On the best Sobolev trace constant and extremals in domains with holes
We study the dependence on the subset A ⊂ Ω of the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measu...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2006 |
| País: | Argentina |
| Institución: | Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales |
| Repositorio: | Biblioteca Digital (UBA-FCEN) |
| Idioma: | inglés |
| OAI Identifier: | paperaa:paper_00074497_v130_n7_p565_FernandezBonder |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00074497_v130_n7_p565_FernandezBonder |
| Access Level: | acceso abierto |
| Palabra clave: | Eigenvalue optimization problems p-capacity p-Laplacian Sobolev trace constant |
| Sumario: | We study the dependence on the subset A ⊂ Ω of the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measure. In the case that Ω is a ball we prove that there exists an optimal hole that is spherically symmetric. In the case p = 2 we prove that every optimal hole is spherically symmetric. Then, we study the behavior of the best constant when the hole is allowed to have zero Lebesgue measure. We show that this constant depends continuously on the subset and we discuss when it is equal to the Sobolev trace constant without the vanishing restriction. © 2005 Elsevier SAS. All rights reserved. |
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