Remarks on Weyl-type bounds for Steklov eigenvalues

We prove that, for $n\ge 3$, there is no constant $C_n>0$ depending only on $n$ such that the Steklov eigenvalues ${\sigma }_k(\mathrm{\Omega })$ satisfy $\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }{\mathrm{\mid }}^{\frac{1}{n-1}}{\sigma }_k(\mathrm{\Omega })\le C_n{k}^{\frac{1}{n-1}}$ for every smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 $\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{\frac{1}{n}}{\sigma }_k(\mathrm{\Omega })\le C_n{k}^{\frac{1}{n-1}}$ for every smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and every $k\ge 1$. This estimate, combined with the isoperimetric bound of Colbois, El Soufi and Girouard \cite{colboisgirouard_steklov}, implies the bound $\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }{\mathrm{\mid }}^{\frac{1}{n-1}}{\sigma }_k(\mathrm{\Omega })\le C_n{k}^{\frac{1}{n-1}+\frac{n-2}{n(n-1{)}^2}}$, where the exponent of $k$ turns out to be sharp.