Minimization of the lowest eigenvalue for a vibrating beam

Quanyi Liang; Kairong Liu; Gang Meng; Zhikun She

doi:10.3934/dcds.2018085

Article Contents

2018, Volume 38, Issue 4: 2079-2092. Doi: 10.3934/dcds.2018085

This issue Previous Article The return times property for the tail on logarithm-type spaces Next Article Invariance entropy, quasi-stationary measures and control sets

Minimization of the lowest eigenvalue for a vibrating beam

1.
LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, China
2.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
3.
LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, China

* Corresponding author: Zhikun She; Email: zhikun.she@buaa.edu.cn
* Corresponding author: Zhikun She; Email: zhikun.she@buaa.edu.cn

Received: April 30, 2017

Published: June 2018

The third author is supported by the National Natural Science Foundation of China (Grant No. 11671378) and the Fund of UCAS. The fourth author is supported by the National Natural Science Foundation of China (Grants No. 11371047 and No. 11422111)

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Abstract

In this paper we solve the minimization problem of the lowest eigenvalue for a vibrating beam. Firstly, based on the variational method, we establish the basic theory of the lowest eigenvalue for the fourth order measure differential equation (MDE). Secondly, we build the relationship between the minimization problem of the lowest eigenvalue for the ODE and the one for the MDE. Finally, with the help of this built relationship, we find the explicit optimal bound of the lowest eigenvalue for a vibrating beam.

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Mathematics Subject Classification: Primary: 34L15; Secondary: 34L40.

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Figure 1. Function $\mathbf{L}(r)$ of $r$

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