Thumbnail

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

View/Open
Type
Article accepté pour publication ou publié
Date
2016
Journal name
Annales de l'Institut Henri Poincaré (C) Non Linear Analysis
Volume
33
Number
2
Pages
409-450
Publication identifier
Metadata
Show full item record
Author(s)
Dambrine, Marc
Kateb, Djalil
Lamboley, Jimmy
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.
Subjects / Keywords
Wentzell eigenvalues; eigenvalue estimates; Faber-Krahn inequality; Shape optimization; Shape derivatives; Stability; Quantitative isoperimetric inequality