American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021181
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Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

 SISSA, Via Bonomea 265 - 34136 Trieste, Italy

* Corresponding author: Sandro Zagatti

Received  June 2021 Revised  October 2021 Early access November 2021

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.
Citation: Sandro Zagatti. Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021181
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