Viscosity solution methods and the discrete Aubry-Mather problem

Diogo A. Gomes

doi:10.3934/dcds.2005.13.103

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2005, Volume 13, Issue 1: 103-116. Doi: 10.3934/dcds.2005.13.103

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Viscosity solution methods and the discrete Aubry-Mather problem

Diogo A. Gomes^1,

1.
Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa

Received: March 31, 2004

Revised: October 31, 2004

Published: March 2005

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Abstract

In this paper we study a discrete multi-dimensional version of Aubry-Mather theory using mostly tools from the theory of viscosity solutions. We set this problem as an infinite dimensional linear programming problem. The dual problem turns out to be a discrete analog of the Hamilton-Jacobi equations. We present some applications to discretizations of Lagrangian systems.

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Mathematics Subject Classification: 49L25, 37J40, 37J50, 70H20.

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