Motivated by generalized impedence boundary conditions, we consider for $\Omega \subset \mathbb R^d$ the first non-trivial eigenvalue for Wentzell boundary conditions:
\[
\begin{cases}
-\Delta u = 0 & \text{in } \Omega\\
-\beta \Delta_\tau u + \partial_n u = \lambda_1(\Omega)u & \text{on } \partial \Omega
\end{cases}
\]
where $\beta$ is a nonnegative real number. This eigenvalue interpolates between the Steklov eigenvalue ($\beta = 0$) and the Laplace-Beltrami eigenvalue ($\beta = +\infty$). In the case $\beta = 0$, Brock proved a Faber-Krahn type result, namely that the ball maximizes $\lambda_1$ among domains of fixed volume. We investigate the similar question in the general case β ≥ 0. To that end we generalize Brock’s approach, and obtain an isoperimetric inequality involving $λ_1$, but which is weaker than a Faber- Krahn type result when β > 0. This is natural since a similar Faber-Krahn result for the other extremal case β = +∞ is valid only under strong topological assumptions (Hersch inequality). Therefore, we investigate first and second order optimality conditions for this problem (maximizing $λ_1$ under volume constraint) and prove that the ball is a local maximum when the ambient space is of dimension 2 or 3, for any β. This is related to the question of quantitative isoperimetric inequalities, and a particular difficulty here is that the eigenvalue of the ball is not simple.http://cvgmt.sns.it/seminar/411/
