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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| Published in | Advanced nonlinear studies Vol. 17; no. 2; pp. 387 - 409 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
De Gruyter
01.05.2017
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1536-1365 2169-0375 |
| DOI | 10.1515/ans-2017-0014 |
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| Abstract | 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. |
|---|---|
| AbstractList | We prove the
W
loc
2
s
,
p
${W_{{\mathrm{loc}}}^{2s,p}}$
local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of
ℝ
N
${\mathbb{R}^{N}}$
. 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. 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. We prove the Wloc2s,p${W_{{\mathrm{loc}}}^{2s,p}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ℝN${\mathbb{R}^{N}}$. 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. |
| Author | Biccari, Umberto Warma, Mahamadi Zuazua, Enrique |
| Author_xml | – sequence: 1 givenname: Umberto surname: Biccari fullname: Biccari, Umberto email: umberto.biccari@deusto.es organization: DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; andFacultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain – sequence: 2 givenname: Mahamadi surname: Warma fullname: Warma, Mahamadi email: mahamadi.warma1@upr.edu organization: Department of Mathematics, College of Natural Sciences, University of Puerto Rico (Rio Piedras Campus), PO Box 70377, San Juan, PR 00936-8377, USA – sequence: 3 givenname: Enrique surname: Zuazua fullname: Zuazua, Enrique email: enrique.zuazua@deusto.es organization: DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; andFacultad 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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| Snippet | 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... We prove the W loc 2 s , p ${W_{{\mathrm{loc}}}^{2s,p}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional... We prove the Wloc2s,p${W_{{\mathrm{loc}}}^{2s,p}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional... |
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| SubjectTerms | 35B65 35R11 35S05 Dirichlet Boundary Condition Fractional Laplacian Local Regularity Weak Solutions |
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| Title | Local Elliptic Regularity for the Dirichlet Fractional Laplacian |
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