Abstract
In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R n. If λ k+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 j p,k denotes the k-th positive zero of the standard Bessel function J p (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.
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Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science
H. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA