Solutions of the divergence operator on John domains

If Ω ⊂ Rn is a bounded domain, the existence of solutions u ∈ W01, p (Ω) of div u = f for f ∈ Lp (Ω) with vanishing mean value and 1 < p < ∞, is a basic result in the analysis of the Stokes equations. It is known that the result holds when Ω is a Lipschitz domain and that it is not val...

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Detalles Bibliográficos
Autores principales:	Acosta Rodriguez, Gabriel, Duran, Ricardo Guillermo
Publicado:	2006
Materias:	Divergence operator John domains Singular integrals
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v206_n2_p373_Acosta http://hdl.handle.net/20.500.12110/paper_00018708_v206_n2_p373_Acosta
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00018708_v206_n2_p373_Acosta
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spelling	paper:paper_00018708_v206_n2_p373_Acosta2023-06-08T14:21:43Z Solutions of the divergence operator on John domains Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Divergence operator John domains Singular integrals If Ω ⊂ Rn is a bounded domain, the existence of solutions u ∈ W01, p (Ω) of div u = f for f ∈ Lp (Ω) with vanishing mean value and 1 < p < ∞, is a basic result in the analysis of the Stokes equations. It is known that the result holds when Ω is a Lipschitz domain and that it is not valid for domains with external cusps. In this paper we prove that the result holds for John domains. Our proof is constructive: the solution u is given by an explicit integral operator acting on f. To prove that u ∈ W01, p (Ω) we make use of the Calderón-Zygmund singular integral operator theory and the Hardy-Littlewood maximal function. For domains satisfying the separation property introduced in [S. Buckley, P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (5) (1995) 577-593], and 1 < p < n, we also prove a converse result, thus characterizing in this case the domains for which a continuous right inverse of the divergence exists. In particular, our result applies to simply connected planar domains because they satisfy the separation property. © 2005 Elsevier Inc. All rights reserved. Fil:Acosta, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Durán, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2006 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v206_n2_p373_Acosta http://hdl.handle.net/20.500.12110/paper_00018708_v206_n2_p373_Acosta
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Divergence operator John domains Singular integrals
spellingShingle	Divergence operator John domains Singular integrals Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Solutions of the divergence operator on John domains
topic_facet	Divergence operator John domains Singular integrals
description	If Ω ⊂ Rn is a bounded domain, the existence of solutions u ∈ W01, p (Ω) of div u = f for f ∈ Lp (Ω) with vanishing mean value and 1 < p < ∞, is a basic result in the analysis of the Stokes equations. It is known that the result holds when Ω is a Lipschitz domain and that it is not valid for domains with external cusps. In this paper we prove that the result holds for John domains. Our proof is constructive: the solution u is given by an explicit integral operator acting on f. To prove that u ∈ W01, p (Ω) we make use of the Calderón-Zygmund singular integral operator theory and the Hardy-Littlewood maximal function. For domains satisfying the separation property introduced in [S. Buckley, P. Koskela, Sobolev-Poincaré implies John, Math. Res. Lett. 2 (5) (1995) 577-593], and 1 < p < n, we also prove a converse result, thus characterizing in this case the domains for which a continuous right inverse of the divergence exists. In particular, our result applies to simply connected planar domains because they satisfy the separation property. © 2005 Elsevier Inc. All rights reserved.
author	Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo
author_facet	Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo
author_sort	Acosta Rodriguez, Gabriel
title	Solutions of the divergence operator on John domains
title_short	Solutions of the divergence operator on John domains
title_full	Solutions of the divergence operator on John domains
title_fullStr	Solutions of the divergence operator on John domains
title_full_unstemmed	Solutions of the divergence operator on John domains
title_sort	solutions of the divergence operator on john domains
publishDate	2006
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v206_n2_p373_Acosta http://hdl.handle.net/20.500.12110/paper_00018708_v206_n2_p373_Acosta
work_keys_str_mv	AT acostarodriguezgabriel solutionsofthedivergenceoperatoronjohndomains AT duranricardoguillermo solutionsofthedivergenceoperatoronjohndomains
_version_	1768542152143405056

Solutions of the divergence operator on John domains

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