Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations
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.
scientific article; zbMATH DE number 917144
scientific article