Mathematical Sciences: The Monge Problem and the Calculus of Variations

数学科学：蒙日问题和变分法

• 批准号: 9622734
• 负责人: Wilfrid Gangbo
• 金额: $ 8.92万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622734&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Monge; Problem; Calculus

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.

1781年Gangbo 9622734提出了一个在经济学和工程学中自然出现的问题：假设在X和Y两个位置有两个相等的质量，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猜想存在一个最优策略，即问题允许一个极小值，并且存在一个标量势函数u，使得质量沿Du(X)方向从X的x传输到Y的y。在大约两百年的时间里，蒙格的猜想没有得到严格的证明。Appell给出了势函数u存在的形式证明，其中他引入了u作为Monge问题的拉格朗日乘子。康托洛维奇引入了一个我们称之为蒙格-康托洛维奇问题的问题，从而使阿佩尔的证明变得严谨。Monge-Kantorovich问题是Monge问题的一种松弛形式，它基于二元性论证。在与L.C.埃文斯正在进行的一项工作中，我们研究了原始的Monge问题，并对很快完全解决它抱有很大的期望。最近，在与R.McCann的合作中，我们证明了给定一个一般代价函数c(x-y)是严格凸的或严格凹的距离函数，Monge-Kantorovich问题允许唯一的最优解是一个映射，例如T从X到Y。我想研究一般代价函数最优映射T的光滑性。众所周知，当代价函数是欧氏距离的平方时，Monge-Kantorovich问题是通过对理想流体的欧拉方程进行离散化得到的。通过应用这些技巧，我期望给出欧拉方程在任意维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，我们考虑了两个例子，第一个例子与环境有关。在这里，我们考虑的流体是湖中运动的水，我们假设我们知道水在初始时间的状态，比方说0。我们想要在以后的时间预测湖泊的状态，比如，知道什么进化规律支配着水。水的后期状态将取决于风等因素。水由粒子组成，这些粒子会从一个位置移动到另一个位置，自然会为这个过程花费最少的能量。我们说，粒子的运动是以最佳方式进行的。当涉及最少功的运动在最后时刻10很难确定时，人们通常通过研究连续时刻1、2等的水的状态来离散化问题。10.这种离散化对应于Monge-Kantorovich问题。离散问题解的光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光滑性重要性的第二个例子是在经济学中。假设我们确定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，如果移除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。如果知道最优策略是以平稳的方式依赖于数据，那么所需的修改将是轻微的。