The nonparametric integral of the calculus of variations as a Weierstrass integral. II: Some applications
type: Item
instance of: scholarly article
title: The nonparametric integral of the calculus of variations as a Weierstrass integral. II: Some applications
zbMATH Open document ID: 0582.49032
DOI: 10.1016/0022-247X(85)90293-8
author: Primo Brandi
author: Anna Salvadori
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: The authors study the non-parametric integrals of the calculus of variations as a Cesari integral in a very general setting. This approach consists in replacing the classical Lebesgue integrals of the calculus of variations with the Cesari integral of suitable set functions [see \textit{L. Cesari}, Trans. Am. Math. Soc. 102, 94-113 (1962; Zbl 0115.269), and ibid. 102, 114-146 (1962; Zbl 0115.270)]. As it is well-known, the Cesari theory of integration is a far reaching extension of both the Weierstrass integral of interval functions and the Burkill one for rectangle functions. The advantages it allows, over the Lebesgue one, are well-known and they are already conspicuous even in the elementary case of the length of a curve. The Cesari integral of the calculus of variations in the non-parametric case was introduced by the reviewer [see Atti Accad. Sci. Lett. Arti Palermo 19, 50-82 (1960; Zbl 0138.369); Ann. Sc. Norm. Super. Pisa, Cl. Sci. Fis. Mat., III. Ser. 22, 229-263 (1968; Zbl 0162.432), and ibid. 355-376 (1968; Zbl 0172.153)] in terms of a suitable associated parametric integral. Successively, many other authors studied this subject (see the references in the present papers). The general and abstract setting adopted by the authors in these notes is the following one. Let K be a metric space, E, \(E_ 1\) and \(E_ 2\) be three Banach spaces and let \(\{\) \(I\}\) be a collection of subsets of a topological space A. Given the function \(f: K\times E\to {\mathbb{R}}\), the set functions \(\phi\) : \(\{\) \(I\}\) \(\to E_ 1\), \(\lambda\) : \(\{\) \(I\}\) \(\to {\mathbb{R}}^+\) and the mappings \(\kappa\) : \(A\times \{I\}\to K\) and \(g: A\times \{I\}\to E_ 2\), let us consider the set function \(\Phi\) : \(\{\) \(I\}\) \(\to {\mathbb{R}}\) defined by \[ \Phi (I)=\lambda (I)f[\kappa (s_ I,I), \phi (I)\cdot g(s'\!_ I,I)/\lambda (I)] \] where \(s_ I,s'\!_ I\in I\) and \(e_ 1\cdot e_ 2\) denotes a bilinear form of \(E_ 1\times E_ 2\) into E. The Cesari integral of \(…
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 28B20
zbMATH DE Number: 3931884
zbMATH Keywords: non-parametric integrals
zbMATH Keywords: Cesari integral
zbMATH Keywords: existence
zbMATH Keywords: Tonelli type theorem
zbMATH Keywords: Burkill integral
zbMATH Keywords: weighted generalized area
reviewed by: Calogero Vinti
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3931884