## Relatório de pesquisa 28/09

Minimizing Orbits in the Discrete Aubry-Mather Model, Eduardo Garibaldi and Philippe Thieullen, submitted September 24, 2009.

Abstract

We consider a generalization of the Frenkel-Kontorova model in
higher dimension. We give a wider applicability to Aubry’s theory
by studying models with vector-valued states over a one dimensional
chain. This theory has a lot of similarities with Mather’s twist
approach over a multidimensional torus. Weakening the standard
hypotheses used in one dimensional, we investigate properties (like
boundness of jumps and definability of a rotation vector) of a special
class of strong ground states: the calibrated configurations.

The main mathematical tool is to cast the study 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 in the same spirit of
Hamilton-Jacobi methods. With reduced hypotheses, we reproduce in this
discrete setting some classical results of the Lagrangian Aubry-Mather
theory. In particular, we obtain a graph property for the Aubry set,
representation formulas for calibrated sub-actions and the existence of
separating sub-actions.

Mathematics Subject Classifications
(2000):

Keywords:

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September 24, 2009

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