JournalsaihpcVol. 33, No. 2pp. 409–450

An extremal eigenvalue problem for the Wentzell–Laplace operator

  • M. Dambrine

    Université de Pau et des Pays de l'Adour, France
  • D. Kateb

    Université de Technologie de Compiègne, France
  • J. Lamboley

    CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, France
An extremal eigenvalue problem for the Wentzell–Laplace operator cover
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Abstract

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell–Laplace operator of a domain Ω, involving only geometrical information. 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.

Cite this article

M. Dambrine, D. Kateb, J. Lamboley, An extremal eigenvalue problem for the Wentzell–Laplace operator. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 409–450

DOI 10.1016/J.ANIHPC.2014.11.002