Local Elliptic Regularity for the Dirichlet Fractional Laplacian

We prove the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of . The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp...

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Bibliographic Details
Published inAdvanced nonlinear studies Vol. 17; no. 2; pp. 387 - 409
Main Authors Biccari, Umberto, Warma, Mahamadi, Zuazua, Enrique
Format Journal Article
LanguageEnglish
Published De Gruyter 01.05.2017
Subjects
35B65
35R11
35S05
Dirichlet Boundary Condition
Fractional Laplacian
Local Regularity
Weak Solutions
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ISSN1536-1365
2169-0375
DOI10.1515/ans-2017-0014

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Summary:We prove the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of . 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.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2017-0014