Equimeasurable rearrangements of functions and fourth order boundary value problems
type: Item
instance of: scholarly article
title: Equimeasurable rearrangements of functions and fourth order boundary value problems
zbMATH Open document ID: 0851.34021
author: Philip Savoye
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-11-26T00:00:00Z
full work available at URL: contents.htm
review text: The boundary value problem (1) \((p(x) y'')''+ \lambda y''+ F(x) y= 0\), \(x\in [0, l]\), (2) \(y(0)= y(l)= y''(0)= y''(l)= 0\), arises in the study of the buckling of a beam which is supported on an elastic foundation. The functions \(p(x)\), \(F(x)\) are the stiffness and elastic destructive force per unit length. The rearrangements of functions \(f_{\pm m}\) that are equimeasurable with \(f\) and periodic in \([0, l]\) with period equal to \({l\over n}\) and satisfy the symmetry requirement are introduced. Some theorems on these functions follow. The \(n\)th eigenvalue of the problem defined by (1) and (2) above is shown to have a lower bound expressible in the form (3) \(\lambda_n(p, F)\geq \lambda_n(\overline p_{-n}, F^*_{+ n}(\lambda, \xi))\). In the above \(\overline p_{- n}\) is the rearrangement which is equimeasurable with \(p(x)\), periodic in \([0, l]\) with period equal to \({l\over n}\) and increasing in the interval \([0, {l\over 2n}]\). The function \(F^*_{+ n}(x, \xi)\) equals \(\overline F_n(x) H^*_n(x, \xi)\) and \(H^*_n(x, \xi)\) is the periodic generalization of the step function with period \({l\over n}\).
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 42C20
zbMATH DE Number: 894684
zbMATH Keywords: boundary value problem
zbMATH Keywords: buckling of a beam
MaRDI profile type: Publication
OpenAlex ID: W2058553037
cites work: A Prufer Transformation for the Equation of the Vibrating Beam
cites work: Buckling of columns and rearrangements of functions
cites work: Rearrangements of functions and lower bounds for eigenvalues of differential equations
cites work: Q3937910
cites work: Q3136598
cites work: Q5722416
cites work: Q3067863
DOI: 10.1216/RMJM/1181072116
description: scientific article; zbMATH DE number 894684