Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations
type: Item
instance of: scholarly article
title: Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations
zbMATH Open document ID: 0856.49035
DOI: 10.1007/BF02104722
published in: Siberian Mathematical Journal
publication date: 1997-03-03T00:00:00Z
review text: The author investigates families of one-dimensional variational problems with a given integrand \(f = f(t, u, \dot u)\) which is strictly convex and coercive with respect to \(\dot u\). The concrete variational problem is determined by a quadruple \(\{a, b, A, B\}\) where \( (a,b)\) gives the interval of integration but \(A\) and \(B\) are the boundary values of \(u\) at \(t = a\) and \(t = b\), respectively. Let the set \(U\) consist of those pairs \((t_0, d_0) \in \mathbb{R}^2\) for which there exists a quadruple \(\{a,b,A,B\}\) such that the corresponding solution \(u \in W^1_1 (a,b)\) of the variational problem has \(u(t_0) = d_0\), \(\dot u(t_0) = \infty\). It is shown that the set \(U\) for a given \(f \in C^1 (\mathbb{R}^3)\) is a set of the first category.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 917144
zbMATH Keywords: regularity of solutions
zbMATH Keywords: one-dimensional variational problems
author: M. A. Sychev
reviewed by: Uldis Raitums
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 917144