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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I28-R145-paper_00018708_v206_n2_p373_Acosta_oai2024-08-16 Acosta, G. Durán, R.G. Muschietti, M.A. 2006 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. application/pdf http://hdl.handle.net/20.500.12110/paper_00018708_v206_n2_p373_Acosta info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar Adv. Math. 2006;206(2):373-401 Divergence operator John domains Singular integrals Solutions of the divergence operator on John domains info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v206_n2_p373_Acosta_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (UBA) |
| topic |
Divergence operator John domains Singular integrals |
| spellingShingle |
Divergence operator John domains Singular integrals Acosta, G. Durán, R.G. Muschietti, M.A. 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. |
| format |
Artículo Artículo publishedVersion |
| author |
Acosta, G. Durán, R.G. Muschietti, M.A. |
| author_facet |
Acosta, G. Durán, R.G. Muschietti, M.A. |
| author_sort |
Acosta, G. |
| 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 |
http://hdl.handle.net/20.500.12110/paper_00018708_v206_n2_p373_Acosta https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00018708_v206_n2_p373_Acosta_oai |
| work_keys_str_mv |
AT acostag solutionsofthedivergenceoperatoronjohndomains AT duranrg solutionsofthedivergenceoperatoronjohndomains AT muschiettima solutionsofthedivergenceoperatoronjohndomains |
| _version_ |
1809356757660598272 |