Singular minimisers in the calculus of variations in one dimension
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.''
scientific article; zbMATH DE number 4071759
scientific article