A variational approach to second order mean field games with density constraints: the stationary case

par Francisco Silva (Université de Limoges)

vendredi 27 mars 2015, 10:30 → 11:30 Europe/Paris

XLIM Salle X.203

In this talk we consider second order stationary Mean Field Game systems under density constraints on a bounded domain $\text{Undefined control sequence R}$. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order $q^{\prime }\in ]1,d/(d-1)[$, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou-Brenier, we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order $q^{\prime }\ge d/(d-1)$, the previous arguments do not apply and we prove the existence by means of an approximation argument. This is a joint work with A. Richárd Mészáros (Université d'Orsay).