节点域内半径的容量估计与改进下界
Capacity estimates and improved lower bounds for the inner radius of nodal domains
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中文总结 AI 辅助
该研究针对闭光滑流形上拉普拉斯特征函数节点域，给出了不同维数下其内半径的下界估计，并证明了$\mathbb{T}^d$上存在内半径更小的节点域序列。
中文摘要 AI 辅助
我们证明，对于每个维数为$d$的闭光滑流形$(M,g)$，存在常数$c(g)$：当$d=3$时，拉普拉斯特征函数的特征值为$\text{lambda}$的任意节点域$\Omega_\lambda$都包含一个半径至少为$c(g)\lambda^{-1/2}\log\log(\lambda)^{-1/2}$的测地球；当$d>3$时，该半径为$c(g)\lambda^{-1/2}\log(\lambda)^{-\frac{d-3}{2}}$，且该球以特征函数在节点域内绝对值取最大值的任意点为中心。此外，我们证明对于任意$d\geq3$，$\mathbb{T}^d$上存在$\lambda$节点域序列，其内半径为$o(\lambda^{-1/2})$量级。
英文摘要
We show that for every closed, smooth manifold $(M,g)$ of dimension $d$, there exists $c(g)$ such that any nodal domain $Ω_λ$ of a Laplace eigenfunction with eigenvalue $λ$ contains a geodesic ball of radius at least $c(g) λ^{-1/2} \log\log(λ)^{-1/2}$ if $d=3$ and $c(g) λ^{-1/2} \log(λ)^{-\frac{d-3}{2}}$ if $d >3$. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any $d \geq 3$, there exist sequences of $λ$-nodal domains on $\mathbb{T}^d$ whose inner radius is of order $o(λ^{-1/2})$.
