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
$ \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.
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.,

show all references

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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