Homogenization of fully overdamped Frenkel-Kontorova models

Journal of Differential Equations

Volumn 246, Issue 3, 2009, Pages 1057-1097

  Forcadel, N ^a,b,c   Imbert, C ^d   Monneau, R ^a  

^a UNIVERSITÉ PARIS EST   (France)

^b ECOLE POLYTECHNIQUE   (France)

^c ENSTA PARISTECH   (France)

^d UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

Cumulative distribution function; Frenkel Kontorova models; Hamilton Jacobi equations; Hull function; Particle systems; Periodic homogenization; Slep ev formulation

Indexed keywords

EID: 55949110481     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.06.034     Document Type: Article

References (19)

1. Aubry S. The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Phys. D 7 (1983) 240-258
2. Aubry S. Exact models with a complete Devil's staircase. J. Phys. C 16 (1983) 2497-2508
3. Aubry S., and Le Daeron P.Y. The discrete Frenkel-Kontorova model and its extensions. Phys. D 8 (1983) 381-422
4. Bacaër N. Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations. M2AN Math. Model. Numer. Anal. 35 (2001) 1185-1195
5. Barles G. Solutions de viscosité des équations de Hamilton-Jacobi. Math. Appl. (Berlin) vol. 17 (1994), Springer-Verlag, Paris
6. Barles G. Some homogenization results for non-coercive Hamilton-Jacobi equations. Calc. Var. Partial Differential Equations 30 4 (2007) 449-466
7. Boccardo L., and Murat F. Remarques sur l'homogénéisation de certains problèmes quasi-linéaires. Port. Math. 41 1-4 (1984) 535-562
8. Braun O.M., and Kivshar Y.S. The Frenkel-Kontorova Model, Concepts, Methods and Applications (2004), Springer-Verlag
9. Crandall M.G., Ishii H., and Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1 (1992) 1-67
10. R. De La Llave, KAM theory for equilibrium states in 1-D statistical mechanics models, preprint 2005-310, University of Texas at Austin, 2005
11. Evans L.C. The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 3-4 (1989) 359-375
12. N. Forcadel, C. Imbert, R. Monneau, Homogenization of some systems of particles with two-body interactions and dislocation dynamics, Discrete Contin. Dyn. Syst. Ser. A, in press
13. Floria L.M., and Mazo J.J. Dissipative dynamics of the Frenkel-Kontorova model. Adv. in Phys. 45 6 (1996) 505-598
14. Hu B., Qin W.-X., and Zheng Z. Rotation number of the overdamped Frenkel-Kontorova model with ac-driving. Phys. D 208 (2005) 172-190
15. Imbert C., and Monneau R. Homogenization of first order equations with (u / ε)-periodic Hamiltonians. Part I: Local equations. Arch. Ration. Mech. Anal. 187 1 (2008) 49-89
16. H.-C. Kao, S.-C. Lee, W.-J. Tzeng, Exact solution of Frenkel-Kontorova models with a complete devil's staircase in higher dimensions, preprint
17. Katok A., and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems (1995), Cambridge Univ. Press, Cambridge
18. Kontorova T., and Frenkel Y.I. On the theory of plastic deformation and doubling. Zh. Eksp. Teor. Fiz. 8 (1938) 89-, 1340-, 1349- (in Russian)
19. P.-L. Lions, G.C. Papanicolaou, S.R.S. Varadhan, Homogeneization of Hamilton-Jacobi equations, unpublished preprint, 1986