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Inequalities for Eigenvalues of a General Clamped Plate Problem

Published online by Cambridge University Press:  20 November 2018

K. Ghanbari
Affiliation:
Mathematics Department, Sahand University of Technology, Tabriz, Iran e-mail: kghanbari@sut.ac.ir bijanbijan_shekarbeigi@yahoo.com
B. Shekarbeigi
Affiliation:
Mathematics Department, Sahand University of Technology, Tabriz, Iran e-mail: kghanbari@sut.ac.ir bijanbijan_shekarbeigi@yahoo.com
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Abstract

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Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$. Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem:

$$\left\{ \begin{align} & {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\ & u{{|}_{\partial D}}\,=\,\frac{\partial u}{\partial n}\,{{|}_{\partial D}}\,=\,0, \\ \end{align} \right.$$

where $V(x)$ is a nonnegative potential, and $\rho (x)\,\in \,C(\overset{-}{\mathop{D}}\,)$ is positive. We prove the following inequalities:

$$\begin{align} & {{\mu }_{k+1}}\le \frac{1}{k}\sum\limits_{i=1}^{k}{\mu i+}{{[\frac{8(n+2)}{{{n}^{2}}}{{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}]}^{1/2}}\times \frac{1}{k}{{\sum\limits_{i=1}^{k}{[{{\mu }_{i}}({{\mu }_{k+1}}-{{\mu }_{i}})]}}^{1/2}}, \\ & \frac{{{n}^{2}}{{k}^{2}}}{8(n+2)}\le {{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}[\sum\limits_{i=1}^{k}{\frac{\mu _{i}^{1/2}}{{{\mu }_{k+1}}-{{\mu }_{i}}}}]\times \sum\limits_{i=1}^{k}{\mu _{i}^{1/2}}. \\ \end{align}$$

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Chen, Z.-C. and Qian, C.-L., Estimates for discrete spectrum of Laplacian operators with any order. J. China Univ. Sci. Tech. 20(1990), no. 3, 259266.Google Scholar
[2] Cheng, Q. M. and Yang, H., Inequalities for eigenvalues of a clamped plate problem. Trans. Amer. Math. Soc. 358(2005), no. 6, 26252635. doi:10.1090/S0002-9947-05-04023-7Google Scholar
[3] Evans, L. C., Partial Differential Equations. Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.Google Scholar
[4] Hile, G. N. and Yeh, R. Z., Inequalities for eigenvalues of the biharmonic operator. Pacific J. Math. 112(1984), no. 1, 115133.Google Scholar
[5] Hook, S. M., Domain independent upper bounds for eigenvalues of elliptic operators. Trans. Amer. Math. Soc. 318(1990), no. 2, 615642. doi:10.2307/2001323Google Scholar
[6] Payne, L. E., Pólya, G., and Weinberger, H. F., On the ratio of consecutive eigenvalues. J. Math. and Phys. 35(1956), 289298.Google Scholar