# On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

## TL;DR

This paper investigates the regularity properties of solutions to anisotropic fractional Laplacian equations, adapting classical techniques to establish H"older and $C^{1, eta}$ regularity for solutions.

## Contribution

It extends the De Giorgi method and geometric techniques to anisotropic fractional Laplacian equations, providing new regularity results for viscosity solutions.

## Key findings

- Established $C^{eta}$-regularity for solutions of class $C^{2}$.
- Derived an ABP estimate and Harnack inequality for viscosity solutions.
- Proved interior $C^{1, eta}$ regularity for solutions.

## Abstract

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^{\gamma}$-regularity for solutions of class $C^{2}$ and use the geometry found in [Caffarelli, Leit\~ao, and Urbano, Math. Ann. 360, 2014] to get an ABP estimate, a Harnack inequality and the interior $C^{1, \gamma}$ regularity for viscosity solutions.