二维空间形式中某些凸域上第二诺伊曼特征函数的临界点

中文摘要 AI 辅助

本文研究二维空间形式中凸域上第二诺伊曼特征函数的临界点。从谱与几何条件、明确的定量位置限制、热点常数三个互补角度进行探讨。证明若凸域Ω包含于半球且\(\mu_{2}(\Omega)\leq 2\)，则其二诺伊曼特征函数无内部临界点；建立统一基于直径的准则；当可能存在内部临界点时，得出定量位置限制；用纯分析方法研究凸域上的热点常数\(\mathfrak{C}(\Omega)\)，改进其欧几里得上界并首次得到非欧几里得空间形式中凸域的相应热点常数，结果定量衡量了热点猜想可能的错误程度。

英文摘要

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $μ_2(Ω)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $Ω$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the \emph{hot spots constant} $\mathfrak{C}(Ω)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(Ω)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(Ω) < 4$ for $Ω\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(Ω) < 11.2$ for $Ω\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure ``how wrong'' the \emph{hot spots conjecture} can be.

发表机构

• Nanjing Audit University（南京审计大学）
• Changzhou University（常州大学）
• Nanjing University（南京大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

40pages, 1 figure