Applications of Monge-Kantorovich Theory and Michell Trusses

建议：DMS-9970520首席研究员威尔弗里德·冈博摘要：1904年，在他关于框架结构中经济极限的杰出论文中，米歇尔解决了以下难题：给定一个具有特定作用点的力(载荷和反力)的平衡系统，并给出其单轴应力的允许范围的弹性材料，在平面上设计具有给定材料的杆的桁架，使得给定力产生的轴向应力在允许的限度内，同时杆所用的材料的总体积尽可能小。在过去的几年里，米歇尔的问题及其许多变体和扩展引起了工程师和数学家的极大关注。在共面的情况下，米歇尔正式证明了，在最优布局中，杆件沿着作为拉格朗日乘子产生的位移的主应变的线排列。这些线在平面上形成了一个正交的坐标系。任何满足这些条件的最优设计称为米歇尔桁架。钢博计划研究最佳应力支承的几何和拓扑性质。研究这个问题的方法类似于应用于Monge-Kantorovich问题的方法。Gangbo还计划利用Monge-Kantorovich理论，该理论最近被公认为建设性地研究半地转浅水问题的有力工具，这是一个三维自由边界问题。米歇尔桁架问题最初的动机是设计一种由能够支撑特定力阵列的梁组成的结构，并通过使用最少的给定弹性材料来最大限度地节约成本。当然，在尽可能提高效率的同时，人们还必须设计出确保结构安全的设计。钢博感兴趣的是为这种体积最小的框架设计确定好的模型。确定框架杆件的终点是这一问题中最困难的部分。该提案的第二部分在于研究浅水流动，即垂直长度尺度与水平长度尺度(例如洋流流动)相比较小的流动。如果一个人在给定的时间收集数据，人们希望预测这些数据将如何随着时间的推移而演变。例如，如果测量今天的水的压力，就会预测明天的水的压力。在地转模式中，气压是一个需要知道的重要量。例如，人们可以从水的压力来推断水的速度。Gangbo将使用所谓的Monge-Kantorovich理论，该理论最近作为一个强大的工具出现在数学的各个领域。

Proposal: DMS-9970520Principal Investigator: Wilfrid GangboAbstract: In 1904, in his remarkable paper on the limits of economy in framed structures, Michell attacked the following difficult problem: given an equilibrium system of forces (loads and reactions) with specified points of application, and given an elastic material with allowable range for its uniaxial stress, to design a truss in the plane with bars of the given material such that the given forces produce axial stresses within the allowable limits, while the total volume of material used for the bars is as small as possible. Michell's problem and its many variants and extensions have attracted a lot of attention in the last few years among engineers as well as mathematicians. In the coplanar case, Michell formally proved that, in the optimal layout, bars are arranged along the lines of principal strains of a displacement that arises as a Lagrange multiplier. These lines form an orthogonal system of coordinates in the plane. Any optimal design satisfying these conditions is called a Michell truss. Gangbo plans to study geometrical and topological properties of supports of optimal stresses. Techniques for studying this problem are similar to those that apply to the Monge-Kantorovich problem. Gangbo also plans to make use of the Monge-Kantorovich theory, which has recently been recognized as a powerful tool for constructively studying the semigeostrophic shallow water problem, which is a three-dimensional free boundary problem. The Michell truss problem was originally motivated by the desire to engineer a structure built of beams capable of supporting a specified array of forces, and to do this with maximum economy by using the least quantity of a given elastic material. Of course, while trying to be as efficient as possible, one must still produce a design that ensures a safe structure. Gangbo's interest is to determine good models for the design of such least-volume frames. The determination of the endpoints of the bars of the frame is the most difficult part of this problem. The second part of the proposal consists in studying shallow-water flow, meaning flow in which the vertical length scale is small in comparison to the horizontal length scale (e.g., the flow of ocean currents). If one collects data at a given time, one hopes to predict how that data will evolve over time. For instance, if one measures the pressure of the water today, one would like to forecast its pressure tomorrow. In the geostrophic model, pressure is an important quantity to know. For instance, one can infer the speed of the water from its pressure. Gangbo will use the so-called Monge-Kantorovich theory, which has recently surfaced as a powerful tool in various fields of mathematics.