# An extremal eigenvalue problem for the Wentzell-Laplace operator

Abstract : 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.
Keywords :
Document type :
Journal articles

Cited literature [24 references]

https://hal.archives-ouvertes.fr/hal-00937113
Contributor : Jimmy Lamboley Connect in order to contact the contributor
Submitted on : Tuesday, September 30, 2014 - 5:03:31 PM
Last modification on : Tuesday, January 18, 2022 - 3:24:07 PM
Long-term archiving on: : Friday, April 14, 2017 - 12:30:36 PM

### Files

DKL-Wentzell-revised-version.p...
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00937113, version 2
• ARXIV : 1401.7098

### Citation

Marc Dambrine, Djalil Kateb, Jimmy Lamboley. An extremal eigenvalue problem for the Wentzell-Laplace operator. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2015. ⟨hal-00937113v2⟩

### Metrics

Les métriques sont temporairement indisponibles