# Mean Field Games with density constraints: pressure equals price

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In the first part of this talk I will do a brief introduction to the recent theory of Mean Field Games (MFG) initiated by J.-M. Lasry and P.-L. Lions. The main objective of the MFG theory is the study of the limit behavior of Nash equilibria for symmetric differential games with a very large number of “small” players. In its simplest form, as the number of players tends to infinity, limits of Nash equilibria can be characterized in terms of the solution of a coupled system of a Hamilton-Jacobi and Fokker-Planck (or continuity) equations. The first equation describes the evolution of the value function of a typical agent, while the second one characterizes the evolution of the agents’ density. In the second part, I will introduce a variational MFG model, where we impose a density constraint. From the modeling point of view, imposing this new constraint means that we are aiming to avoid congestion among the agents. We will see that a weak solution of the system contains an extra term, an additional price imposed on the saturated zones. I will show that this price corresponds to the pressure field from the models of incompressible Euler equations à la Brenier. If time permits, I will discuss the regularity properties of the pressure variable, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model. The talk is based on a joint work with P. Cardaliaguet (Paris Dauphine) and F. Santambrogio (Paris-Sud, Orsay).

# Two dimensional water waves in holomorphic coordinates

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We consider this problem expressed in position-velocity potential holomorphic coordinates. We will explain the set up of the problem(s) and try to present the advantages of choosing such a framework. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. The talk will try to be self contained.

# Reactive Processes in Inhomogeneous Media

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We study fine details of spreading of reactive processes in multidimensional

inhomogeneous media. In the real world, one often observes a transition from one equilibrium (such as unburned areas in forest fires) to another (burned areas)to happen over short spatial as well as temporal distances. We demonstrate that this phenomenon also occurs in one of the simplest models of reactive processes, reaction-diffusion equations with ignition reaction functions, under very general hypotheses.

Specifically, in up to three spatial dimensions, the width (both in space and time) of the zone where the reaction occurs turns out to remain uniformly bounded in time for fairly general classes of initial data. This bound even becomes independent of the initial data and of the reaction function after an initial time interval. Such results have recently been obtained in one dimension, in which one can even completely characterize the long term dynamics of general solutions to the equation, but are new in dimensions two and three. An indication of the added difficulties is the fact that three dimensions turns out to indeed be the borderline case, as the bounded-width result is in fact false for general inhomogeneous media in four and more dimensions.

# A Degenerate Isoperimetric Problem and its Relation to Traveling Waves for a Bi-Stable Hamiltonian System

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I will discuss a simple-looking isoperimetric problem for curves in the plane where length is measured with respect to a degenerate metric. One motivation for the study is that geodesics for this problem, appropriately parametrized, lead to traveling waves associated with a Hamiltonian system based on a bi-stable potential. This is joint work with Stan Alama, Lia Bronsard, Andres Contreras and Jiri Dadok.