On the Lavrentiev phenomenon and the validity of Euler-Lagrange equations for a class of integral functionals
The paper under review deals with the problem of establishing when for a functional of the calculus of variations of the form \[ F(u)= \int^ 1_ 0 f(t, u, u') dt\qquad (u\in W^{1,1}(0,1; \mathbb{R}^ n)), \] subjected to the boundary conditions \(u(0)= \alpha\), \(u(1)= \beta\), the equality \[ \inf\{ F(u): u\in W^{1,1}\}= \inf\{ F(u): u\in W^{1,\infty}\}\tag{1} \] holds. This is related with the nonoccurrence of the so-called \textit{Lavrentiev phenomenon} (\textit{M. Lavrentieff} [Ann. Mat. Pura Appl. 4, 7-28 (1926; JFM 53.0481.02)], where a first example of a strict inequality in (1) was shown). The main result of the paper is an approximation result in energy for a function \(u\in W^{1,p}(0,1; \mathbb{R}^ n)\) by means of Lipschitz continuous functions, provided there exist \(\varepsilon> 0\) and \(C> 0\) such that the integrand \(f\) satisfies the inequality \[ | f(t, s, u'(t))- f(t, u(t), u'(t))|\leq C(1+ | u'(t)|^{p+ \varepsilon})| u(t)- s|^{\varepsilon/(p- 1)} \] in a neighbourhood of the graph of \(u\). In Section 4 the question of the validity of Euler-Lagrange equation for local minima is also studied, and assumptions weaker than the classical ones are found (Theorem 4.2).
scientific article; zbMATH DE number 617441