An extremal eigenvalue problem for the Wentzell-Laplace operator

Marc Dambrine (1) , Djalil Kateb (2) , Jimmy Lamboley (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Wentzell eigenvalues; eigenvalue estimates; Faber-Krahn inequality; Shape optimization; Shape derivatives; Stability; Quantitative isoperimetric inequality

Domaines: Optimisation et contrôle [math.OC]

Marc Dambrine, Djalil Kateb, Jimmy Lamboley. An extremal eigenvalue problem for the Wentzell-Laplace operator. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2015, 33 (2), pp.409-450. ⟨10.1016/j.anihpc.2014.11.002⟩. ⟨hal-00937113v2⟩