Inequalities for eigenvalues of a clamped plate problem
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- by Qing-Ming Cheng and Hongcang Yang PDF
- Trans. Amer. Math. Soc. 358 (2006), 2625-2635 Request permission
Abstract:
Let $D$ be a connected bounded domain in an $n$-dimensional Euclidean space $\mathbb {R}^n$. Assume that \[ 0 < \lambda _1 <\lambda _2 \le \cdots \le \lambda _k \le \cdots \] are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: \[ \left \{ \begin {aligned} &\Delta ^2 u =\lambda u, \ \text { in $D$,} &u|_{\partial D}=\left . \frac {\partial u}{\partial n}\right |_{\partial D}=0. \end {aligned} \right . \] 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: \[ \lambda _{k+1} \le \frac 1k\sum _{i=1}^k \lambda _i +\left [\frac {8(n+2)}{n^2} \right ]^{1/2} \frac 1k\sum _{i=1}^k \biggl [ \lambda _i(\lambda _{k+1} -\lambda _i) \biggl ]^{1/2}. \] Further, a more explicit inequality of eigenvalues is also obtained.References
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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
- 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 - © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 2625-2635
- MSC (2000): Primary 35P15, 58G25
- DOI: https://doi.org/10.1090/S0002-9947-05-04023-7
- MathSciNet review: 2204047