Optimal Transportation and Action-Minimizing Measures
Description
In this book we describe recent developments in the theory of optimal transportation, and some of its applications to fluid dynamics. Moreover we explore new variants of the original problem, and we try to figure out some common (and sometimes unexpected) features in this emerging variety of problems .
In Chapter 1 we study the optimal transportation problem on manifolds with geometric costs coming from Tonelli Lagrangians, while in Chapter 2 we consider a generalization of the classical transportation problem called the optimal irrigation problem. Then, Chapter 3 is about the Brenier variational theory of incompressible flows, which concerns a weak formulation of the Euler equations viewed as a geodesic equation in the space of measure-preserving diffeomorphism. Chapter 4 is devoted to the study of regularity and uniqueness of solutions of Hamilton-Jacobi equations applying the Aubry-Mather theory. Finally, the last chapter deals with a DiPerna-Lions theory for martingale solutions of stochastic differential equations.
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Content
1. The optimal transportation problem - Optimal transportation on non-compact manifolds, costs obtained from Lagrangians, interpolation and absolute continuity, displacement convexity.- 2. The irrigation problem - Dynamic cost on traffic plans, syncronization, stability.- 3. Variational models for the incompressible Euler equations - Arnold's least action problem, Brenier's variationals models, gap phenomena, necessary and sufficient optimality conditions, regularity of the pressure.- 4. On the structure of the Aubry set and Hamilton-Jacobi equation - Structure of the Mather quotient set, estimate of its Hausdorff dimensions, applications in dynamics.