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

Qing-Ming Cheng; Ichikawa, Takamichi; Mametsuka, Shinji
December 2009
Calculus of Variations & Partial Differential Equations;Dec2009, Vol. 36 Issue 4, p507
Academic Journal
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 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.


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