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Acta Mathematica Scientia

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

New Bounds on Eigenvualues of Laplacian

  • Zhengchao JiEmail author
Article
  • 1 Downloads

Abstract

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 

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Notes

Acknowledgements

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.

References

  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

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