Biryuk, Andrey; Gomes, Diogo A. An introduction to the Aubry-Mather theory. (English) Zbl 1243.49001 São Paulo J. Math. Sci. 4, No. 1, 17-63 (2010). 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. Cited in 12 Documents 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 Keywords:Aubry-Mather theory; viscosity solutions; Hamilton-Jacobi equations Citations:Zbl 0886.58037; Zbl 0986.37056 PDFBibTeX XMLCite \textit{A. Biryuk} and \textit{D. A. Gomes}, São Paulo J. Math. Sci. 4, No. 1, 17--63 (2010; Zbl 1243.49001) Full Text: DOI