# American Institute of Mathematical Sciences

April  2018, 38(4): 2079-2092. doi: 10.3934/dcds.2018085

## 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

Received  May 2017 Published  January 2018

Fund Project: 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).

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.

Citation: Quanyi Liang, Kairong Liu, Gang Meng, Zhikun She. Minimization of the lowest eigenvalue for a vibrating beam. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2079-2092. doi: 10.3934/dcds.2018085
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##### References:
Function $\mathbf{L}(r)$ of $r$
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