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

 

Inequalities for eigenvalues of a clamped plate problem


Authors: Qing-Ming Cheng and Hongcang Yang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2625-2635
MSC (2000): Primary 35P15, 58G25
Published electronically: October 31, 2005
MathSciNet review: 2204047
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a connected bounded domain in an $ n$-dimensional Euclidean space $ \mathbb{R}^n$. Assume that

$\displaystyle 0 < \lambda_1 <\lambda_2 \le \cdots \le \lambda_k \le \cdots $

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

$\displaystyle \left \{ \aligned&\Delta^2 u =\lambda u, \text{ in $D$,}\\ &u\ve... ...rac {\partial u}{\partial n}\right \vert _{\partial D}=0. \endaligned \right . $

Then, we give an upper bound of the $ (k+1)$-th eigenvalue $ \lambda_{k+1}$ in terms of the first $ k$ eigenvalues, which is independent of the domain $ D$, that is, we prove the following:

$\displaystyle \lambda_{k+1} \le \frac 1k\sum_{i=1}^k \lambda_i +\left [\frac {8... ...ac 1k\sum_{i=1}^k \biggl[ \lambda_i(\lambda_{k+1} -\lambda_i) \biggl ]^{1/2}. $

Further, a more explicit inequality of eigenvalues is also obtained.


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Additional Information

Qing-Ming Cheng
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan
Email: cheng@ms.saga-u.ac.jp

Hongcang Yang
Affiliation: Academy of Mathematics and Systematical Sciences, CAS, Beijing 100080, People’s Republic of China
Email: yanghc@math03.math.ac.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04023-7
PII: S 0002-9947(05)04023-7
Keywords: Eigenvalue, inequality of eigenvalue, biharmonic operator, clamped plate problem
Received by editor(s): December 10, 2002
Received by editor(s) in revised form: July 13, 2004
Published electronically: October 31, 2005
Additional Notes: The first author’s research was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science
The second author’s research was partially supported by the NSF of China and the Fund of CAS
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.