Universal bounds for eigenvalues of the polyharmonic operators
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- by Jürgen Jost, Xianqing Li-Jost, Qiaoling Wang and Changyu Xia PDF
- Trans. Amer. Math. Soc. 363 (2011), 1821-1854 Request permission
Abstract:
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-Pólya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a Euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by Ashbaugh. Finally, we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere.References
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Additional Information
- Jürgen Jost
- Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
- Email: jost@mis.mpg.de
- Xianqing Li-Jost
- Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
- Email: xli-jost@mis.mpg.de
- Qiaoling Wang
- Affiliation: Departamento de Matemática, University of Brasilia, 70910-900, Brasília-DF, Brazil
- Email: wang@mat.unb.br
- Changyu Xia
- Affiliation: Departamento de Matemática, University of Brasilia, 70910-900, Brasília-DF, Brazil
- Email: xia@mat.unb.br
- Received by editor(s): August 19, 2008
- Published electronically: November 8, 2010
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 1821-1854
- MSC (2010): Primary 35P15, 53C20
- DOI: https://doi.org/10.1090/S0002-9947-2010-05147-5
- MathSciNet review: 2746667