Abstract Gangbo 9622734 In 1781 G. Monge formulated a question which occurs naturally in economics and engineering: Given two equal masses at two locations X and Y, find the "best strategy" to move the mass from the first location to second one, where optimality is measured against a cost function. Monge conjectured that there exists a best strategy, i.e. the problem admits a minimizer and that there exists a scalar potential function u such that mass is transported from x in X to y in Y along the direction -Du(x). For about two hundred years, no rigorous proof of Monge's conjecture was given. Appell presented a formal proof of the existence of the potential function u, where he introduced u as a Lagrange multiplier of Monge's problem. Kantorovich made Appell's proof rigorous by introducing a problem we call the Monge-Kantorovich problem. The Monge-Kantorovich problem, which is a relaxation of the Monge problem, is based on a duality argument. In a work in progress with L.C. Evans, we study the original Monge problem and have great expectation that soon we will completely solve it. Recently, in a joint work with R. McCann, we proved that given a general cost function c(x-y) which is either strictly convex or astrictly concave function of the distance, the Monge-Kantorovich problem admits a unique optimal solution which is a map, say T from X to Y. Among other things I would like to study the smoothness properties of the optimal map T for general cost functions. It is known that when the cost function is the square of the euclidian distance, the Monge-Kantorovich problem is obtained by discretizing the Euler equation of an ideal fluid. I expect to give existence results for the Euler equation in any dimensional space by applying these techniques. To illustrate the importance of the mass transport problem and the regularity of its solutions we consider two examples, the first one being related to environment. Here, the fluid we consider is water in motion in a lake, and we assume that we know the state of the water at an initial time, say 0. We want to predict the state of the lake at a later time, say, 10 knowing what evolution laws govern the water. The later state of the water will depend on factors like the wind. The water consists of particles which will move from one location to another and will naturally spend the least energy for this process. We say that the motion of particles is made in an optimal way. When the motion involving the least work is hard to determine at the final time 10, one usually discretizes the problem by studying the state of the water at successive times, 1, 2, etc... 10. This discretization corresponds to the Monge-Kantorovich problem. The smoothness of the solutions of the discrete problem will guarantee that a slight error of measurement of the initial state will not affect our prediction much. A second example we use to illustrate the importance of smothness of optimal strategies is in economics. Assume that we determine the cheapest way of transporting materials in a network between suppliers and customers. One of the issues is to know if by removing one supplier and one customer we will be forced to make a significant revision in our strategy of tranportation. If the know that the optimal strategy depends on the data in a smooth way then the revision needed would be slight.
