Bounds on eigenvalues of Dirichlet Laplacian

被引:0
|
作者
Qing-Ming Cheng
Hongcang Yang
机构
[1] Saga University,Department of Mathematics, Faculty of Science and Engineering
[2] International Centre for Theoretical Physics,Department of Mathematics
[3] University of California,Academy of Mathematics and Systematical Sciences
[4] CAS,undefined
来源
Mathematische Annalen | 2007年 / 337卷
关键词
35P15; 58G25;
D O I
暂无
中图分类号
学科分类号
摘要
In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space Rn. If λk+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥  41 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\geq 41, \lambda_{k+1}\leq k^{\frac2n}\lambda_1$$\end{document} and, for any n and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k, \lambda_{k+1}\leq C_{0}(n,k) k^{\frac2n}\lambda_1$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0(n,k)\leq {j^{2}_{n/2,1}}/{j^{2}_{n/2-1,1}}$$\end{document}, where jp,k denotes the k-th positive zero of the standard Bessel function Jp(x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.
引用
收藏
页码:159 / 175
页数:16
相关论文
共 50 条
  • [31] Sharp upper bounds for the Laplacian graph eigenvalues
    Pan, YL
    LINEAR ALGEBRA AND ITS APPLICATIONS, 2002, 355 : 287 - 295
  • [32] On the bounds for the largest Laplacian eigenvalues of weighted graphs
    Sorgun, Sezer
    Buyukkose, Serife
    DISCRETE OPTIMIZATION, 2012, 9 (02) : 122 - 129
  • [33] Upper bounds for the sum of Laplacian eigenvalues of graphs
    Du, Zhibin
    Zhou, Bo
    LINEAR ALGEBRA AND ITS APPLICATIONS, 2012, 436 (09) : 3672 - 3683
  • [34] Upper bounds on the (signless) Laplacian eigenvalues of graphs
    Das, Kinkar Ch.
    Liu, Muhuo
    Shan, Haiying
    LINEAR ALGEBRA AND ITS APPLICATIONS, 2014, 459 : 334 - 341
  • [35] Sharp lower bounds on the Laplacian eigenvalues of trees
    Das, KC
    LINEAR ALGEBRA AND ITS APPLICATIONS, 2004, 384 : 155 - 169
  • [36] Bounds for the Largest Two Eigenvalues of the Signless Laplacian
    Kolotilina L.Y.
    Journal of Mathematical Sciences, 2014, 199 (4) : 448 - 455
  • [37] Some Results on the Bounds of Signless Laplacian Eigenvalues
    Shuchao Li
    Yi Tian
    Bulletin of the Malaysian Mathematical Sciences Society, 2015, 38 : 131 - 141
  • [38] Bounds for the extremal eigenvalues of gain Laplacian matrices
    Kannan, M. Rajesh
    Kumar, Navish
    Pragada, Shivaramakrishna
    LINEAR ALGEBRA AND ITS APPLICATIONS, 2021, 625 : 212 - 240
  • [39] Some Results on the Bounds of Signless Laplacian Eigenvalues
    Li, Shuchao
    Tian, Yi
    BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, 2015, 38 (01) : 131 - 141
  • [40] Bounds on Signless Laplacian Eigenvalues of Hamiltonian Graphs
    Andelic, Milica
    Koledin, Tamara
    Stanic, Zoran
    BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, 2021, 52 (03): : 467 - 476