Interior regularity for fractional systems

TL;DR

This paper investigates the interior regularity of solutions to elliptic fractional systems, establishing H"older estimates that generalize classical results and are stable across different fractional orders.

Contribution

It introduces new interior regularity results for fractional elliptic systems with nonlocal gradient dependence, extending classical theories to fractional orders.

Findings

Proves interior H"older estimates for fractional systems

Results are stable as the fractional order varies

Recovers classical elliptic system regularity results

Abstract

We study the regularity of solutions of elliptic fractional systems of order 2s, $s \in (0, 1)$, where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural conditions on the system we prove interior H\"older estimates in the spirit of [1]. Our results are stable in s allowing us to recover the classic results for elliptic systems due to S. Hildebrandt and K. Widman [6] and M. Wiegner [9].