An optimization problem related to the best Sobolev trace constant in thin domains

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Let Ω ⊂ ℝN be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) Rightwards arrow with hook sign Lq (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem SA = inf||u||W 1,p(ω)p/||u||Lq(∂ω) for...

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Autores principales:	Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B.
Formato:	JOUR
Materias:	Calculus of variations Optimal design Sobolev trace embedding
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_02191997_v10_n5_p633_Bonder
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	todo:paper_02191997_v10_n5_p633_Bonder2023-10-03T15:11:03Z An optimization problem related to the best Sobolev trace constant in thin domains Bonder, J.F. Rossi, J.D. SchÖnlieb, C.-B. Calculus of variations Optimal design Sobolev trace embedding Let Ω ⊂ ℝN be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) Rightwards arrow with hook sign Lq (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem SA = inf\|\|u\|\|W 1,p(ω)p/\|\|u\|\|Lq(∂ω) for functions that verify u\|A = 0. It is known that there exists an optimal hole that minimizes the best constant SA among subsets of Ω of the prescribed volume. In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain. © 2008 World Scientific Publishing Company. 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_02191997_v10_n5_p633_Bonder
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Calculus of variations Optimal design Sobolev trace embedding
spellingShingle	Calculus of variations Optimal design Sobolev trace embedding Bonder, J.F. Rossi, J.D. SchÖnlieb, C.-B. An optimization problem related to the best Sobolev trace constant in thin domains
topic_facet	Calculus of variations Optimal design Sobolev trace embedding
description	Let Ω ⊂ ℝN be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) Rightwards arrow with hook sign Lq (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem SA = inf\|\|u\|\|W 1,p(ω)p/\|\|u\|\|Lq(∂ω) for functions that verify u\|A = 0. It is known that there exists an optimal hole that minimizes the best constant SA among subsets of Ω of the prescribed volume. In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain. © 2008 World Scientific Publishing Company.
format	JOUR
author	Bonder, J.F. Rossi, J.D. SchÖnlieb, C.-B.
author_facet	Bonder, J.F. Rossi, J.D. SchÖnlieb, C.-B.
author_sort	Bonder, J.F.
title	An optimization problem related to the best Sobolev trace constant in thin domains
title_short	An optimization problem related to the best Sobolev trace constant in thin domains
title_full	An optimization problem related to the best Sobolev trace constant in thin domains
title_fullStr	An optimization problem related to the best Sobolev trace constant in thin domains
title_full_unstemmed	An optimization problem related to the best Sobolev trace constant in thin domains
title_sort	optimization problem related to the best sobolev trace constant in thin domains
url	http://hdl.handle.net/20.500.12110/paper_02191997_v10_n5_p633_Bonder
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An optimization problem related to the best Sobolev trace constant in thin domains

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