Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space


Authors: Qing-Ming Cheng and Hongcang Yang
Journal: Proc. Amer. Math. Soc. 139 (2011), 461-471
MSC (2010): Primary 35P15, 58G40
Published electronically: September 23, 2010
MathSciNet review: 2736329
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35P15, 58G40

Retrieve articles in all journals with MSC (2010): 35P15, 58G40


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

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10484-7
PII: S 0002-9939(2010)10484-7
Keywords: Eigenvalue, universal inequality for eigenvalues, hyperbolic space, biharmonic operator and a clamped plate problem.
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.