Optimal Shapes and Masses, and Optimal Transportation Problems

Abstract

1 Introduction 2 Some classical problems 2.1 The isoperimetric problem 2.2 The Newton’s problem of optimal aerodynamical profiles 2.3 Optimal Dirichlet regions 2.4 Optimal mixtures of two conductors 3 Mass optimization problems 4 Optimal transportation problems 4.1 The optimal mass transportation problem: Monge and Kantorovich formulations 4.2 The PDE formulation of the mass transportation problem 5 Relationships between optimal mass and optimal transportation 6 Further results and open problems 6.1 A vectorial example 6.2 A p-Laplacian approximation 6.3 Optimization of Dirichlet regions 6.4 Optimal transporting distances References