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Universal inequalities for eigenvalues of a system of elliptic equations

Published online by Cambridge University Press:  25 March 2009

Qing-Ming Cheng  and
Hong-Cang Yang
Qing-Ming Cheng
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan (cheng@ms.saga-u.ac.jp)
Hong-Cang Yang
Affiliation:
Academy of Mathematics and Systematical Sciences, Chinese Academy of Sciences, Beijing 100080, People's Republic of China (yanghc@math03.math.ac.cn)
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Abstract

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume that

are eigenvalues of an eigenvalue problem of a system of n elliptic equations:

In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 2 , April 2009 , pp. 273 - 285
Copyright
Copyright © Royal Society of Edinburgh 2009

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