Optimal Transportation in the presence of a prescribed pressure field

The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the Lagrangian L(v, x, t) from the set of orbits in Rn to a set of measure-valued orbits. The Eulerian, dual formulation leads an optimization problem on the set of sub-solutions of the corresponding Hamilton-Jacobi equation. Finally, the Monge problem and its Kantorovich relaxation are obtained by reducing the optimization problem to the set of measure preserving mappings and two point distribution measures subjected to an appropriately defined cost function. In this paper we concentrate on mechanical Lagrangians L = |v|2/2+P (x, t) leading, in general, to a non-homogeneous cost function. The main results yield existence of a unique flow of homomorphisms which transport the optimal measure valued orbit of the extended Lagrangian, as well as the existence of an optimal solution to the dual Euler problem and its relation to the Mongeand Kantorovich formulations.