On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

Abstract

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in Silvestre (Indiana Univ Math J 55:1155–1174, 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 et al. (Math Ann 360(3–4): 681–714, 2014) to get an ABP estimate, a Harnack inequality and the interior \(C^{1, \gamma }\) regularity for viscosity solutions.