Density constraints in optimal transport, PDEs and mean field games.

Summary Motivated by questions posed by F. Santambrogio, this thesis is dedicated to the study of mean-field games and models involving optimal transport with density constraints. In order to study second order MFG models in the spirit of F. Santambrogio's work, we introduce as an elementary brick a diffusive model of crowd motion with density constraints (generalizing in a sense the work of Maury et al.). The model is described by the evolution of the density of the crowd, which can be seen as a curve in Wasserstein space. From a PDE point of view, it corresponds to a modified Fokker-Planck equation, with an additional term, the gradient of a pressure (only in the saturated zone) in the drift. By going through the dual equation and using well known parabolic estimates, we prove the uniqueness of the density and pressure pair. Motivated initially by the splitting algorithm (used in the existence result above), we study fine properties of the Wasserstein projection below a given threshold. Integrating this question into a larger class of problems involving optimal transport, we demonstrate BV estimates for optimizers. Other possible applications (in partial transport, shape optimization and degenerate parabolic problems) of these BV estimates are also discussed. In this sense, MFG systems are obtained as first order optimality conditions for two convex problems in duality. In these systems an additional term appears, interpreted as a price to be paid when agents pass through saturated areas. First, taking advantage of the elliptic regularity results, we show the existence and characterization of second order stationary MFG solutions with density constraints. As an additional result, we characterize the subdifferential of a functional introduced by Benamou-Brenier to give a dynamic formulation of the optimal transport problem. Second, (based on a penalty technique) we show that a class of first order MFG systems with density constraints is well posed. An unexpected connection with the incompressible Euler equations à la Brenier is also given.