Acta Mathematica Scientia

, Volume 39, Issue 2, pp 545–550 | Cite as

New Bounds on Eigenvualues of Laplacian

  • Zhengchao JiEmail author


In this paper, we investigate non-zero positive eigenvalues of the Laplacian with Dirichlet boundary condition in an n-dimentional Euclidean space ℝn, then we obtain an new upper bound of the (k + 1)-th eigenvalue λk+1, which improve the previous estimate which was obtained by Cheng and Yang, see (1.8).

Key words

eigenvalues Laplacian Euclidean space recursion formula 

2010 MR Subject Classification

35P15 58G25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author would like to thank Professor Kefeng Liu and Professor Hongwei Xu for their continued support, advice and encouragement. Thanks also to Professor En-Tao Zhao for helpful discussions.


  1. [1]
    Shao Z Q, Hong J X. The eigenvalue problem for the Laplacian equations. Acta Math Sci, 2007, 27B(2): 329–337MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Huang G Y, Chen W Y. Universal bounds for eigenvalues of Laplacian operator of any order. Acta Math Sci, 2010, 30B(3): 939–948MathSciNetzbMATHGoogle Scholar
  3. [3]
    Hu Y X, Xu H W. An eigenvalue pinching theorem for compact hypersurfaces in a sphere. J Geom Anal, 2017, 27(3): 2472–2489MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Hu Y X, Xu H W, Zhao E T. First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures. Ann Global Anal Geom, 2015, 48(1): 23–35MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Li P, Yau S T. On the Schödinger equation and the eigenvalue problem. Comm Math Phys, 1983, 88(3): 309–318MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Stewarson K, Waechter R T. On hearing the shape of a drum: further results. Proc Camb Philol Soc, 1971, 69(2): 353–369CrossRefzbMATHGoogle Scholar
  7. [7]
    Payne L E, Pölya G, Weinberger H F. Sur le quotient de deux fréquence propers consécutives. C R Acad Sci Paris, 1955, 241: 917–919MathSciNetzbMATHGoogle Scholar
  8. [8]
    Hile G N, Potter M H. Inequalities for eigenvalues of the Laplacian. Indiana Univ Math J, 1980, 29(4): 255–306MathSciNetCrossRefGoogle Scholar
  9. [9]
    Yang H C. An estimate of the difference between consecutive eigenvalues. preprint. 1991, IC/91/60 of the Intl Centre for Theoretical PhysicsGoogle Scholar
  10. [10]
    Cheng Q M, Yang H C. Bounds on eigenvaluesof Dirichlet Laplacian. Math Ann, 2007, 337(1): 159–175MathSciNetCrossRefGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Center of Mathematical SciensesZhejiang UniversityHangzhouChina

Personalised recommendations