Singular minimisers in the calculus of variations in one dimension
type: Item
instance of: scholarly article
title: Singular minimisers in the calculus of variations in one dimension
zbMATH Open document ID: 0656.49005
DOI: 10.1007/BF00251459
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: Consider the basic problem of the calculus of variations: to choose an absolutely continuous function u: [a,b]\(\to {\mathbb{R}}\) which minimizes the integral functional \(\int^{b}_{a}f(x,u(x),u'(x))dx\) subject to the constraints \(u(a)=\alpha\), \(u(b)=\beta.\) This paper presents several general constructions of the following type: Let u: [a,b]\(\to {\mathbb{R}}\) be any increasing continuous function for which u'(x) exists in \(]0,+\infty]\) for every \(x\in [a,b]\), the set \(E=\{x:\) \(u'(x)=\infty \}\) has measure zero, and u is \(C^{\infty}\) outside E. Let \(\rho\) : [a,b]\(\to {\mathbb{R}}\) be any positive-valued continuous function which is smooth outside E. Then there exists a nonnegative \(C^{\infty}\) function f(u,p): \({\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) satisfying \(f_{pp}(x,p)>0\), for which u(\(\cdot)\) solves the variational problem described above while \(f_ p(u(x),u'(x))=\rho (x)\). The paper also includes specific examples, including one which appears to be the first published instance in which the Euler equation fails even though \(f_ u\) is integrable along the solution arc u(\(\cdot).\) Writing on a related topic, \textit{A. C. Heinricher} and \textit{V. J. Mizel} [Arch. Ration. Mech. Anal. 102, No.1, 57-93 (1988; Zbl 0655.49003)] call the current paper ``a profound improvement of the original examples of Ball and Mizel.''
Mathematics Subject Classification ID: 49K05
Mathematics Subject Classification ID: 49J05
Mathematics Subject Classification ID: 49K30
zbMATH DE Number: 4071759
zbMATH Keywords: singular solutions
zbMATH Keywords: calculus of variations
zbMATH Keywords: integral functional
zbMATH Keywords: Euler equation
author: A. M. Davie
reviewed by: P. D. Loewen
MaRDI profile type: Publication
cites work: Singular minimizers for regular one-dimensional problems in the calculus of variations
cites work: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation
cites work: Regularity Properties of Solutions to the Basic Problem in the Calculus of Variations
cites work: Q3732431
full work available at URL: https://doi.org/10.1007/bf00251459
OpenAlex ID: W1967781275
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description: scientific article; zbMATH DE number 4071759