American Institute of Mathematical Sciences

April  2022, 42(4): 2005-2025. doi: 10.3934/dcds.2021181

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 Published  April 2022 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 and Continuous Dynamical Systems, 2022, 42 (4) : 2005-2025. doi: 10.3934/dcds.2021181
References:
 [1] L. Cesari, Optimization - Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, 1983. [3] B. Dacorogna, Direct Method in the Calculus of Variations, second edition, Springer, New York, 2008. [4] G. D. Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [5] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1972. [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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References:
 [1] L. Cesari, Optimization - Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5. [2] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, 1983. [3] B. Dacorogna, Direct Method in the Calculus of Variations, second edition, Springer, New York, 2008. [4] G. D. Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [5] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1972. [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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