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Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian
SISSA, Via Bonomea 265 - 34136 Trieste, Italy |
$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $ |
$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $ |
$ \overline{f} = \overline{f}(x,p,q,\xi) $ |
$ f $ |
$ \xi $ |
$ p_i, q_i, \xi_i $ |
$ p,q,\xi $ |
$ p_i $ |
$ \xi_i $ |
$ \{f> \overline{f}\} $ |
$ q_i $ |
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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