Geodesics and PDE methods in transport models.

Summary This thesis is dedicated to the study of optimal transportation problems, alternative to the Monge-Kantorovich problem: they appear naturally in practical applications, such as the design of optimal transportation networks or the modeling of urban traffic problems. In particular, we consider problems where the cost of transportation has a nonlinear dependence on the mass: typically in such problems, the cost of moving a mass m for a length ℓ is φ (m) ℓ, where φ is an assigned function, thus obtaining a total cost of type Σ φ (m) ℓ. Two important cases are discussed in detail in this work: the case where the function φ is subadditive (branched transport), so that the mass has an interest in traveling together, so as to reduce the total cost. the case where φ is superadditive (congested transport), where on the contrary, the mass tends to diffuse as much as possible. In the case of branched transport, we introduce two new models: in the first one, the transport is described by curves of probability measures that minimize a geodesic type functional (with a coefficient that penalizes the measures that are not atomic). The second one is more in the spirit of the formulation of Benamou and Brenier for the Wasserstein distances, in particular, the transport is described by pairs of ``measurement curves--velocity fields'', linked by the continuity equation, which minimize an adequate (non-convex) energy. For both models, we demonstrate the existence of minimal configurations and the equivalence with other existing formulations in the literature. For the case of congested transportation, we review two already existing models, in order to prove their equivalence: while the first of these models can be considered as a Lagrangian approach to the problem and it has interesting links with equilibrium issues for urban traffic, the second one is a convex optimization problem with divergence constraints by The proof of the equivalence between the two models constitutes the main body of the second part of this thesis and contains various elements of interest, including: the flow theory of sparsely regular vector fields (DiPerna-Lions), the Dacorogna and Moser construction for transport applications, and in particular the regularity results (which we prove here) for a highly degenerate elliptic equation, which does not seem to have been studied much.