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On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian. (English) Zbl 1462.35115
Summary: In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [L. Silvestre, Indiana Univ. Math. J. 55, No. 3, 1155–1174 (2006; Zbl 1101.45004)], we adapt the De Giorgi technique to achieve the $$C^{\gamma }$$-regularity for solutions of class $$C^2$$ and use the geometry found in [L. A. Caffarelli et al., Math. Ann. 360, No. 3–4, 681–714 (2014; Zbl 1304.35730)] to get an ABP estimate, a Harnack inequality and the interior $$C^{1, \gamma}$$ regularity for viscosity solutions.
##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35D40 Viscosity solutions to PDEs 35R09 Integro-partial differential equations 35R11 Fractional partial differential equations 35J61 Semilinear elliptic equations 35J70 Degenerate elliptic equations
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