# Regularity for solutions of nonlocal, nonsymmetric equations

### Héctor Chang Lara

University of Texas at Austin, Department of Mathematics, 1 University Station C1200, Austin, TX 78712, United States### Gonzalo Dávila

University of Texas at Austin, Department of Mathematics, 1 University Station C1200, Austin, TX 78712, United States

## 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 parts of the kernels have a fixed homogeneity *σ* and the skew symmetric parts have strictly smaller homogeneity *τ*. We prove a weak ABP estimate and $C^{1,\alpha }$ regularity. Our estimates remain uniform as we take $\sigma \rightarrow 2$ and $\tau \rightarrow 1$ so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.

## Cite this article

Héctor Chang Lara, Gonzalo Dávila, Regularity for solutions of nonlocal, nonsymmetric equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, pp. 833–859

DOI 10.1016/J.ANIHPC.2012.04.006