Isodiametric-type upper bounds for Steklov eigenvalues

In this paper, we establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, valid for every $k\in \mathbb{N}$: $$D(\mathrm{\Omega }){\sigma }_k(\mathrm{\Omega })\le C(n)k,$$ $$D(\mathrm{\Omega }){\sigma }_k(\mathrm{\Omega })\le C(n)k^2{\left(\frac{\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{\frac{1}{n}}}{D(\mathrm{\Omega })}\right)}^{\frac{n}{n-1}},$$ where $C(n)>0$ depends only on $n$. The first estimate is sharp in its linear dependence on $k$, while the second is sharp both in the exponent of the volume-diameter ratio and, once that exponent is fixed, in its quadratic dependence on $k$. In particular, the second estimate gives an affirmative answer to Open Question 4.37 in [5].