TITRE: "Mouvements of measures for optimal transportation problems"

ABSTRACT: Starting from the work of Brenier where a dynamic formulation for 
optimal transportation problems was proposed, we investigate minimum problems 
of the form
$$\min\Big\{\int_01 \Psi(\sigma)\,dt\ :\ -{\rm div\,}\sigma=f\Big\}$$
where $\Psi$ is a l.s.c. functional defined on measures. As an application we 
study the mouvement of a measure $\rho_t$ which satisfies the continuity 
equation
$$\partial_t\rho+{\rm div}_x(\rho v)=0$$
and minimizes some suitable cost functional $F(\rho,v)$ assuming fixed values 
$\rho_{t=0}=\rho_0$ and $\rho_{t=1}=\rho_1$. Some numerical computations are 
also provided, following the scheme proposed by Benamou and Brenier.