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

  • * Corresponding author: Sandro Zagatti

    * Corresponding author: Sandro Zagatti
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  • 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.

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


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    [6] K. Wang and Y. Li, Existence and monotonicity of minimizers of a nonconvex variational problem with a second-order lagrangian, Discrete Continuous Dynam. Systems, 25 (2009), 687-699.  doi: 10.3934/dcds.2009.25.687.
    [7] S. Zagatti, Uniqueness and continuous dependence on boundary data for integro-extremal minimizer of the functional of the gradient, J. Convex Analysis, 14 (2007), 705-727. 
    [8] S. Zagatti, Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations, Calc. Var. and PDE's, 31 (2008), 511-519.  doi: 10.1007/s00526-007-0124-7.
    [9] S. Zagatti, Minimization of non quasiconvex functionals by integro-extremization method, Discrete Continuous Dynam. Systems - A, 21 (2008), 625-641.  doi: 10.3934/dcds.2008.21.625.
    [10] S. Zagatti, The minimum problem for one-dimensional non-semicontinuous functionals, to appear 2021.,
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