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 solutionsof 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. Gives for the first time a generalized approachto many problems of different nature, in the context of optimal transportation INDICE: 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.
- ISBN: 978-88-7642-330-7
- Editorial: Birkhaüser
- Encuadernacion: Rústica
- Páginas: 250
- Fecha Publicación: 01/07/2008
- Nº Volúmenes: 1
- Idioma: Inglés