An extremal eigenvalue problem for the Ventcel-Laplace operator

Abstract: We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Ventcel-Laplace operator of a domain Ω, involving only geometrical informations. We provide such an upper bound, by generalizing Brock’s inequality concerning Steklov eigenvalue, and we conjecture a Faber-Krahn type inequality which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Ventcel 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