The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry–Mather theory and spaces of Delone sets.
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Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in mathematics ETH Zürich. Birkhäuser, Basel (2005)
Aubry, S., Le Daeron, P.Y.: The discrete Frenkel–Kontorova model and its extensions: I. Exact results for the ground states. Phys. D 8, 381–422 (1983)
Auslander, L., Hahn, F.: Real functions coming from flows on compact spaces and concepts of almost periodicity. Trans. Am. Math. Soc. 106, 415–426 (1963)
Bellissard, J., Benedetti, R., Gambaudo, J.M.: Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261, 1–41 (2006)
Davini, A., Siconolfi, A.: Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case. Math. Ann. 345, 749–782 (2009)
Davini, A., Siconolfi, A.: Metric techniques for convex stationary ergodic Hamiltonians. Calc. Var. Partial Differ. Equ. 40, 391–421 (2011)
Davini, A., Siconolfi, A.: Weak KAM theory topics in the stationary ergodic setting. Calc. Var. Partial Differ. Equ. 44, 319–350 (2012)
de la Llave, R., Su, X.: KAM theory for quasi-periodic equilibria in one-dimensional quasi-periodic media. SIAM J. Math. Anal. 44, 3901–3927 (2012)
Fathi, A.: Solutions KAM faibles conjuguées et barrières de Peierls. Comptes Rendus des Séances de l’Académie des Sciences, Série I, Mathématique 325, 649–652 (1997)
Fathi, A.: The weak KAM theorem in Lagrangian dynamics. Cambridge University Press (in press)
Frenkel, Ya I.: On the theory of plastic deformation and twinning I. Zh. Eksp. Teor. Fiz. 8, 89–95 (1938)
Frenkel, Ya I., Kontorova, T.A.: On the theory of plastic deformation and twinning II. Zh. Eksp. Teor. Fiz. 8, 1340–1349 (1938)
Frenkel, Ya I., Kontorova, T.A.: On the theory of plastic deformation and twinning III. Zh. Eksp. Teor. Fiz. 8, 1349–1359 (1938)
Gambaudo, J.M., Guiraud, P., Petite, S.: Minimal configurations for the Frenkel–Kontorova model on a quasicrystal. Commun. Math. Phys. 265, 165–188 (2006)
Garibaldi, E., Thieullen, Ph: Minimizing orbits in the discrete Aubry–Mather model. Nonlinearity 24, 563–611 (2011)
Gomes, D.A.: Viscosity solution methods and the discrete Aunbry–Mather problem. Discrete Contin. Dyn. Syst. Ser. A 13, 103–116 (2005)
Gomes, D.A.: Generalized Mather problem and selection principles for viscosity solutions and Mather measures. Adv. Calc. Var. 1, 291–307 (2008)
Gomes, D.A., Oliveira, E.R.: Mather problem and viscosity solutions in the stationary setting. São Paulo J. Math. Sci. 6, 301–334 (2012)
Kellendonk, J.: Pattern-equivariant functions and cohomology. J. Phys. A Math. Gen. 36, 5765–5772 (2003)
Kellendonk, J., Putnam, I.F.: Tilings, \(C^*\)-algebras and \(K\)-theory. In: Baake, M., Moody, R.V. (eds.) Directions in Mathematical Quasicrystals, CRM Monograph Series 13, pp. 177–206. AMS, Providence (2000)
Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone sets and quasicrystals. Ergod. Theory Dyn. Syst. 23, 831–867 (2003)
Lions, P.L., Souganidis, P.E.: Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Commun. Pure Appl. Math. 56, 1501–1524 (2003)
Mañé, R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9, 273–310 (1996)
Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982)
Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)
Tuy, H.: Topological minimax theorems: old and new. Vietnam J. Math. 40, 391–405 (2012)
van Erp, T.S.: Frenkel–Kontorova model on quasi-periodic substrate potential. PhD thesis, Katholieke Universiteit Nijmegen (1999)
Villani, C.: Optimal transport: old and new. Grundlehren der mathematischen Wissenschaften 338. Springer, Berlin (2008)
Work is supported by FAPESP 2009/17075-8, Brazilian-French Network in Mathematics CAPES-COFECUB 661/10, MAth AmSud 38889TM—DCS and ANR WKBHJ “Weak KAM” ANR-12-BS01-0020.
Communicated by Dmitry Dolgopyat.
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Garibaldi, E., Petite, S. & Thieullen, P. Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments. Ann. Henri Poincaré 18, 2905–2943 (2017). https://doi.org/10.1007/s00023-017-0589-7