
An extremal eigenvalue problem for the Wentzell-Laplace operator
Dambrine, Marc; Kateb, Djalil; Lamboley, Jimmy (2016), An extremal eigenvalue problem for the Wentzell-Laplace operator, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33, 2, p. 409-450. 10.1016/j.anihpc.2014.11.002
Type
Article accepté pour publication ou publiéDate
2016Journal name
Annales de l'Institut Henri Poincaré (C) Non Linear AnalysisVolume
33Number
2Pages
409-450
Publication identifier
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Show full item recordAbstract (EN)
We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.Subjects / Keywords
Wentzell eigenvalues; eigenvalue estimates; Faber-Krahn inequality; Shape optimization; Shape derivatives; Stability; Quantitative isoperimetric inequalityRelated items
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