三维至六维加权 Neumann 与 Steklov 特征值的等周界
Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文确定了三维至六维有界欧几里得域上质量归一化第一加权 Neumann 特征值和 Steklov 特征值的尖锐等周界，并解决了 Vinokurov 提出的问题 1.12。
AI中文摘要：
对于 $3\le n\le6$，我们确定了有界欧几里得域上质量归一化的第一加权 Neumann 特征值的尖锐等周界。对于 $\overline\Omega$ 上的每个有限非负无原子测度 $\mu$，\begin{equation*} \overline\lambda_1^N(\Omega,\mu)\le\frac{n(n-1)}{n-2}\omega_n\sin^2\vartheta_n\left(\frac{|\Omega|}{\omega_n}\right)^{(n-2)/n}, \end{equation*} 其中 $\omega_n$ 是单位球的体积，$\vartheta_n\in(\pi/2,\pi)$ 是正则旋转调和映射剖面第一驻定半径处的角度。该界在球上由光滑正径向密度达到。事实上，我们肯定地回答了 Vinokurov 提出的问题 1.12 \cite{Vinokurov2026}。此外，对于具有有限边界测度的可容许域，我们还证明了尖锐的严格 Steklov 界 \begin{equation*} \sigma_1(\Omega)|\partial\Omega|\\,|\Omega|^{(2-n)/n}<\frac{n(n-1)}{n-2}\omega_n^{2/n}\sin^2\vartheta_n, \end{equation*} 其常数可由 $C^1$ 穿孔域逼近。
英文摘要：
For $3\le n\le6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $μ$ on $\overlineΩ$, \begin{equation*} \overlineλ_1^N(Ω,μ)\le\frac{n(n-1)}{n-2}ω_n\sin^2\vartheta_n\left(\frac{|Ω|}{ω_n}\right)^{(n-2)/n}, \end{equation*} where $ω_n$ is the volume of the unit ball and $\vartheta_n\in(π/2,π)$ is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound \begin{equation*} σ_1(Ω)|\partialΩ|\,|Ω|^{(2-n)/n}<\frac{n(n-1)}{n-2}ω_n^{2/n}\sin^2\vartheta_n, \end{equation*} whose constant is approached by $C^1$ perforated domains.
