Abstract:
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.
Registro:
Documento: |
Artículo
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Título: | An optimization problem related to the best Sobolev trace constant in thin domains |
Autor: | Bonder, J.F.; Rossi, J.D.; SchÖnlieb, C.-B. |
Filiación: | Departamento de Matemática, FCEyN UBA (1428), Buenos Aires, Argentina IMDEA Matematicas, C-IX Campus UAM, Madrid, Spain DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090, Vienna, Austria
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Palabras clave: | Calculus of variations; Optimal design; Sobolev trace embedding |
Año: | 2008
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Volumen: | 10
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Número: | 5
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Página de inicio: | 633
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Página de fin: | 650
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DOI: |
http://dx.doi.org/10.1142/S0219199708002922 |
Título revista: | Communications in Contemporary Mathematics
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Título revista abreviado: | Commun. Contemp. Math.
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ISSN: | 02191997
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02191997_v10_n5_p633_Bonder |
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Citas:
---------- APA ----------
Bonder, J.F., Rossi, J.D. & SchÖnlieb, C.-B.
(2008)
. An optimization problem related to the best Sobolev trace constant in thin domains. Communications in Contemporary Mathematics, 10(5), 633-650.
http://dx.doi.org/10.1142/S0219199708002922---------- CHICAGO ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B.
"An optimization problem related to the best Sobolev trace constant in thin domains"
. Communications in Contemporary Mathematics 10, no. 5
(2008) : 633-650.
http://dx.doi.org/10.1142/S0219199708002922---------- MLA ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B.
"An optimization problem related to the best Sobolev trace constant in thin domains"
. Communications in Contemporary Mathematics, vol. 10, no. 5, 2008, pp. 633-650.
http://dx.doi.org/10.1142/S0219199708002922---------- VANCOUVER ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B. An optimization problem related to the best Sobolev trace constant in thin domains. Commun. Contemp. Math. 2008;10(5):633-650.
http://dx.doi.org/10.1142/S0219199708002922