Minimizing orbits in the discrete Aubry–Mather model

Author： Garibaldi Eduardo Thieullen Philippe

ISSN： 0951-7715

Source： Nonlinearity, Vol.24, Iss.2, 2011-02, pp. : 563-611

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Abstract

We consider a generalization of the Frenkel–Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one-dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual 'total ordering' of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study of the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax–Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton–Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as 'ground states' obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the noncrossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.