New identities for Weak KAM theory

Abstract

This paper records for the Hamiltonian H = 1/2 |p|² + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.

Keywords

Weak KAM theory / Effective Hamiltonian / Hamiltonian dynamics

References

[1] Bernardi O., Cardin F., Guzzo M.. New estimates for Evans’ variational approach to weak KAM theory. Comm. in Contemporary Math., 2013, 15: 1250055

[2] Evans L. C.. Some new PDE methods for weak KAM theory. Calculus of Variations and Partial Differential Equations, 2003, 17: 159-177

[3] Evans L. C.. Further PDE methods for weak KAM theory. Calculus of Variations and Partial Differential Equations, 2009, 35: 435-462

[4] Evans L. C., Gomes D.. Effective Hamiltonians and averaging for Hamiltonian dynamics I. Archive Rational Mech. and Analysis, 2001, 157: 1-33

[5] Fathi A.. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sr. I Math., 1997, 324: 1043-1046

[6] Fathi, A., Weak KAM theorem in Lagrangian dynamics, Cambridge Studies in Advanced Mathematics, to be published.

[7] Gomes D., Iturriaga R., Sanchez-Morgado H., Yu Y.. Mather measures selected by an approximation scheme. Proc. Amer. Math. Soc., 2010, 138: 3591-3601

[8] Gomes D., Sanchez-Morgado H.. A stochastic Evans-Aronsson problem. Trans. Amer. Math. Soc., 2014, 366: 903-929

[9] Lions P.-L., Papanicolaou G., Varadhan S. R. S.. Homogenization of Hamilton–Jacobi equation. Comm. Pure Appl. Math., 1987, 56: 1501-1524

[10] Mather J.. Minimal measures. Comment. Math. Helvetici, 1989, 64: 375-394

[11] Mather J.. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Zeitschrift, 1991, 207: 169-207

[12] Yu Y.. L∞ variational problems and weak KAM theory. Comm. Pure Appl. Math., 2007, 60: 1111-1147