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  • Book
  • Jul 2008

Optimal Transportation and Action-Minimizing Measures


  • Gives for the first time a generalized approach to many problems of different nature, in the context of optimal transportation

Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 8)

Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)

About this book

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.


  • Euler equation
  • Hamilton-Jacobi equation
  • irrigation problem
  • optimal transport
  • partial differential equations

Authors and Affiliations

  • Labo. J.-A. Dieudonne, Universit√© Nice CNRS UMR 6621, Vice CX, France

    Alessio Figalli

Bibliographic Information