Cordes-Nirenberg type results for nonlocal equations with deforming kernels

TL;DR

This paper extends Cordes-Nirenberg results to nonlocal elliptic equations with deforming kernels linked to Monge-Ampère solutions, establishing stability and regularity under natural assumptions.

Contribution

It introduces a stability lemma and derives Hölder regularity estimates for solutions of nonlocal equations with deforming kernels, expanding classical results to a new nonlocal setting.

Findings

Established stability of ellipticity class under kernel deformation

Proved Hölder regularity of solution gradients

Extended Cordes-Nirenberg results to nonlocal equations with Monge-Ampère related kernels

Abstract

We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Amp\`ere equation. Under a natural integrability assumption on the Monge-Amp\`ere solution, we prove a stability lemma allowing the ellipticity class to vary. Using a compactness method, we then derive H\"older regularity estimates for the gradient of the solutions.