Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

摘要

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 331:445-460, 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 mu and eta. First of all, we study properties of functions a (i) and b (i) and the operators mu and eta. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.