$C^{1,\alpha}$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations With Rough Kernels

TL;DR

This paper establishes interior regularity results for fully nonlinear nonlocal elliptic equations with rough kernels, extending understanding of solution smoothness without kernel regularity assumptions.

Contribution

It proves a $C^{1,eta}$ interior regularity theorem for nonlinear nonlocal equations without kernel regularity, and explores applications to linear theory and higher regularity.

Findings

Proved $C^{1,eta}$ regularity for solutions of nonlinear nonlocal equations.

Extended regularity results to equations with rough kernels.

Provided applications to linear theory and higher regularity cases.

Abstract

We prove a $C^{1,\alpha}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.