Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space
Authors:
QingMing Cheng and Hongcang Yang
Journal:
Proc. Amer. Math. Soc. 139 (2011), 461471
MSC (2010):
Primary 35P15, 58G40
Published electronically:
September 23, 2010
MathSciNet review:
2736329
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Abstract: In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an dimensional hyperbolic space. It is well known that, for a bounded domain in the 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 Yangtype universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the 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 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 when the domain tends to the hyperbolic space, then all of the eigenvalues tend to .
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Additional Information
QingMing Cheng
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 8408502, Japan
Email:
cheng@ms.sagau.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/S000299392010104847
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 GrantinAid 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.
