The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case

LetΩ⊂ R N be a bounded domain. We study the best constant of the Sobolev trace embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. That is we deal with the minimization problem S T A = inf ||u|| W 1,∞ (Ω)/||u||L ∞ (∂Ω) for functions...

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Autor principal: Rossi, Julio Daniel
Publicado: 2008
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00192082_v52_n4_p1111_Bonder
http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00192082_v52_n4_p1111_Bonder
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spelling paper:paper_00192082_v52_n4_p1111_Bonder2023-06-08T14:40:16Z The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case Rossi, Julio Daniel LetΩ⊂ R N be a bounded domain. We study the best constant of the Sobolev trace embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. That is we deal with the minimization problem S T A = inf ||u|| W 1,∞ (Ω)/||u||L ∞ (∂Ω) for functions that verify u|A = 0. We find that there exists an optimal hole that minimizes the best constant S T A among subsets of Ω of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points x* 0 on ∂Ω with respect to the arc-length metric in Ω and the best holes are of the form A* =Ω\\B d (x* 0 , r*) where the ball is taken again with respect of the arc-length metric. A similar analysis can be performed for the best constant of the embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (Ω) with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points x* 0 on ∂Ω and the best holes are of the form A*=Ω\\ B d (x* 0 ,r*). © 2009 University of Illinois. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2008 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00192082_v52_n4_p1111_Bonder http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description LetΩ⊂ R N be a bounded domain. We study the best constant of the Sobolev trace embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. That is we deal with the minimization problem S T A = inf ||u|| W 1,∞ (Ω)/||u||L ∞ (∂Ω) for functions that verify u|A = 0. We find that there exists an optimal hole that minimizes the best constant S T A among subsets of Ω of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points x* 0 on ∂Ω with respect to the arc-length metric in Ω and the best holes are of the form A* =Ω\\B d (x* 0 , r*) where the ball is taken again with respect of the arc-length metric. A similar analysis can be performed for the best constant of the embedding W 1,∞ (Ω) {right arrow, hooked} L ∞ (Ω) with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points x* 0 on ∂Ω and the best holes are of the form A*=Ω\\ B d (x* 0 ,r*). © 2009 University of Illinois.
author Rossi, Julio Daniel
spellingShingle Rossi, Julio Daniel
The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
title_short The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
title_full The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
title_fullStr The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
title_full_unstemmed The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case
title_sort best constant and extremals of the sobolev embeddings in domains with holes: the l ∞ case
publishDate 2008
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00192082_v52_n4_p1111_Bonder
http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder
work_keys_str_mv AT rossijuliodaniel thebestconstantandextremalsofthesobolevembeddingsindomainswithholesthelcase
AT rossijuliodaniel bestconstantandextremalsofthesobolevembeddingsindomainswithholesthelcase
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