Regularity for Fully Nonlinear Equations Driven by Spatial-Inhomogeneous Nonlocal Operators

Abstract

We consider a class of nonlocal operators that are not necessarily spatially homogeneous and impose mild assumptions on its kernel near zero. Furthermore, we prove Holder regularity for a large class of fully nonlinear integro-differential equations. In particular, the results cover the case when the kernel K(x,y) is comparable to \(|x-y|^{-d-\alpha } \ln \left (|x-y|^{-1}\right )\) for |x−y|<r0, where 0 < α < 2.