Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space
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- by Qing-Ming Cheng and Hongcang Yang PDF
- Proc. Amer. Math. Soc. 139 (2011), 461-471 Request permission
Abstract:
In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an $n$-dimensional hyperbolic space. It is well known that, for a bounded domain in the $n$-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yang-type universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the $n$-dimensional Euclidean space, which is sharper than the other ones. For a domain in a unit sphere, Wang and Xia (2007) have also given a universal inequality for eigenvalues. For a bounded domain in the $n$-dimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing “nice” trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to $\tfrac {(n-1)^4}{16}$ when the domain tends to the hyperbolic space, then all of the eigenvalues tend to $\tfrac {(n-1)^4}{16}$.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, The Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
- Email: yanghc@math03.math.ac.cn
- Received by editor(s): January 27, 2009
- Published electronically: September 23, 2010
- Additional Notes: The first author’s research was partially supported by a Grant-in-Aid for Scientific Research from JSPS
The second author’s research was partially supported by the NSF of China and the Fund of the Chinese Academy of Sciences. - Communicated by: Matthew J. Gursky
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 461-471
- MSC (2010): Primary 35P15, 58G40
- DOI: https://doi.org/10.1090/S0002-9939-2010-10484-7
- MathSciNet review: 2736329