标题：Singularity of the n-th eigenvalue of high dimensional Sturm-Liouville problems
作者：Hu, Xijun; Liu, Lei; Wu, Li; Zhu, Hao
作者机构：[Hu, Xijun; Liu, Lei; Wu, Li] Shandong Univ, Dept Math, Jinan 250100, Shandong, Peoples R China.; [Zhu, Hao] Nankai Univ, Chern Inst Math, Tianjin 3 更多
通讯作者地址：[Zhu, H]Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China.
来源：JOURNAL OF DIFFERENTIAL EQUATIONS
关键词：High dimensional Sturm-Liouville problems; The n-th eigenvalue;; Continuity; Singularity; Multiplicity
摘要：It is natural to consider continuous dependence of the n-th eigenvalue on d-dimensional (d >= 2) Sturm- Liouville problems after the results on 1-dimensional case by Kong, Wu and Zettl . In this paper, we find all the boundary conditions such that the n-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the n-th eigenvalue. This renders a precise description of the jump phenomena of the n-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into 2d + 1 layers and show that the n-th eigenvalue is continuously dependent on Sturm-Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions. (C) 2018 Elsevier Inc. All rights reserved.