Differential equations methods for the Monge-Kantorovich mass transfer problem

We demonstrate under some assumptions on f(+), f(-) that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure mu(+) = f(+)dx onto mu(-) = f(-)dy call be constructed by studying the p-Laplacian equation -div(/Du(p)/(p-2)Du(p)) = f(+) - f(-) the limit as p --> infinity. The idea is to show u(p) --> u, where u satisfies /Du/ less than or equal to 1, -div(aDu) = f(+) - f(-) for some density a greater than or equal to 0, and then to build a Row by solving a nonautonomous ODE involving a, Du. f(+) and f(-).