Remarks on Weyl-type bounds for Steklov eigenvalues

Abstract

We prove that, for $n\ge 3$, there is no constant $C_n>0$ depending only on $n$ such that the Steklov eigenvalues ${\sigma }_k(\Omega )$ satisfy $|\partial \Omega {|}^{\frac{1}{n-1}}{\sigma }_k(\Omega )\le C_n{k}^{\frac{1}{n-1}}$ for every smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ and every $k\ge 1$, providing a negative answer to an open problem posed by Girouard and Polterovich \cite{GiPo}. On the other hand, we prove that there exists a constant $C_n>0$ depending only on $n$ such that $|\Omega {|}^{\frac{1}{n}}{\sigma }_k(\Omega )\le C_n{k}^{\frac{1}{n-1}}$ for every smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ and every $k\ge 1$. This estimate, combined with the isoperimetric bound of Colbois, El Soufi and Girouard \cite{colboisgirouard_steklov}, implies the bound $|\partial \Omega {|}^{\frac{1}{n-1}}{\sigma }_k(\Omega )\le C_n{k}^{\frac{1}{n-1}+\frac{n-2}{n(n-1{)}^2}}$, where the exponent of $k$ turns out to be sharp.

16 pages. Revised and expanded since the original version: added a sharp volume-normalised upper bound and determined the optimal exponent in the universal perimeter-normalised bound. Revised title, abstract, and introduction