Regularity for solutions of non local, non symmetric equations

TL;DR

This paper establishes regularity results for solutions of fully nonlinear integro differential equations with nonsymmetric kernels, extending classical elliptic PDE theory to nonlocal, nonsymmetric settings with uniform estimates.

Contribution

It introduces a regularity framework for nonlocal, nonsymmetric equations, including a weak ABP estimate and $C^{1,eta}$ regularity, valid as parameters approach classical limits.

Findings

Proves a weak ABP estimate for nonsymmetric kernels.

Establishes $C^{1,eta}$ regularity for solutions.

Results are uniform as the order approaches classical differential operators.

Abstract

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric part of the kernels have a fixed homogeneity $\sigma$ and the skew symmetric part have strictly smaller homogeneity $\tau$. We prove a weak ABP estimate and $C^{1,\alpha}$ regularity. Our estimates remain uniform as we take $\sigma \to 2$ and $\tau \to 1$ so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.