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Zbl 1180.35389
Cheng, Qing-Ming; Ichikawa, Takamichi; Mametsuka, Shinji
Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere.
(English)
[J] Calc. Var. Partial Differ. Equ. 36, No. 4, 507-523 (2009). ISSN 0944-2669; ISSN 1432-0835/e

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 {\it Q.-M. Cheng} and {\it 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 $i = 0, 1, \dots , l$ 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.
MSC 2000:
*35P15 Estimation of eigenvalues for PD operators
35J91
35J35 Higher order elliptic equations, variational problems

Keywords: poly-Laplacian; $n$-dimensional unit sphere; estimates for eigenvalues

Citations: Zbl 1122.35086

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