Isodiametric-type upper bounds for Steklov eigenvalues

Abstract

Let $n\ge 2$. We establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $\Omega$ in Euclidean space ${\mathbb{R}}^n$ and hyperbolic space ${\mathbb{H}}^n$. In Euclidean space, we prove \[ D(Ω)σ_k(Ω)\leq C(n)k, \] with optimal dependence on $k$. For domains in either ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, we establish \[ D(Ω)σ_k(Ω)\leq C(n)k^2\left(\frac{|Ω|^{\frac{1}{n}}}{D(Ω)}\right)^{\frac{n}{n-1}}. \] The exponent of the volume-diameter ratio is optimal in both settings, and with this exponent fixed, the quadratic dependence on $k$ is also optimal. Interpolating the preceding volume-diameter estimate with a volume-normalized Steklov bound, we further obtain \[ D(Ω)σ_k(Ω)\leq C(n)k^{1+\frac{1}{2n-1}} \] in hyperbolic space. These results answer Open Question 4.37 affirmatively and resolve the hyperbolic part of Open Question 4.35, both posed in [5].

arXiv comment16 pages. Revised and expanded version. New results on Steklov eigenvalue bounds in hyperbolic space. All comments are welcome