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## 4. Steklov eigenvalues

1. ### Non-existence of a maximizer for the first Steklov eigenvalue with perimeter constraint

Let $\sigma_1(\Omega)$ denote the first non-trivial 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 $P(\Omega)^{1/(n-1)}\sigma_1(\Omega)$.
A topological perturbation argument can be used to prove that the disk is not a maximizer: make a small hole in the center and compute the resulting perturbation of the Steklov eigenvalue.

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 $$\sigma_1(\Omega)\frac{|\partial\Omega|}{|\Omega|^{(n-2)/n}}\leq \sigma_1(B)\frac{|\partial B|}{|B|^{(n-2)/n}}.$$

Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.