Small perturbation solutions for nonlocal elliptic equations

TL;DR

This paper proves that solutions to certain nonlocal elliptic equations are smooth when the solutions are sufficiently small, extending classical results to a nonlocal context.

Contribution

It introduces a small perturbation theorem for nonlocal elliptic equations, generalizing Savin's classical result to nonlocal operators.

Findings

Solutions are in $C^{\sigma+\alpha}$ for small solutions

Extends classical second-order results to nonlocal equations

Provides a new regularity result for nonlocal elliptic problems

Abstract

We present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in $C^{\sigma+\alpha}$ for any $\alpha\in (0,1)$ as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.