Mathematische Annalen

, Volume 337, Issue 1, pp 159–175

Bounds on eigenvalues of Dirichlet Laplacian

Article

DOI: 10.1007/s00208-006-0030-x

Cite this article as:
Cheng, QM. & Yang, H. Math. Ann. (2007) 337: 159. doi:10.1007/s00208-006-0030-x

Abstract

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 \(k\geq 41, \lambda_{k+1}\leq k^{\frac2n}\lambda_1\) and, for any n and \(k, \lambda_{k+1}\leq C_{0}(n,k) k^{\frac2n}\lambda_1\) with \(C_0(n,k)\leq {j^{2}_{n/2,1}}/{j^{2}_{n/2-1,1}}\), 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.

Mathematics Subject Classification (2000)

35P1558G25

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringSaga UniversitySagaJapan
  2. 2.International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  4. 4.Academy of Mathematics and Systematical SciencesCASBeijingChina