An introduction to the Aubry-Mather theory

Resumo

This paper is a self-contained introduction to the Aubry-Mather theory and its connections with the theory of viscosity solutionsof Hamilton-Jacobi equations. Our starting point is Ma~ne's variationalapproach using holonomic measures [Mn96]. We present the Legendre-Fenchel-Rockafellar theorem from convex analysis and discuss the basictheory of viscosity solutions of rst order Hamilton-Jacobi equations.We apply these tools to study the Aubry-Mather problem following theideas in [EG01]. Finally, in the last section, we present a new proof ofthe invariance under the Euler-Lagrange ow of the Mather measuresusing ideas from calculus of variations.