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An extremal eigenvalue problem for the Wentzell-Laplace operator

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DKL-Wentzell-revised-version.pdf (718.7Kb)
Date
2016
Dewey
Analyse
Sujet
Wentzell eigenvalues; eigenvalue estimates; Faber-Krahn inequality; Shape optimization; Shape derivatives; Stability; Quantitative isoperimetric inequality
Journal issue
Annales de l'Institut Henri Poincaré (C) Non Linear Analysis
Volume
33
Number
2
Publication date
2016
Article pages
409-450
DOI
http://dx.doi.org/10.1016/j.anihpc.2014.11.002
URI
https://basepub.dauphine.fr/handle/123456789/17305
Collections
  • CEREMADE : Publications
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Author
Dambrine, Marc
Kateb, Djalil
Lamboley, Jimmy
Type
Article accepté pour publication ou publié
Abstract (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.

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