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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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder |
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todo:paper_00192082_v52_n4_p1111_Bonder2023-10-03T14:16:33Z The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case Bonder, J.F. Rossi, J.D. Schönlieb, C.-B. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder |
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Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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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. |
| format |
JOUR |
| author |
Bonder, J.F. Rossi, J.D. Schönlieb, C.-B. |
| spellingShingle |
Bonder, J.F. Rossi, J.D. Schönlieb, C.-B. The best constant and extremals of the sobolev embeddings in domains with holes: The L ∞ case |
| author_facet |
Bonder, J.F. Rossi, J.D. Schönlieb, C.-B. |
| author_sort |
Bonder, J.F. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_00192082_v52_n4_p1111_Bonder |
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