Higher integrability for nonlinear nonlocal equations with irregular kernel

TL;DR

This paper establishes higher regularity for solutions to nonlinear nonlocal equations with irregular kernels, using a novel approach based on Bessel potential spaces and nonlocal gradients.

Contribution

It introduces a new regularity result for nonlinear nonlocal equations with minimal kernel regularity assumptions, connecting Bessel potential and Sobolev-Slobodeckij spaces.

Findings

Higher integrability of solutions in Bessel potential spaces

Regularity results extend to Sobolev-Slobodeckij spaces

Method based on nonlocal gradient characterization

Abstract

We prove a higher regularity result for weak solutions to nonlinear nonlocal equations along the integrability scale of Bessel potential spaces $H^{s,p}$ under a mild continuity assumption on the kernel. By embedding, this also yields regularity in Sobolev-Slobodeckij spaces $W^{s,p}$. Our approach is based on a characterization of Bessel potential spaces in terms of a certain nonlocal gradient-type operator and a perturbation approach commonly used in the context of local elliptic equations in divergence form.