Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Q in an n-dimensional Euclidean space R-n. If lambda(k+1) is the (k+l)th eigenvalue of Dirichlet Laplacian on Q, then, we prove that, 2 2 for n >= 41 and k >= 41,lambda(k+1) <= k(2)/(n) lambda(1) and, for any n and k,lambda(k+1) <= C-0(n, k)(2)/(n) lambda(1) with C-0 (n, k) <= j(n)(2)/2,1/j(n/2-1,1)(2) 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 Polya, we know that our estimates are optimal in the sense of order of k.