Regularity theory for nonlocal equations with VMO coefficients

TL;DR

This paper establishes higher regularity results for nonlinear nonlocal equations with VMO coefficients, demonstrating higher differentiability and Hölder continuity, which are novel in the nonlocal context compared to local equations.

Contribution

It introduces new higher regularity results for nonlocal equations with VMO coefficients, including differentiability and Hölder regularity, extending the understanding beyond local elliptic equations.

Findings

Higher differentiability in fractional Sobolev spaces

Higher Hölder regularity achieved via embedding

Results are of purely nonlocal nature

Abstract

We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher H\"older regularity for such nonlocal equations.