*Phd Thesis*

**Inserted:** 5 oct 2015

**Last Updated:** 5 oct 2015

**Year:** 2015

**Notes:**

Université Paris-Sud, advisor: Filippo Santambrogio

**Abstract:**

Motivated by some questions raised by F. Santambrogio, this thesis is devoted to the study of Mean Field Games and models involving optimal transport with *density constraints*.

To study second order MFG models in the spirit of the model proposed by F. Santambrogio, as a possible first step we introduce and show the well-posedness of a diffusive crowd motion model with density constraints (generalizing in some sense the works by B. Maury et al.). The model is described by the evolution of the people’s density, that can be seen as a curve in the Wasserstein space. From the PDE point of view, this corresponds to a modified Fokker-Planck equation, with an additional gradient of a pressure (only living in the saturated zone $\{\rho = 1\}$) in the drift. We provide a uniqueness result for the pair density and pressure $(\rho, p)$ by passing through the dual equation and using some well-known parabolic estimates.

Initially motivated by the splitting algorithm (used for the above existence result), we study some fine properties of the Wasserstein projection below a given threshold. Embedding this question into a larger class of variational problems involving optimal transport, we show *BV estimates* for the optimizers. Other possible applications (for partial optimal transport, shape optimization and degenerate parabolic problems) of these BV estimates are also discussed.

Changing the point of view, we also study *variational Mean Field Game* models with density constraints. In this sense, the MFG systems are obtained as first order optimality conditions of two convex problems in duality. In these systems an additional term appears, interpreted as a price to be paid when agents pass through saturated zones. Firstly, profiting from the regularity results of elliptic PDEs, we give the existence and characterization of the solutions of stationary second order MFGs with density constraints. As a byproduct we characterize the subdifferential of a convex functional introduced initially by Benamou-Brenier to give a dynamic formulation of the optimal transport problem. Secondly, (based on a penalization technique) we prove the well-posedness of a class of first order evolutive MFG systems with density constraints. An unexpected connection with the incompressible Euler’s equations *à la Brenier* is also given.

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