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Advanced Nonlinear Studies

Editor-in-Chief: Shair Ahmad

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Volume 17, Issue 4

Issues

Addendum: 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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/ Mahamadi Warma
  • 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
  • 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: 2017-08-03 | DOI: https://doi.org/10.1515/ans-2017-6020

An erratum for this article can be found here: https://doi.org/10.1515/ans-2017-0014

Abstract

In [1], for 1<p<, we proved the Wloc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (-Δ)s on an arbitrary bounded open set of N. Here we make a more precise and rigorous statement. In fact, for 1<p<2 and s12, local regularity does not hold in the Sobolev space Wloc2s,p, but rather in the larger Besov space (Bp,22s)loc.

Keywords: Fractional Laplacian; Dirichlet Boundary Condition; Weak Solutions; Local Regularity

MSC 2010: 35B65; 35R11; 35S05

References

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    U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud. 17 (2017), 387–409. Web of ScienceGoogle Scholar

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    U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, preprint (2017). Google Scholar

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    G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. CrossrefGoogle Scholar

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    X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations 50 (2014), 723–750. Web of ScienceCrossrefGoogle Scholar

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About the article

Received: 2017-05-10

Accepted: 2017-05-10

Published Online: 2017-08-03

Published in Print: 2017-10-01


Citation Information: Advanced Nonlinear Studies, Volume 17, Issue 4, Pages 837–839, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-6020.

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