Article Contents
Article Contents

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

• * Corresponding author: Sandro Zagatti
• We study the minimum problem for functionals of the form

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

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.

Mathematics Subject Classification: Primary: 49J45, 49K15.

 Citation:

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