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

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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.

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EBS and RAL thank the Analysis research group of UFC for fostering a pleasant and productive scientific atmosphere. EBS supported by CAPES-Brazil.

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Correspondence to R. Leitão.

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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

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Santos, E.B.d., Leitão, R. On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian. Partial Differ. Equ. Appl. 2, 25 (2021).

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