Differential equations methods for the Monge-Kantorovich mass transfer problem

We demonstrate that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measurein the limit as p → ∞.The idea is to show u p → u, where u satisfiesfor some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du, f + and f -.