A sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues

Abstract

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in ${\mathbb{R}}^N$. More precisely, if $N\ge 2$ and $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a bounded Lipschitz domain, then $$\prod _{j=1}^N\ddot{I}{ƒ}_j(\hat{I}\text{c◯})\le \frac{\ddot{I}{‰}_N}{|\hat{I}\text{c◯}|},$$ where $0=\ddot{I}{ƒ}_0(\hat{I}\text{c◯})<\ddot{I}{ƒ}_1(\hat{I}\text{c◯})\le \ddot{I}{ƒ}_2(\hat{I}\text{c◯})\le \cdots$ are the Steklov eigenvalues of $\hat{I}\text{c◯}$, and $\ddot{I}{‰}_N$ denotes the volume of the unit ball in ${\mathbb{R}}^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.

10 pages. This is the submitted version