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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Universal bounds for eigenvalues of the polyharmonic operators


Authors: Jürgen Jost, Xianqing Li-Jost, Qiaoling Wang and Changyu Xia
Journal: Trans. Amer. Math. Soc. 363 (2011), 1821-1854
MSC (2010): Primary 35P15, 53C20
Published electronically: November 8, 2010
MathSciNet review: 2746667
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05147-5
PII: S 0002-9947(2010)05147-5
Keywords: Universal bounds, eigenvalues, polyharmonic operator, Riemannian manifolds, Euclidean space, spheres
Received by editor(s): August 19, 2008
Published electronically: November 8, 2010
Article copyright: © Copyright 2010 American Mathematical Society