空间形式上Neumann特征值的精确平移倒数和
Sharp shifted reciprocal sums of Neumann eigenvalues on space forms
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中文总结 AI 辅助
本文研究空间形式上带Lipschitz边界有界域的Neumann特征值，证明了第2到第n+1个特征值的倒数和满足精确平移倒数不等式，等号对应两等体积测地球不交并，正面回应了相关猜想。
中文摘要 AI 辅助
设$\u2114_κ^n$为截面曲率$κ\in\{-1,0,1\}$的空间形式，即$\u2114_{-1}^n=\u210D^n$、$\u2114_{0}^n=\u211D^n$、$\u2114_{1}^n=\u2120^n$。设$Ω\subset\u2114_κ^n$为带Lipschitz边界的非空有界开集，且当$κ=1$时满足$0<|Ω|<|\u2120^n|$。记Neumann谱为$0=μ_0(Ω)\leqμ_1(Ω)\leq\cdots$，令$B_R^κ\subset\u2114_κ^n$为体积等于$|Ω|/2$的测地球。我们证明了精确的平移倒数不等式：$\sum_{j=2}^{n+1}\frac1{μ_j(Ω)} \geq \frac{n}{μ_1(B_R^κ)} = \frac{n}{μ_2(B_R^κ\sqcup B_R^κ)}$，等号成立当且仅当$Ω$是两个等体积测地球的不交并。该结果对BucurMartinetNahon2025的注记11中的猜想给出了肯定回答。
英文摘要
Let $\mathbb{M}_κ^n$ be the space form of sectional curvature $κ\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $Ω\subset\mathbb{M}_κ^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|Ω|<|\mathbb{S}^n|$ when $κ=1$. Write $0=μ_0(Ω)\leqμ_1(Ω)\leq\cdots$ for the Neumann spectrum, and let $B_R^κ\subset\mathbb{M}_κ^n$ be a geodesic ball of volume $\vertΩ\vert/2$. We prove the sharp shifted reciprocal inequality \[ \sum_{j=2}^{n+1}\frac1{μ_j(Ω)} \geq \frac{n}{μ_1(B_R^κ)} = \frac{n}{μ_2(B_R^κ\sqcup B_R^κ)}. \] Equality holds if and only if $Ω$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.
