Regularity for fully non linear equations with non local drift

TL;DR

This paper establishes Holder regularity for solutions of nonlocal integro-differential equations with non-symmetric kernels, incorporating drift terms with controlled order, using a localized Aleksandrov-Bakelman-Pucci estimate.

Contribution

It extends regularity results to equations with non-symmetric kernels and drift terms of order less than or equal to the diffusion, including a new localized ABP estimate.

Findings

Holder regularity for solutions with non-symmetric kernels

Uniform estimates as the order approaches two

Applicable to equations like fractional Laplacian plus drift

Abstract

We prove Holder regularity for solutions of non divergence integro-differential equations with non necessarily even kernels. The even/odd decomposition of the kernel can be understood as a sum of a diffusion and a drift term. In our case we assume that such drift have the order smaller than or equal to the diffusion and at least one. For example we can say something about the following equation $\Delta^{1/2}u + |Du| = f$. The main step relies in a localized version of the Aleksandrov-Bakelman-Pucci estimate. Our estimates are also uniform as the order of the equation goes to two.