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.

抽象的Gangbo 9622734在1781年G. Monge提出了一个自然发生在经济学和工程中的问题：在两个位置x和y处两个相等的质量X和Y，找到“最佳策略”，将质量从第一个位置转移到第二个位置，在第二个位置，该位置可以衡量针对成本功能的最优性。 Monge猜想存在最佳策略，即问题是最小化的，并且存在标量电位函数U，使得质量从X in x In x中的x转移到y，沿y沿y方向-du（x）。大约两百年来，没有任何严格的蒙格猜想证明。阿佩尔（Appell）向潜在功能U的存在提供了正式的证明，他将U作为Monge问题的Lagrange乘法器介绍。坎托维奇（Kantorovich）通过引入一个问题，我们称之为Monge-Kantorovich问题，从而使Appell的证明很严格。 Monge-Kantorovich问题是对Monge问题的放松，是基于双重性论点。在与L.C.进行的工作中埃文斯（Evans），我们研究了原始的蒙加问题，并寄予厚望，很快我们将完全解决它。最近，在与R. McCann的联合合作中，我们证明了一般成本函数C（X-Y）是严格凸出的距离或距离的划分局部函数，Monge-Kantorovich问题承认了一个独特的最佳解决方案，该解决方案是X到Y的唯一最佳解决方案，例如，从X到Y。除其他外，我还想研究一般成本范围的平滑性属性。众所周知，当成本函数是欧几里得距离的平方时，Monge-Kantorovich问题是通过离散理想流体的Euler方程来获得的。我希望通过应用这些技术在任何维空间中给出Euler方程的存在结果。 为了说明大众运输问题的重要性及其解决方案的规律性，我们考虑了两个示例，第一个示例与环境有关。在这里，我们认为的液体是在湖中运动的水，我们假设我们在初始时间（例如0）知道水的状态。我们想在以后的时间预测湖泊状态，例如，10知道哪种进化法律控制水。后来的水状态将取决于风等因素。水由颗粒组成，这些颗粒将从一个位置移到另一个位置，并且自然会花费最少的能量来进行此过程。我们说颗粒的运动是以最佳方式进行的。当在最后时间很难确定涉及最少工作的运动时，通常会通过在连续时间，1、2等研究水状态来分散问题。离散问题解决方案的平滑度将确保初始状态的测量略有误差不会对我们的预测产生很大的影响。我们用来说明最佳策略的重要性的第二个例子是经济学。假设我们确定供应商和客户之间网络中运输材料的最便宜方式。问题之一是要知道，通过删除一个供应商和一个客户，我们将被迫对我们的货币策略进行重大修订。如果知道最佳策略以平滑的方式取决于数据，那么所需的修订将很小。