Weak KAM Theory: The connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation

Proceeding of the International Congress of Mathematicans, ICM 2014

Volumn 3, Issue , 2014, Pages 597-621

  Fathi, Albert ^a  

^a NONE   (France)

Author keywords

Aubry Mather; Hamilton Jacobi; Hamiltonian; Lagrangian; Weak KAM

Indexed keywords

MECHANICS;

HAMILTON - JACOBI EQUATIONS; HIGHER DIMENSIONS; KAM THEORY; LAGRANGIAN SYSTEM; VISCOSITY SOLUTIONS;

VISCOSITY;

EID: 85078511210     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (24)

1. Arnold, V.I., Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian (Russian), Uspehi Mat. Nauk 18 (1963), 13-40.
2. Aubry, S., The devil's staircase transformation in incommensurate lattices, in The Riemann problem, complete integrability and arithmetic applications, Bures-surYvette/New York, 1979/1980. Lecture Notes in Math. 925, Springer, Berlin-New York, 1982. 221-245.
3. Bardi, M. and Cappuzzo-Dolceta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations and Applications, Birkhäuser Boston Inc., Boston, MA, 1997.
4. Barles, G., Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications 17, Springer-Verlag, Paris, 1994.
5. Buttazzo, G., Giaquinta, M., and Hildebrandt, S., One-dimensional variational problems, Oxford Lecture Series in Mathematics and its Applications 15, Oxford Univ. Press, New York, 1998.
6. Cannarsa, P. and Sinestrari, C., Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications 58, Birkhäuser, Boston, 2004.
7. Clarke, F.H., Methods of dynamic and nonsmooth optimization, CBMS-NSF Reg. Conf. Series in App. Math. 57, SIAM, Philadelphia, PA, 1989.
8. Crandall, M.G. and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277(1983), 1-42.
9. Dias Carneiro, M.J., On minimizing measures of the action of autonomous Lagrangians, Nonlinearity 8 (1995), 1077-1085.
10. Dugundji, J., Topology, Allyn and Bacon Inc., Boston, 1970.
11. E, Weinan, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math. 52 (1999), 811-828.
12. Evans, L. C. and Gomes, D. Effective Hamiltonians and averaging for Hamiltonian dynamics. I-II, Arch. Ration. Mech. Anal. 157 (2001), 1-33 & 161 (2002), 271-305.
13. Fathi, A., Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math. 324(9) (1997), 1043-1046.
14. Fathi, A., Weak KAM Theorem and Lagrangian Dynamics, Preprint, http://tinyurl.com/ AFnotesWeakKam, http://tinyurl.com/oh4qjv3.
15. Fathi, A. and A. Figalli, Optimal transportation on non-compact manifolds, Israel J. Math. 175 (2010), 1-59.
16. Fleming, W.H., The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations 5 (1969), 515-530.
17. Kolmogorov, A.N., Théorie générale des systèmes dynamiques et mécanique classique, in Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1 315-333. Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co 1957.
18. Lions, P.-L., Papanicolau, G., and Varadhan, S.R.S., Homogenization of Hamilton-Jacobi equation, Unpublished preprint, (1987).
19. Mañé, R., Lagrangian flows: the dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), 141-153.
20. Mather, J.N., Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 457-467.
21. Mather, J.N., Action minimizing measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169-207.
22. Mather, J.N., Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993), 1349-1386.
23. Moser, J., A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1824-1831.
24. Poincaré, H., Les Méthodes nouvelles de la Mécanique Céleste, 3 tomes, Paris, Grauthier-Viltars, 1892-1899.