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Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere. (English) Zbl 1180.35389
Summary: 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 Q.-M. Cheng and H. Yang [Math. Ann. 331, No. 2, 445–460 (2005; Zbl 1122.35086)] is included in ours. 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.
MSC:
35P15Estimation of eigenvalues and upper and lower bounds for PD operators
35J91Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J35Higher order elliptic equations, variational problems
References:
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