One-dimensional infinite-horizon variational problems arising in continuum mechanics
type: Item
instance of: scholarly article
title: One-dimensional infinite-horizon variational problems arising in continuum mechanics
zbMATH Open document ID: 0672.73010
DOI: 10.1007/BF00251430
author: Arie Leizarowitz
author: Victor J. Mizel
published in: Archive for Rational Mechanics and Analysis
publication date: 1989-01-01T00:00:00Z
review text: The authors study a variational problem for real valued functions defined on an infinite semiaxis of the line. They seek a ``minimal solution'' to the following problem: minimize the functional \[ (1)\quad I(w(\cdot))=\int^{\infty}_{0}f(w(s),\dot w(s),\quad \ddot w(s))ds, \] \[ (2)\quad w\in A_ x=\{v\in W^{2,1}_{loc}(0,\infty):\quad (v(0),\dot v(0))=\underline x\}. \] Here \(W^{2,1}_{loc}\subset C^ 1\) denotes the Sobolev space of functions possessing a locally integrable second derivative, and \(f=f(w,p,r)\) is a smooth function satisfying \(f_{rr}\geq 0,\) \(f(w,p,r)\geq a| w|^{\alpha}-b| p|^{\beta}+c| r|^{\gamma}-d;\) \(a,b,c,d>0;\) \(\alpha,\gamma \in (1,\infty),\) \(\beta \in [1,\infty),\) \(\alpha >\beta,\) \(\gamma >\beta\) and \(\underline x\) is a given value. The organization of the paper is as follows: In Section 1 the connection of the variational problem (1),(2) with the equilibrium problem of a long slender bar of polymetric material under tension is discussed. In Section 2 a fixed endpoint variational problem is analyzed. In Section 3 a criterion is given for the solution of (1),(2) to be minimal. Section 4 gives the proof the existence of a minimal energy solution. The Section 5 is containing the main result of the paper: there always exists a periodic minimal solution of (1),(2). In Section 7 an interesting analytic result is established.
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74A99
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49K27
zbMATH DE Number: 4100677
zbMATH Keywords: equilibrium problem
zbMATH Keywords: long slender bar
zbMATH Keywords: polymetric material
zbMATH Keywords: tension
zbMATH Keywords: fixed endpoint variational problem
zbMATH Keywords: existence
zbMATH Keywords: minimal energy solution
zbMATH Keywords: periodic minimal solution
reviewed by: István Ecsedi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4100677