Topics in Optimal Transport and Nonlinear Partial Differential Equations

Loyola University Of Chicago, Chicago IL

Investigators

Abstract

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.