Fully nonlinear integro-differential equations with deforming kernels

TL;DR

This paper establishes a regularity theory for a class of nonlinear integro-differential equations with spatially deforming kernels, including ABP estimates, Harnack inequalities, and Hölder and $C^{1,eta}$ regularity results.

Contribution

It introduces a novel regularity framework for integro-differential equations with kernels shaped by Monge-Ampère solutions, extending classical theories to more complex kernel structures.

Findings

Proved an ABP estimate for the equations.

Established a Harnack inequality for solutions.

Derived Hölder and $C^{1,eta}$ regularity results.

Abstract

We develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Amp\`{e}re equation. We prove an ABP estimate and a Harnack inequality and derive H\"{o}lder and $C^{1,\alpha}$ regularity results for solutions.