  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.
