On the Monge mass transfer problem

The Monge mass transfer problem, as proposed by Monge in 1781, is to move points from one mass distribution to another so that a cost functional is minimized among all measure preserving maps. The existence of an optimal mapping was proved by Sudakov in 1979, using probability theory. A proof based on partial differential equations was recently found by Evans and Gangbo. In this paper we give a more elementary and shorter proof by constructing an optimal mapping directly from the potential functions of Monge and Kantorovich.