Minimizers of Dirichlet Functionals on the n-Torus and the Weak KAM Theory
Wolansky, G.
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 521-545 / Harvested from Numdam
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@article{AIHPC_2009__26_2_521_0,
     author = {Wolansky, G.},
     title = {Minimizers of Dirichlet Functionals on the $n$-Torus and the Weak KAM Theory},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {26},
     year = {2009},
     pages = {521-545},
     doi = {10.1016/j.anihpc.2007.09.007},
     zbl = {1173.35047},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2009__26_2_521_0}
}

Wolansky, G. Minimizers of Dirichlet Functionals on the $n$-Torus and the Weak KAM Theory. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) pp. 521-545. doi : 10.1016/j.anihpc.2007.09.007. http://gdmltest.u-ga.fr/item/AIHPC_2009__26_2_521_0/

[1] Arriola E. A., Soler J., A Variational Approach to the Schrödinger-Poisson System: Asymptotic Behaviour, Breathers, and Stability, J. Stat. Phys. 103 (5-6) (2001) 1069-1106. | MR 1851367 | Zbl 0999.82062

[2] Aubry S., The Twist Map, the Extended Frenkel-Kontrovna Model and the Devil's Staircase, Physica D 7 (1983) 240-258. | MR 719055 | Zbl 0559.58013

[3] Bernard P., Buffoni B., Optimal Mass Transportation and Mather Theory, Preprint, http://arxiv.org/abs/math.DS/0412299. | MR 2283105 | Zbl pre05129006

[4] Evans L. C., A Survey of Partial Differential Equations Methods in Weak KAM Theory, Comm. Pure Appl. Math. 57 (4) (2004) 445-480. | MR 2026176 | Zbl 1040.37046

[5] Evans L. C., Gomes D., Effective Hamiltonians and Averaging for Hamiltonian Dynamics. I, Arch. Ration. Mech. Anal. 157 (1) (2001) 1-33. | MR 1822413 | Zbl 0986.37056

[6] Evans L. C., Gomes D., Effective Hamiltonians and Averaging for Hamiltonian Dynamics. II, Arch. Ration. Mech. Anal. 161 (4) (2002) 271-305. | MR 1891169 | Zbl 1100.37039

[7] Fathi A., The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics Series, vol. 88, Cambridge University Press, 2003.

[8] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1977. | MR 473443 | Zbl 0361.35003

[9] Gomes D. A., Oberman A. M., Computing the Effective Hamiltonian Using a Variational Approach, SIAM J. Control Optim. 43 (3) (2004) 792-812. | MR 2114376 | Zbl 1081.49024

[10] Govin M., Chandre C., Jauslin H. R., Kolmogorov-Arnold-Moser-Renormalization-Group Analysis of Stability in Hamiltonian Flows, Phys. Rev. Lett. 79 (1997) 3881-3884.

[11] L. Granieri, On action minimizing measures for the Monge-Kantorovich problem, Preprint, July 2004. | MR 2346457 | Zbl 1133.37027

[12] Hedlund G. A., Geodesics on a 3 Dimensional Riemannian Manifolds With Periodic Coefficients, Ann. of Math. 33 (1932) 719-739. | JFM 58.1256.01 | MR 1503086

[13] Hiriart-Urruty J. B., Lemarechal C., Convex Analysis and Minimization Algorithms II, Grundlehren der Mathematischen Wissenschaften, vol. 306, Springer-Verlag, 1993, (Chapter 10). | MR 1295240 | Zbl 0795.49002

[14] Illner R., Zweifel P. F., Lange H., Global Existence, Uniqueness and Asymptotic Behaviour of Solutions of the Wigner-Poisson and Schrödinger-Poisson Systems, Math. Meth. Appl. Sci. 17 (1994) 349-376. | MR 1273317 | Zbl 0808.35116

[15] Keller J. B., Semiclassical Mechanics, SIAM Rev. 27 (4) (1985) 485-504. | MR 812451 | Zbl 0581.70012

[16] Luigi D. P., Stella G. M., Granieri L., Minimal Measures, One-Dimensional Currents and the Monge-Kantorovich Problem, Calc. Var. Partial Differential Equations 27 (1) (2006) 1-23. | MR 2241304 | Zbl 1096.37033

[17] Mañè R., On the Minimizing Measures of Lagrangian Dynamical Systems, Nonlinearity 5 (1992) 623-638. | MR 1166538 | Zbl 0799.58030

[18] Mather J. N., Existence of Quasi-Periodic Orbits for Twist Homeomorphisms on the Annulus, Topology 21 (1982) 457-467. | MR 670747 | Zbl 0506.58032

[19] Mather J. N., Minimal Measures, Comment. Math. Helv. 64 (1989) 375-394. | MR 998855 | Zbl 0689.58025

[20] Moser J., Monotone Twist Mappings and the Calculus of Variations, Ergodic Theory Dynam. Systems 6 (1986) 401-413. | MR 863203 | Zbl 0619.49020

[21] Rubinstein J., Wolansky G., Eikonal Functions: Old and New, in: Givoli D., Grote M. J., Papanicolaou G. (Eds.), A Celebration of Mathematical Modeling: the Joseph B. Keller Anniversary Volume, Kluwer, 2004. | MR 2160990

[22] Siburg K. F., The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Mathematics, vol. 1844, Springer, 2004. | MR 2076302 | Zbl 1060.37048

[23] Villani C., Topics in Optimal Transportation, Graduate Studies in Math., vol. 58, Amer. Math. Soc., 2003. | MR 1964483 | Zbl 1106.90001

[24] Wolansky G., Optimal Transportation in the Presence of a Prescribed Pressure Field, Preprint, arXiv: math-ph/0306070 v5.

[25] G. Wolansky, On time reversible description of the process of coagulation and fragmentation, Arch. Rat. Mech., submitted for publication. | Zbl 1169.76052

[26] Markowich P., Rein G., Wolansky G., Existence and Nonlinear Stability of Stationary States of the Schrödinger-Poisson System, J. Stat. Phys. 106 (2002) 1221-1239. | MR 1889607 | Zbl 1001.82107