Topics in Optimal Transport and Nonlinear Partial Differential Equations

运输问题涉及研究质量从一个地方到另一个地方的运动。最优运输是研究如何做到这一点，使运输成本尽可能小。这是一个非常广泛的应用领域，从城市交通网络的规划到分子运输。本项目考虑了经典最优运输问题的几个重要推广，这些推广以前没有被研究过。例如，要考虑的一个自然问题是，如果有两个或更多的实体将相同的初始质量运输到另一个位置，每个实体都有自己的成本最小化，会发生什么。这是最优运输问题的“博弈论”版本。另一个问题是与时间相关的运输成本。这考虑了移动对象的成本取决于移动时间的各种情况。加班费或交通拥挤时期的交通费就属于这一类。寻求最小化最坏可能成本（而不是某些平均成本）的传输问题在应用程序中也很重要，这里将予以考虑。由于最佳运输在许多经济、生物、工程和其他科学领域都有出现，因此为本项目所考虑的问题开发的数学理论可能会产生重大的实际影响。此外，该项目将研究在图像处理和理解由于火焰或等离子体传播而引起的锋面边界运动中出现的几个新的数学模型。最优运输问题的简化版涉及找到一种度量，称为“运输计划”，在运输计划给出边际的约束下，使给定成本密度的积分最小化。这是一个单人运输问题。在这个项目中，将考虑对这个被广泛研究的问题的几个概括。例如，如果有两个玩家，每个人都有自己的成本密度，每个人都选择自己的运输计划，那么他们就会通过成本密度，更重要的是，通过给定的边际来联系在一起。原始问题的另一个推广是考虑依赖于时间的成本密度。当这种情况发生时，交通计划和地图也将取决于时间。然而，另一条研究路线是由新兴的l -∞最优运输领域的问题驱动的，其中最小化的成本被表示为成本函数的本质最高，而不是成本密度的积分。博弈模型中输运映射的最优输运微分方程应导致广义蒙日-安培系统。时间相关的动态规划和可能的博弈论输运问题将导致测量空间中的哈密顿-雅可比方程。在非零和情况下，将建立一个广义纳什均衡的存在性。最后，分析了拟凸函数研究和随机微分对策理论中出现的几种偏微分方程模型。本课题将运用非线性泛函分析、博弈论、由动态规划引起的非线性偏微分方程、一阶和二阶偏微分方程的黏性解理论和现代变分学方法来解决这些问题。

Transport problems involve the study of the movement of mass from one location to another. Optimal transport is the study of exactly how to do this so that the cost of transportation is the smallest it can be. This is a very broad field with many applications, from the planning of urban transportation networks to molecular transport. The present project considers several important generalizations of the classical optimal transport problem which have not been previously studied. For example, a natural question to consider is what happens if there are two or more entities transporting the same initial mass to another location, each having its own costs to minimize. This is the "game theoretic" version of the optimal transport problem. Another problem deals with time-dependent transportation costs. This considers various cases when moving objects is more or less expensive depending on the time of movement. Overtime payments or transport in congested periods fall into this category. Transport problems in which one seeks to minimize the worst possible costs (instead of some average cost) are also important in applications and are considered here. Since optimal transport arises in many economic, biological, engineering, and other scientific areas, the mathematical theory developed for the problems considered in this project could have a significant practical impact. In addition, the project will study several new mathematical models arising in image processing and in understanding the motion of frontal boundaries due to flame or plasma propagation.The relaxed version of the optimal transport problem involves finding a measure, called a "transport plan," that minimizes the integral of a given cost density, subject to the constraint that the transport plan has given marginals. This is a one-player transport problem. Several generalizations of this much studied problem will be considered in this project. For example, if there are two players, each with their own cost density and each choosing their own transport plan, then they are coupled through the cost densities and, more importantly, through the given marginals. Another generalization of the original problem is to consider cost densities that depend on time. When that happens, the transport plan and map will also depend on time. Yet another line of investigation is driven by issues in the emerging area of L-infinity optimal transport, where the cost to be minimized is represented as the essential supremum of a cost function rather than the integral of a cost density. Optimal transport differential equations for the transport maps in the game model should lead to generalized Monge-Ampere systems. Dynamic programming of the time-dependent and perhaps the game-theoretic transport problem should lead to Hamilton-Jacobi equations in measure spaces. Existence of a generalized Nash equilibrium in the non-zero-sum case will be established. Finally, several PDE models arising in the study of quasiconvex functions and in the theory of stochastic differential games will be analyzed. The techniques used to tackle the problems proposed in this project will involve nonlinear functional analysis, game theory, and nonlinear partial differential equations arising from dynamic programming, as well as the theory of viscosity solutions for first and second order partial differential equations and modern methods in the calculus of variations.