The nonparametric integral of the calculus of variations as a Weierstrass integral. II: Some applications
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 \(\Phi\) over A, if it exists, is called the non- parametric Cesari integral of the calculus of variations. In the first paper the authors prove some existence results which extend those previously obtained by \textit{M. Boni} and \textit{P. Brandi} [Atti Sem. Mat. Fis. Univ. Modena 23, 308-327 (1974; Zbl 0331.49007)]. Moreover, they also state a Tonelli type theorem with the consequent representation result in terms of a Lebesgue-Stieltjes integral. These results are very general and sharp and, due to this fact, the authors can get several applications to particular problems of calculus of variations to which the second paper is devoted. Among others they recover, as particular cases, the existence theorem given by \textit{M. Boni} [Rend. Ist. Mat. Univ. Trieste 6, 51-70 (1974; Zbl 0295.49007)] in the case of continuous curves of bounded variation and extend the existence result proved by \textit{D. Candeloro} and \textit{P. Pucci} [Boll. Unione Mat. Ital., V. Ser., B 18, 1-24 (1981; Zbl 0461.49027)] for continuous surfaces of bounded variation in the sense of Tonelli, to the strict Burkill integral. Moreover the authors observe that the generality of their formulation, in particular the presence of the bilinear form, allows to include the integrals of the calculus of variations depending on second order differential elements. These classical integrals, as it is well-known, were introduced and studied by \textit{S. Cinquini} [Rend. Circ. Mat. Palermo, II. Ser. 6, 271-288 (1957; Zbl 0082.318)] (see also the references in the present papers). A further application is finally devoted to the definition of the weighted generalized area for \(L_ 1\) surfaces and to the formulation of a related Tonelli-type result, perfectly analogous to the classic one.
scientific article; zbMATH DE number 3931884
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