In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain in an n-dimensional Euclidean space R n . If λ k+1 is the (k+1)th eigenvalue of Dirichlet Laplacian on , then, we prove that, for n ≥ 41 and k ≥ 41, λ k+1 ≤ k 2 n λ 1 and, for any n and k, λ k+1 ≤ C 0 (n, k)k 2 n λ 1 with C 0 (n, k) ≤ 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. Mathematics Subject Classification (2001) 35P15 · 58G25Q.-M.

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In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λhypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with polynomial area growth and H−λ ≥ 0, which are generalizations of the results due to Huisken [19], . We also define a F -functional and study F -stability of λ-hypersurfaces, which extend a result of Colding-Minicozzi [11]. Lower bound growth and upper bound growth of the area for complete and non-compact λ-hypersurfaces are also studied.2001 Mathematics Subject Classification: 53C44, 53C42. Key words and phrases: the weighted volume-preserving mean curvature flow, the weighted area functional, F -stability, weak stability, λ-hypersurfaces.

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Abstract. In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we solve partially a conjecture proposed in [7]. For a domain in the unit sphere, we give an important improvement on the results of Wang and Xia [16].