# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021181
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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
##### References:

show all references