Calculus of Variations and Geometric Measure Theory
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M. Dambrine - D. Kateb - J. Lamboley

An extremal eigenvalue problem for the Wentzell-Laplace operator

created by lamboley on 06 Jun 2016


Published Paper

Inserted: 6 jun 2016
Last Updated: 6 jun 2016

Journal: Annales de l'IHP, Analyse non linéaire
Volume: 33
Number: 2
Pages: 409-450
Year: 2016


We consider the question of giving an upper bound for the rst 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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