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

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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.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, Universidade Federal do Ceará-UFC, Fortaleza, CE, 60455-760, Brazil

    E. B. dos Santos & R. Leitão

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  1. E. B. dos Santos
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  2. R. Leitão
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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). https://doi.org/10.1007/s42985-021-00083-x

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