Optimal mass transportation and Mather theory

Patrick Bernard (1, 2) , Boris Buffoni (3)

1 IF - Institut Fourier
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 EPFL - Ecole Polytechnique Fédérale de Lausanne

We study optimal transportation of measures on compact manifolds for costs defined from convex Lagrangians. We prove that optimal transportation can be interpolated by measured Lipschitz laminations, or geometric currents. The methods are inspired from Mather theory on Lagrangian systems. We make use of viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of Fathi's approach to Mather theory.

Mots clés: Lagrangian systems; Mather theory; Fathi's Weak KAM theory; Hamilton-Jacobi equations.; Hamilton-Jacobi equations; Otimal transportation on manifolds

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Patrick Bernard, Boris Buffoni. Optimal mass transportation and Mather theory. Journal of the European Mathematical Society, 2007, 9, pp.85-121. ⟨10.4171/JEMS/74⟩. ⟨hal-00003587v3⟩