Mathematical Sciences: The Monge Problem and the Calculus of Variations

1781年G. Monge提出了一个在经济学和工程学中很常见的问题：给定X和Y两个位置的两个相等的质量，找到将质量从第一个位置移动到第二个位置的“最佳策略”，其中最优性是根据成本函数来衡量的。Monge推测存在一个最佳策略，即问题允许一个最小化，并且存在一个标量势函数u，使得质量沿-Du(x)方向从x中的x传递到y中的y。大约两百年来，蒙日猜想没有得到严格的证明。阿佩尔给出了势函数u存在的形式化证明，他将u作为蒙日问题的拉格朗日乘子引入。Kantorovich通过引入一个我们称之为Monge-Kantorovich问题的问题使Appell的证明变得严格。Monge- kantorovich问题，是Monge问题的一个松弛，是基于对偶论证的。在与L.C. Evans正在进行的一项工作中，我们研究了原始的Monge问题，并期望很快就能完全解决它。最近，在与R. McCann的联合工作中，我们证明了给定一个一般代价函数c(X -y)，它要么是距离的严格凸函数，要么是距离的严格凹函数，Monge-Kantorovich问题承认一个唯一的最优解，它是一个映射，比如从X到y的T。除此之外，我想研究一般代价函数的最优映射T的平滑性。已知当代价函数为欧氏距离的平方时，将理想流体的欧拉方程离散得到Monge-Kantorovich问题。我希望通过应用这些技术，给出欧拉方程在任何维度空间中的存在性结果。为了说明质量运输问题的重要性及其解决方案的规律性，我们考虑两个例子，第一个与环境有关。这里，我们考虑的流体是湖中运动的水，我们假设我们知道初始时刻水的状态，比如说0。我们想在以后的时间里预测湖泊的状态，比如说，知道什么进化法则支配着水。水的后期状态将取决于风等因素。水由从一个地方移动到另一个地方的粒子组成，自然会在这个过程中花费最少的能量。我们说粒子的运动是以最优的方式进行的。当在最后时刻10难以确定需要最少功的运动时，人们通常通过研究连续时刻（1、2等）的水的状态来离散问题。10. 这种离散化对应于Monge-Kantorovich问题。离散问题解的平滑性将保证初始状态测量的微小误差不会对我们的预测产生太大影响。我们用来说明最优策略平滑性重要性的第二个例子是在经济学中。假设我们确定了在供应商和客户之间的网络中运输材料的最便宜的方式。其中一个问题是要知道，如果取消一个供应商和一个客户，我们是否会被迫对我们的运输战略进行重大修改。如果知道最优策略以一种平滑的方式依赖于数据，那么所需的修改将是轻微的。

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.