AI 中文总结
该研究证明，当空间维度n≥3时，不存在仅依赖于n的常数Cₙ满足Steklov特征值的Weyl型上界估计
AI 中文摘要
我们证明，对于n≥3，不存在仅依赖于n的常数Cₙ，使得对每个光滑有界域Ω⊂ℝⁿ及每个k≥1，Steklov特征值σₖ(Ω)满足σₖ(Ω)≤Cₙ(k/|∂Ω|)^(1/(n-1))
英文摘要
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 $, 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 $|Ω|^{\frac 1n}σ_k(Ω)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $Ω\subset\mathbb R^n$ and every $k\geq 1$. This estimate, combined with the isoperimetric bound of Colbois, El Soufi and Girouard \cite{colboisgirouard_steklov}, implies the bound $|\partialΩ|^{\frac 1{n-1}}σ_k(Ω)\leq C_nk^{\frac1{n-1}+\frac{n-2}{n(n-1)^2}}$, where the exponent of $k$ turns out to be sharp.
Comments16 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
