发表机构
Hangzhou Dianzi University; Tsinghua University（杭州电子科技大学; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 Langford 和 Laugesen 关于曲率上界下第三 Neumann 特征值的双圆盘猜想，并加强为尖锐的倒数不等式，通过双极点 Green 坐标等方法给出证明。
AI 中文摘要
Langford 和 Laugesen 在高斯曲率上界条件下猜想第三 Neumann 特征值满足尖锐的双圆盘界（Math. Ann. 386 (2023), 2255--2281, 猜想 1.4）。我们证明了该猜想在 Langford 和 Laugesen 所使用的权重正则性下对有界 Lipschitz 膜成立，并将其加强为尖锐的倒数不等式。设 $\Omega\subset\mathbb{C}$ 为有界单连通 Lipschitz 区域，$\omega\in C^2(\Omega)\cap C(\overline{\Omega})$ 在 $\overline{\Omega}$ 上为正，并赋予 $\Omega$ 度量 $g=\omega|dz|^2$。假设 $K_g\le K$，$A=\int_\Omega\omega\\,dx>0$，且当 $K>0$ 时 $KA<4\pi$。将 Neumann 特征值按重数计数排列为 $0=\lambda_0<\lambda_1\le\lambda_2\le\cdots$。若 $D_K(A/2)$ 是面积为 $A/2$ 的常曲率测地圆盘，则 $\frac{1}{\lambda_2(\Omega,g)}+\frac{1}{\lambda_3(\Omega,g)}>\frac{2}{\lambda_1(D_K(A/2))}$ 且 $\lambda_2(\Omega,g)<\lambda_1(D_K(A/2))$。不要求 $\lambda_1$ 的简单性或 $\omega$ 的边界可微性。同样的结论对光滑 Riemann 曲面中相对紧的圆盘型 Lipschitz 区域也成立。在固定面积和曲率上界下，两个界都是尖锐的：对每个可容许的 $K,A$，常曲率模型中面积为 $A$ 的光滑连通区域序列具有固定指标的 Neumann 谱收敛到 $D_K(A/2)\sqcup D_K(A/2)$ 的谱。两个极值在任一连通类中均不达到。证明使用了双极点 Green 坐标、正核比较、两个复矩的同时中心化以及平移倒数变分估计。
英文摘要
Langford and Laugesen conjectured a sharp two-disk bound for the third Neumann eigenvalue under an upper Gaussian-curvature bound (Math. Ann. 386 (2023), 2255--2281, Conjecture 1.4). We prove the conjectured bound for bounded Lipschitz membranes with the weight regularity used by Langford and Laugesen, and strengthen it to a sharp reciprocal inequality. Let $Ω\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain, let $ω\in C^2(Ω)\cap C(\overlineΩ)$ be positive on $\overlineΩ$, and equip $Ω$ with $g=ω|dz|^2$. Suppose $K_g\le K$, $A=\int_Ωω\,dx>0$, and $KA<4π$ when $K>0$. Enumerate the Neumann eigenvalues, counting multiplicity, by $0=λ_0<λ_1\leλ_2\le\cdots$. If $D_K(A/2)$ is the constant-curvature geodesic disk of area $A/2$, then $\frac{1}{λ_2(Ω,g)}+\frac{1}{λ_3(Ω,g)}>\frac{2}{λ_1(D_K(A/2))}$ and $λ_2(Ω,g)<λ_1(D_K(A/2))$. No simplicity of $λ_1$ or boundary differentiability of $ω$ is required. The same conclusions hold for relatively compact disk-type Lipschitz domains in smooth Riemannian surfaces. Both bounds are sharp at fixed area and curvature upper bound: for each admissible $K,A$, a sequence of smooth connected domains of area $A$ in the constant-curvature model has fixed-index Neumann spectra converging to those of $D_K(A/2)\sqcup D_K(A/2)$. Neither extremal value is attained in either connected class. The proof uses two-pole Green coordinates, a positive-kernel comparison, simultaneous centering of two complex moments, and a shifted reciprocal variational estimate.
Comments47 pages, 3 figures
