Optimal Transportation in the presence of a prescribed pressure field

Abstract

The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the {\em Lagrangian} $L(v,x,t)$ from the set of orbits in $\R^n$ to a set of measure-valued orbits. The {\em 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 {\em 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 Monge- and Kantorovich formulations.