Abstract. In this paper we study Mean Field Game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler's equationsà la Brenier. By this observation we manage to obtain a minimal regularity, which allows to write optimality conditions at the level of single agent trajectories and to define a weak notion of Nash equilibrium for our model.
In this manuscript, we propose a structural condition on non-separable Hamiltonians, which we term displacement monotonicity condition, to study second order mean field games master equations.A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a-priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.
In this paper we study the BV regularity for solutions of variational
problems in Optimal Transportation. As an application we recover BV estimates
for solutions of some non-linear parabolic PDE by means of optimal
transportation techniques. We also prove that the Wasserstein projection of a
measure with BV density on the set of measures with density bounded by a given
BV function f is of bounded variation as well. In particular, in the case f = 1
(projection onto a set of densities with an L^\infty bound) we precisely prove
that the total variation of the projection does not exceed the total variation
of the projected measure. This is an estimate which can be iterated, and is
therefore very useful in some evolutionary PDEs (crowd motion,. . .). We also
establish some properties of the Wasserstein projection which are interesting
in their own, and allow for instance to prove uniqueness of such a projection
in a very general framework
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