Minimization of the lowest eigenvalue for a vibrating beam
type: Item
instance of: scholarly article
title: Minimization of the lowest eigenvalue for a vibrating beam
zbMATH Open document ID: 1403.34064
author: Quanyi Liang
author: Kairong Liu
author: Gang Meng
author: Zhikun She
published in: Discrete and Continuous Dynamical Systems
publication date: 2018-08-16T00:00:00Z
review text: Consider a fourth order measure differential equation (FOMDE) described by \[ dy^{(3)}(t)+\lambda y^{\prime \prime }\left( t\right) dt+y(t)d\mu \left(t\right) =0 \tag{1} \] in the interval [0,1] and with boundary conditions \[ y(0)=y^{\prime \prime }\left( 0\right) =y(1)=y^{\prime \prime }(1)=0.\tag{2} \] In this paper, using the well-known variational method, the authors, first, establish the minimization characterization for the lowest eigenvalue \(\lambda _{1}\left( \mu \right) \) of the FOMDE and then they construct a relation between the minimization problem of the lowest eigenvalue for the differential equation \[ y^{\prime \prime \prime \prime }(t)+\lambda y^{\prime \prime}(t)+F(t)y(t)=0,\quad t\in \left[ 0,1\right] \tag{3} \] together with the boundary conditions (2) and the one for FOMDE (1) and use this relationship to find explicit optimal lower bound of the lowest eigenvalue for the differential equation (3) arriving the following result: Theorem. Given \(r>0,\) one has \[ \mathbf{L}(r)=Q^{-1}\left( r\right) . \] Here \[ \mathbf{L}(r):=\inf \left\{ \lambda _{1}\left( F\right) :F\in B_{1}\left[ r\right] \right\} \] for \(r\in \left( 0,+\infty \right)\), \[ B_{1}\left[ r\right] :=\left\{ F\in \mathcal{L}^{1}:\left\| F\right\|_{1}\leq r\right\} \] and the invertible elementary function \(Q:\left( -\infty ,\pi ^{2}\right) \rightarrow \left( 0,+\infty \right) \) is defined as \[ Q\left( x\right) :=\frac{2x^{\frac{3}{2}}\sin x^{\frac{1}{2}}}{1-\cos x^{\frac{1}{2}}-\frac{1}{2}x^{\frac{1}{2}}\sin x^{\frac{1}{2}}}. \]
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 6919326
zbMATH Keywords: eigenvalue
zbMATH Keywords: minimization problem
zbMATH Keywords: fourth order equation
reviewed by: Erdoǧan Şen
MaRDI profile type: Publication
full work available at URL: dcds.2018085
OpenAlex ID: W2783703466
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DOI: 10.3934/DCDS.2018085
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description: scientific article; zbMATH DE number 6919326