The minimum problem for one-dimensional non-semicontinuous functionals

Abstract

We study the minimum problem for functionals of the form

$$\begin{aligned} \mathcal {F}(u) = \int _{I} f(x, u(x), u^\prime (x))\,dx, \end{aligned}$$

where the integrand $f:I\times \mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}$ is not convex in the last variable. We provide an existence result assuming that the lower convex envelope $\overline{f}=\overline{f}(x,p,\xi )$ of f satisfies a suitable affinity condition on the set on which $f>\overline{f}$ and that the map $p_i\mapsto f(x,p,\xi )$ is monotone with respect to one single component $p_i$ of the vector p. We show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the existence of minimizers.