Free access
Issue
ESAIM: M2AN
Volume 42, Number 6, November-December 2008
Page(s) 1047 - 1064
DOI http://dx.doi.org/10.1051/m2an:2008035
Published online 25 September 2008
  1. V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical aspects of classical and celestial mechanics. Springer-Verlag, Berlin (1997). Translated from the 1985 Russian original by A. Iacob, reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems III, Encyclopaedia Math. Sci. 3, Springer, Berlin (1993) MR 95d:58043a].
  2. W.E. Aubry, Mather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math. 52 (1999) 811–828. [CrossRef] [MathSciNet]
  3. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997).
  4. U. Bessi, An analytic counterexample to the KAM theorem. Ergod. Theory Dyn. Syst. 20 (2000) 317–333. [CrossRef]
  5. A. Biryuk and D. Gomes, An introduction to the Aubry-Mather theory. São Paulo Journal of Mathematical Sciences (to appear).
  6. G. Contreras, R. Iturriaga, G.P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values. Geom. Funct. Anal. 8 (1998) 788–809. [CrossRef] [MathSciNet]
  7. L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI, USA (1998).
  8. L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal. 157 (2001) 1–33. [CrossRef] [MathSciNet]
  9. L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch. Ration. Mech. Anal. 161 (2002) 271–305. [CrossRef] [MathSciNet]
  10. A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649–652.
  11. A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043–1046.
  12. A. Fathi, Orbite hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1213–1216.
  13. A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 267–270.
  14. A. Fathi and A. Siconolfi, Existence of Formula critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004) 363–388. [CrossRef] [MathSciNet]
  15. W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (1993).
  16. G. Forni, Analytic destruction of invariant circles. Ergod. Theory Dyn. Syst. 14 (1994) 267–298.
  17. G. Forni, Construction of invariant measures supported within the gaps of Aubry-Mather sets. Ergod. Theory Dyn. Syst. 16 (1996) 51–86.
  18. H. Goldstein, Classical mechanics. Addison-Wesley Publishing Co., Reading, Mass., second edition (1980).
  19. D.A. Gomes, Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems. Calc. Var. Partial Differential Equations 14 (2002) 345–357. [CrossRef] [MathSciNet]
  20. D.A. Gomes, Perturbation theory for viscosity solutions of Hamilton-Jacobi equations and stability of Aubry-Mather sets. SIAM J. Math. Anal. 35 (2003) 135–147 (electronic). [CrossRef] [MathSciNet]
  21. D.A. Gomes, Duality principles for fully nonlinear elliptic equations, in Trends in partial differential equations of mathematical physics, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser, Basel (2005) 125–136.
  22. D.A. Gomes and A.M. Oberman, Computing the effective Hamiltonian using a variational approach. SIAM J. Contr. Opt. 43 (2004) 792–812 (electronic). [CrossRef] [MathSciNet]
  23. D.A. Gomes and E. Valdinoci, Lack of integrability via viscosity solution methods. Indiana Univ. Math. J. 53 (2004) 1055–1071. [CrossRef] [MathSciNet]
  24. À. Haro, Converse KAM theory for monotone positive symplectomorphisms. Nonlinearity 12 (1999) 1299–1322. [CrossRef] [MathSciNet]
  25. A. Knauf, Closed orbits and converse KAM theory. Nonlinearity 3 (1990) 961–973. [CrossRef] [MathSciNet]
  26. P.L. Lions and P. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Math. Appl. 56 (2003) 1501–1524. [CrossRef] [MathSciNet]
  27. P.L. Lions, G. Papanicolao and S.R.S. Varadhan, Homogeneization of Hamilton-Jacobi equations. Preliminary version (1988).
  28. R.S. MacKay, Converse KAM theory, in Singular behavior and nonlinear dynamics, Vol. 1 (Sámos, 1988), World Sci. Publishing, Teaneck, USA (1989) 109–113.
  29. R.S. MacKay and I.C. Percival, Converse KAM: theory and practice. Comm. Math. Phys. 98 (1985) 469–512. [CrossRef] [MathSciNet]
  30. R.S. MacKay, J.D. Meiss and J. Stark, Converse KAM theory for symplectic twist maps. Nonlinearity 2 (1989) 555–570. [CrossRef] [MathSciNet]
  31. R. Mañé, On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5 (1992) 623–638. [CrossRef] [MathSciNet]
  32. R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996) 273–310. [CrossRef] [MathSciNet]
  33. J.N. Mather, Minimal action measures for positive-definite Lagrangian systems, in IXth International Congress on Mathematical Physics (Swansea, 1988), Hilger, Bristol (1989) 466–468.
  34. J.N. Mather, Minimal measures. Comment. Math. Helv. 64 (1989) 375–394. [CrossRef] [MathSciNet]
  35. J.N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169–207. [CrossRef] [MathSciNet]
  36. J. Qian, Two approximations for effective hamiltonians arising from homogenization of Hamilton-Jacobi equations. Preprint (2003).

Recommended for you