Extremum estimates of the \(L^1\)-norm of weights for eigenvalue problems of vibrating string equations based on critical equations
type: Item
instance of: scholarly article
title: Extremum estimates of the \(L^1\)-norm of weights for eigenvalue problems of vibrating string equations based on critical equations
zbMATH Open document ID: 1472.34036
author: Bing Xie
author: Jiangang Qi
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2021-06-17T00:00:00Z
review text: The authors consider the boundary value problem \begin{align*} -y''(x)=\lambda w(x)y(x),\tag{1}\\ y(x)=0=y(1)-hy'(1), \quad 0<h<1\tag{2}, \end{align*} where \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(mes \{x:w(x)>0, \, x\in [0,1]\}>0\) as well as \(mes \{x:w(x)<0, \, x\in [0,1]\}>0\). The expression of the infimum function \(E(\lambda, h)\) of \(||w||_1\) is given if (1) and (2) has nontrivial solution i.e \(\int_0^1|w|\ge E(\lambda,h)\) and \[ E(\lambda, h)=\inf\{||w||_1: w\in \Omega(\lambda)\},\tag{3} \] where \[ \Omega(\lambda)=\left\{w\in L^1[a,b]: \lambda\in \sigma(w)\cap\mathbb R, \int_0^1|w|>0\right\}. \] The authors apply the critical equation weights to solve the extremal problem (3). They obtain the following main result: ``If the problem (1) and (2) has nontrivial solution for \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(h\in(0,1)\), then \[ E(\lambda, h)=\dfrac1{|\lambda|}\in\left\{\dfrac{1-h}h, \dfrac4{1-h}\right\}. \]
reviewed by: Bertin Zinsou
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 7360590
zbMATH Keywords: Lyapunov-type inequality
zbMATH Keywords: extremum
zbMATH Keywords: vibrating string equations
zbMATH Keywords: critical equations
zbMATH Keywords: eigenvalue
MaRDI profile type: Publication
full work available at URL: dcdsb.2020243
OpenAlex ID: W3057081467
cites work: On a Liapounoff Criterion of Stability
cites work: Q3999065
cites work: Estimates for the first eigenvalue in some Sturm-Liouville problems
cites work: Q5352070
cites work: Sturm-Liouville problems involving distribution weights and an application to optimal problems
cites work: Bounds and optimization of the minimum eigenvalue for a vibrating system
cites work: On nonlinear boundary value problem corresponding to \(N\)-dimensional inverse spectral problem
cites work: On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems
cites work: Sharp Estimates for the Eigenvalues of Some Differential Equations
cites work: On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability
cites work: An Introduction to Banach Space Theory
cites work: Lyapunov-type Inequalities
cites work: Extremal norms of the potentials recovered from inverse Dirichlet problems
cites work: Extremal values of eigenvalues of Sturm-Liouville operators with potentials in \(L^1\) balls
cites work: On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures
cites work: Extremal values of smallest eigenvalues of Hill's operators with potentials in \(L^1\) balls
cites work: On the number and complete continuity of weighted eigenvalues of measure differential equations.
DOI: 10.3934/DCDSB.2020243
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description: scientific article; zbMATH DE number 7360590