Weak KAM pairs and Monge-Kantorovich duality. (English) Zbl 1161.37043
Kozono, Hideo (ed.) et al., Asymptotic analysis and singularities. Elliptic and parabolic PDEs and related problems. Papers of the 14th International Research Institute of the Mathematical Society of Japan (MSJ), Sendai, Japan, July, 18–27, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-41-9/hbk). Advanced Studies in Pure Mathematics 47-2, 397-420 (2007).
The dynamics of globally minimizing orbits of Lagrangian systems is often studied using the (Peierls) barrier function, which first appeared in a work by Mather, or using the pairs of weak KAM solutions, introduced by Fathi. The present paper presents several relations between these two theories.
The main result of this paper states that the conjugate pairs of weak KAM solutions are precisely the admissible pairs for the Kantorovich problem when the cost function is the Peierls barrier.
In an earlier work [P. Bernard, B. Buffoni, J. Eur. Math. Soc. (JEMS) 9, No. 1, 85–121 (2007; Zbl 1241.49025)], the authors have already applied Fathi’s weak KAM theory to the optimal transportation problem.
For the entire collection see [Zbl 1130.35004].
The main result of this paper states that the conjugate pairs of weak KAM solutions are precisely the admissible pairs for the Kantorovich problem when the cost function is the Peierls barrier.
In an earlier work [P. Bernard, B. Buffoni, J. Eur. Math. Soc. (JEMS) 9, No. 1, 85–121 (2007; Zbl 1241.49025)], the authors have already applied Fathi’s weak KAM theory to the optimal transportation problem.
For the entire collection see [Zbl 1130.35004].
Reviewer: Maria Saprykina (Stockholm)
MSC:
| 37J50 | Action-minimizing orbits and measures (MSC2010) |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 47H20 | Semigroups of nonlinear operators |
| 70H03 | Lagrange’s equations |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
