Inserted: 18 nov 2018
Last Updated: 18 nov 2018
In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. As an alternative way, we show that solutions of the planning problem can be approximated, via a $\Gamma$-convergence procedure, by solutions of standard mean field games with suitable penalized final couplings. In the same time (relying on the techniques developed recently by Graber and M\'esz\'aros), under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of mean field games with general final couplings and the planning problem as well, both for space and time derivatives.