No abstract

In this paper, we study eigenvalues of Laplacian with any order on a bounded domain in an n-dimensional Euclidean space and obtain estimates for eigenvalues, which are the Yang-type inequalities. In particular, the sharper result of Yang is included here. Furthermore, for lower order eigenvalues, we obtain two sharper inequalities. As a consequence, a proof of results announced by Ashbaugh [1] is also given.

In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331: [445][446][447][448][449][450][451][452][453][454][455][456][457][458][459][460] 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions a i and b i , for i = 0, 1, . . . , l and two operators µ and η. First of all, we study properties of functions a i and b i and the operators µ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues. Mathematics Subject Classification (2000) 35P15 · 58G25( 1.1) where is the Laplacian in M and ν denotes the outward unit normal. It is well known that the spectrum of this eigenvalue problem is real and discrete. 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · −→ ∞.