In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction. 1

We prove the existence of optimal transport maps for the Monge problem when the cost is a Finsler distance on a compact manifold. Our point of view consists in considering the distance as a Mañé potential, and to rely on recent developments in the theory of viscosity solutions of the Hamilton-Jacobi equation. Résumé: On montre l’existence d’une application de transport optimale pour le problème de Monge lorsque le cout est une distance Finslerienne sur une variété compacte. Le nouveau point de vue consiste à considérer la distance comme un potentiel de Mañé, et à exploiter des développements récents sur les solutions de viscostité de l’équation de Hamilton-Jacobi.

We study Brenier’s variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps. 1

Abstract. In recent works L.C. Evans has noticed a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the theory developed in the last years for the optimal mass transportation (or Monge-Kantorovich) problem. In this paper we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Some minimality properties of such curves are also addressed. Keywords. Mather’s minimal measures, Monge-Kantorovich problem, optimal transport problems, normal 1-currents. MSC 2000. 37J50, 49Q20, 49Q15. 1.

Abstract. We prove the existence of C 1 critical subsolutions of the Hamilton-Jacobi equation for a time-periodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system. 1.

Abstract. We study Monge’s optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by d 2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane. 1.

The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way. Let M be a compact connected manifold and consider a C 2 Lagrangian function L: TM × R → R that satisfies the standard hypotheses of the calculus of variations,

The weak KAM theory predicts the survivals of invariant measures of Hamiltonian systems under large perturbations. It is the subject of an extensive research in the last few decades. The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Recently, some interesting links where discovered between these two fields. Here we investigate a new, surprising link involving the metric Monge distance. As a special case we get for any pair of no-negative measures λ +, λ − of equal mass a generalization of the identity W1(λ − , λ +) = lim ε→0 ε −2 inf µ W2(µ + ελ − , µ + ελ +) where Wp is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.

Abstract. In this manuscript we extend De Giorgi’s interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite “entropy”, we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi’s interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in L 1 of our solutions. 1.

In this paper we prove the existence of an optimal transport map on non-compact manifolds for a large class of cost functions that includes the case c(x, y) = d(x, y), under the only hypothesis that the source measure is absolutely continuous with respect to the volume measure. In particular, we assume compactness neither of the support of the source measure nor of that of the target measure. 1