Optimal transportation and applications : lectures given at the C.I.M.E. summer school held in Martina Franca, Italy, September 2-8, 2001 / L. A. Caffarelli, S. Salsa

The book contains 4 lecture notes and an original paper.
The first lecture note by L. Caffarelli is a review of some fundamental relationships between the Monge-Ampere equation and the theory of optimal transportation. Many ideas are presented in a heuristic way and some of the basic proofs are given.
The second lecture note by G. Buttazzo and L. De Pascale is about the relationships between shape and mass optimization problems and optimal transportation problems. After a short introduction and some classical examples of shape optimization problems, mass optimization problems are introduced.
The relationships between mass optimization problems and optimal transportation problems are analyzed in detail as well as the state of the art on the mass optimization problem at the time of the summer school (summer 2001).
The third lecture note by C. Villani is about the application of optimal transportation problems to the trend to equilibrium for dissipative PDE. It contains a short introduction, a section devoted to the study of the fast trend to equilibrium, a section devoted to the study of the slow trend to equilibrium, and another section devoted to some estimates in a mean-field limit problem. A fifth section contains some theoretical background on the differential point of view developed by Benamou, Brenier, McCann, Otto and others.
The last lecture note by Y. Brenier is a systematic presentation of the theory developed by the author in the last few years. First the Monge-Kantorovich theory is revisited in terms of generalized geodesics. Then some generalization of the MK-theory is presented including a relativistic heat equation and a variational interpretation of Moser's lemma. Section 3 is devoted to a definition of generalized harmonic functions and a few results about them. The multi-phase MK-theory with constraints which is considered in relation with the relaxed theory of geodesics on the group of measure preserving maps related to incompressible fluid mechanics follows. Finally the author introduces a theory of generalized extremal surfaces and relates the surfaces to the Maxwell equation and the pressureless Euler-Maxwell equation. ... (MathSciNet)