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Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

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Abstract

In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian \((-\Delta )^s|_{\Omega }\) for \(s>0\) and \(s\in \mathbb {Q}_+\). Here \(\Omega \subset \mathbb {R}^n\) is a bounded open domain, and \(\mathbb {Q}_+\) is the set of all positive rational numbers. Secondly, if \(s\in \mathbb {Q}_+\) and \(s\ge 1\) (in this case, the operator is also called the non-integer poly-Laplacian), then by this universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called Yang type inequality for the corresponding eigenvalue problem, which is the extension to the case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding eigenvalues from the Yang type inequality.

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Acknowledgements

The last version of this paper was done when the first author was visiting the Max-Planck Institute for Mathematics in the Sciences, Leipzig during August of 2016. We would like to thank Professor J. Jost (in Max-Planck Institute for Mathematics in the Sciences, Leipzig) for his invitation and support. We also thank the referee for the comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hua Chen & Ao Zeng

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  1. Hua Chen
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  2. Ao Zeng
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Correspondence to Hua Chen.

Additional information

Communicated by J. Jost.

This work is partially supported by the NSFC Grants 11631011 and 11626251.

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Chen, H., Zeng, A. Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain. Calc. Var. 56, 131 (2017). https://doi.org/10.1007/s00526-017-1220-y

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