Self-improving Inequalities for bounded weak solutions to nonlocal double phase equations

TL;DR

This paper establishes higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal equations with variable growth, demonstrating improved integrability and differentiability even with rough kernels.

Contribution

It introduces a novel fractional Gehring lemma adaptation for nonlinear nonlocal equations with nonuniform growth and rough kernels.

Findings

Solutions exhibit enhanced integrability and differentiability.

Results apply to operators with rough kernels and modulating coefficients.

The method extends nonlocal regularity theory to complex nonlinear settings.

Abstract

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic ``phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work ``Nonlocal self-improving properties" Anal. PDE, 8(1):57--114 for the specific nonlinear setting under investigation in this manuscript.