Higher H\"older regularity for nonlocal equations with irregular kernel

TL;DR

This paper proves that solutions to certain nonlinear nonlocal elliptic equations exhibit higher H"older regularity than expected, even with irregular kernels, highlighting a purely nonlocal phenomenon.

Contribution

It introduces a novel approach using discrete fractional derivatives to establish higher regularity for nonlocal equations with mild kernel continuity assumptions.

Findings

Solutions have better than expected H"older regularity.

The method applies to equations with locally translation invariant kernels.

Regularity results are achieved even with globally irregular kernels.

Abstract

We study the higher H\"older regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as follows. First, we use certain test functions that involve discrete fractional derivatives in order to obtain higher H\"older regularity for homogeneous equations driven by a locally translation invariant kernel, while the global behaviour of the kernel is allowed to be more general. This enables us to deduce the desired regularity in the general case by an approximation argument.