A variational approach to second order mean field games with density constraints: the stationary case
Abstract
In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain
Ω⊂
R
d
. 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
′
∈]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
′
≥d/(d−1)
, the previous arguments do not apply and we prove the existence by means of an approximation argument.