled.

Results apply to fractional Laplacian and similar operators.

Abstract

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover H\"older continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.