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

Luigi Ambrosio, Yann Brenier, Giuseppe Buttazzo, Cédric Villani, Luis A. Caffarelli, Luis A. Caffarelli, Sandro Salsa354040192X, 9783540401926

Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Amp?re and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view.
The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory.

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Table of contents :
ymau60re2553uwqn.pdf……Page 0
1 Optimal Transportation……Page 9
2 The continuous case……Page 10
3 The dual problem……Page 11
4 Existence and Uniqueness……Page 12
5 The potential equation……Page 14
6 Some remarks on the structure of the equation……Page 15
1 Introduction……Page 19
2.1 The isoperimetric problem……Page 21
2.2 The Newton’s problem of optimal aerodynamical profiles……Page 22
2.3 Optimal Dirichlet regions……Page 25
2.4 Optimal mixtures of two conductors……Page 27
3 Mass optimization problems……Page 31
4 Optimal transportation problems……Page 37
4.1 The optimal mass transportation problem: Monge and Kantorovich formulations……Page 38
4.2 The PDE formulation of the mass transportation problem……Page 40
5 Relationships between optimal mass and optimal transportation……Page 41
6.1 A vectorial example……Page 43
6.2 A $p$-Laplacian approximation……Page 45
6.3 Optimization of Dirichlet regions……Page 46
6.4 Optimal transporting distances……Page 48
References……Page 52
1 Some motivations……Page 61
2 A study of fast trend to equilibrium……Page 62
3 A study of slow trend to equilibrium……Page 71
4 Estimates in a mean-field limit problem……Page 78
5 Otto’s differential point of view……Page 87
References……Page 95
2.1 Generalized geodesics……Page 97
2.2 Extension to probability measures……Page 100
2.3 A decomposition result……Page 102
2.4 Relativistic MKT……Page 103
2.5 A relativistic heat equation……Page 104
2.6 Laplace’s equation and Moser’s lemma revisited……Page 106
3.1 Classical harmonic functions……Page 108
Optimality equations……Page 111
Superharmonicity of the Boltzmann entropy……Page 113
A tentative proof……Page 114
4 Multiphasic MKT……Page 115
5 Generalized extremal surfaces……Page 117
5.2 Degenerate quadratic cost functions……Page 119
6 Generalized extremal surfaces in $mathbb{R}^5$ and Electrodynamics……Page 120
6.1 Recovery of the Maxwell equations……Page 121
6.2 Derivation of a set of nonlinear Maxwell equations……Page 122
6.3 An Euler-Maxwell-type system……Page 124
References……Page 126
1 Introduction……Page 128
2 Notation……Page 134
3 Duality and optimality conditions……Page 135
4 Gamma-convergence and Gamma-asymptotic expansions……Page 140
5 1-dimensional theory……Page 141
6 Transport rays and transport set……Page 143
7 A stability result……Page 151
8 A counterexample……Page 155
9 Appendix: disintegration of measures……Page 157
References……Page 163