measuring tape
optimal transportation and action minimizing measures
p 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 . p 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 .