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Estimates of the best Sobolev constant of the embedding of into and related shape optimization problems

✍ Scribed by Nicolas Saintier

Publisher: Elsevier Science
Year: 2008
Tongue: English
Weight: 329 KB
Volume: 69
Category: Article
ISSN: 0362-546X
DOI: 10.1016/j.na.2007.08.026

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✦ Synopsis

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality λ 1 (Ω ) u L 1 (∂Ω ) ≤ u W 1,1 (Ω ) that are independent of Ω . These estimates generalize those of [J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press)] concerning the p-Laplacian to the case p = 1.

We apply our results to prove the existence of an extremal for this embedding. We then study an optimal design problem related to λ 1 , and eventually compute the shape derivative of the functional Ω → λ 1 (Ω ).

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