An optimization problem related to the best Sobolev trace constant in thin domains
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: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_02191997_v10_n5_p633_Bonder |
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| Sumario: | 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. |
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