Partial regularity in nonlocal systems II

TL;DR

This paper proves that solutions to certain nonlinear nonlocal systems are smooth outside a small singular set, with the set's size depending on the dimension and order of the system.

Contribution

It establishes partial regularity results for nonlinear nonlocal systems of order greater than one, detailing the size and nature of singular sets.

Findings

Solutions are $C^{1,eta}$ for all $eta < 2s-1$ outside a singular set

The singular set has Hausdorff dimension less than $n-2$

The singular set is empty when $n=2$

Abstract

Solutions to nonlinear nonlocal systems of order $2s>1$ in $\mathbb{R}^n$ are $C^{1,\alpha}$, for every $\alpha <2s-1$, outside a closed singular set whose Hausdorff dimension is less than $n-2$, and which is empty when $n=2$.