4. Steklov eigenvalues

Nonexistence of a maximizer for the first Steklov eigenvalue with perimeter constraint
Let $\sigma_1(\Omega)$ denote the first nontrivial Steklov eigenvalue for a bounded domain $\Omega\subset\mathbb{R}^n$.Problem 4.1.
[A. Henrot] Prove that there is no $\Omega\subset \mathbb{R}^n$ which maximizes the quantity $P(\Omega)^{1/(n1)}\sigma_1(\Omega)$.
However, it is proved by R. Weinstock in [MR0064989] that in two dimensions the disk is the maximizer among all simply connected domains. In higher dimensions it was proved by D. Bucur, V. Ferone, C. Nitsch and C. Trombetti [arXiv:1710.04587] that the ball is the maximizer among convex domains. In fact, their result is the following stronger inequality $$ \sigma_1(\Omega)\frac{\partial\Omega}{\Omega^{(n2)/n}}\leq \sigma_1(B)\frac{\partial B}{B^{(n2)/n}}. $$
Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.