Regularity for fully nonlinear equations driven by spatial-inhomogeneous nonlocal operators

TL;DR

This paper establishes Hölder regularity for a broad class of fully nonlinear integro-differential equations driven by spatial-inhomogeneous nonlocal operators with mild kernel assumptions, extending regularity results to more general kernels.

Contribution

It proves Hölder regularity for nonlinear equations with nonlocal operators that are not spatially homogeneous, covering kernels with logarithmic modifications near zero.

Findings

Hölder regularity is established for the class of equations.

Results apply to kernels comparable to |x-y|^{-d-α} log(|x-y|^{-1}) near zero.

The work extends regularity theory to inhomogeneous nonlocal operators.

Abstract

In this paper we consider a large class of fully nonlinear integro-differential equations. The class of our nonlocal operators we consider is not spatial homogeneous and we put mild assumptions on its kernel near zero. We prove the H\"older regularity for such equation. In particular, our result covers the case that the kernel $K(x,y)$ is comparable to $|x-y|^{-d-\alpha} \ln (|x-y|^{-1})$ for $|x-y|<c$ where $0<\alpha<2$.