A remark on Weyl-type bounds for Steklov eigenvalues

Abstract

We prove that, for $n\geq 3$, there is no constant $C_n>0$ depending only on $n$ such that the Steklov eigenvalues $σ_k(Ω)$ satisfy $|\partialΩ|^{\frac 1{n-1}}σ_k(Ω)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $Ω\subset\mathbb R^n$ and every $k\geq1 $. On the other hand, we prove that there exists a constant $C_n>0$ depending only on $n$ such that $|Ω|^{\frac 1n}σ_k(Ω)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $Ω\subset\mathbb R^n$ and every $k\geq 1$. This estimate, combined with an isoperimetric bound of Colbois-El Soufi-Girouard, implies the bound $|\partialΩ|^{\frac 1{n-1}}σ_k(Ω)\leq C_nk^{\frac1{n-1}+\frac{n-2}{n(n-1)^2}}$, which we also prove to be sharp.

Disclosure

“é for helpful comments and stimulating exchanges concerning homogenisation methods and the question of the optimal eigenvalue growth rate. AI usage disclosure The author acknowledges the use of ChatGPT 5.6 Sol for mathematical discussions and editorial assistance. In particular, the AI tool contributed to the development of the proofs of the technical Lemmas 2.3, 2.4, 2.5 and 3.1. All AI-assisted arguments and computations were independe”

PDF page 15

Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:07:48.924859Z

Structural counts