We study the minimum problem for functionals of the form begin{equation*} mathcal{F}(u) = int_{I} f(x, u(x), u^pr(x), u^pp(x)),dx, end{equation*} oindent where the integrand $f:I imesR^m imesR^m imes R^m o R$ is not convex in the last variable. We provide an existence result assuming that the lower convex envelope $of=of(x,p,q,\xi)$ 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 ${f&gt;of}$ 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.

Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian / Zagatti, Sandro. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 42:4(2022), pp. 2005-2025. [10.3934/dcds.2021181]

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

#### Abstract

We study the minimum problem for functionals of the form begin{equation*} mathcal{F}(u) = int_{I} f(x, u(x), u^pr(x), u^pp(x)),dx, end{equation*} oindent where the integrand $f:I imesR^m imesR^m imes R^m o R$ is not convex in the last variable. We provide an existence result assuming that the lower convex envelope $of=of(x,p,q,\xi)$ 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 ${f>of}$ 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.
##### Scheda breve Scheda completa Scheda completa (DC)
42
4
2005
2025
https://www.aimsciences.org/article/doi/10.3934/dcds.2021181
Zagatti, Sandro
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/125269
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