Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

We study the minimum problem for functionals of the form

$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $

where the integrand $ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \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,q,\xi) $ of $ f $ with respect to $ \xi $ is regular and enjoys a special dependence with respect to the i-th single components $ p_i, q_i, \xi_i $ of the vector variables $ p,q,\xi $. More precisely, we assume that it is monotone in $ p_i $ and that it satisfies suitable affinity properties with respect to $ \xi_i $ on the set $ \{f> \overline{f}\} $ and with respect to $ q_i $ on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.