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

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