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Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Umberto Biccari
• DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; and Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain
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• Department of Mathematics, College of Natural Sciences, University of Puerto Rico (Rio Piedras Campus), PO Box 70377, San Juan, PR 00936-8377, USA
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/ Enrique Zuazua
• Corresponding author
• DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; and Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country; and Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
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Published Online: 2017-04-21 | DOI: https://doi.org/10.1515/ans-2017-0014

Abstract

We prove the ${W}_{\mathrm{loc}}^{2s,p}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ${ℝ}^{N}$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

MSC 2010: 35B65; 35R11; 35S05

Dedicated to Ireneo Peral on the occasion of his 70th birthday: Gracias Ireneo por tantos años de amistad y ejemplo

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Revised: 2017-04-13

Accepted: 2017-04-19

Published Online: 2017-04-21

Published in Print: 2017-05-01

Funding Source: U.S. Air Force

Award identifier / Grant number: FA9550-15-1-0027

Award identifier / Grant number: FA9550-14-1-0214

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-52347

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ICON

All authors were supported by the Air Force Office of Scientific Research through award no. FA9550-15-1-0027. Umberto Biccari and Enrique Zuazua were supported by Ministerio de Economía y Competitividad (Spain) through grant MTM2014-52347 and by the European Research Council Executive Agency through Advanced Grant DYCON (Dynamic Control). Enrique Zuazua was supported by EOARD-AFOSR through award no. FA9550-14-1-0214 and by Agence Nationale de la Recherche (France) through ICON.

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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