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An introduction to the Aubry-Mather theory. (English) Zbl 1243.49001

Summary: This paper is a self-contained introduction to the Aubry-Mather theory and its connections with the theory of viscosity solutions of Hamilton-Jacobi equations. Our starting point is R. Mañé’s variational approach using holonomic measures [Nonlinearity 9, No. 2, 273–310 (1996; Zbl 0886.58037)]. We present the Legendre-Fenchel-Rockafellar theorem from convex analysis and discuss the basic theory of viscosity solutions of first-order Hamilton-Jacobi equations. We apply these tools to study the Aubry-Mather problem following the ideas of L. C. Evans and D. Gomes [Arch. Ration. Mech. Anal. 157, 1–33 (2001; Zbl 0986.37056). Finally, in the last section, we present a new proof of the invariance under the Euler-Lagrange flow of the Mather measures using ideas from calculus of variations.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37J50 Action-minimizing orbits and measures (MSC2010)
49K10 Optimality conditions for free problems in two or more independent variables
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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