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In this paper, using variational approaches, we investigate the first orderplanning problem arising in the theory of mean field games. We show theexistence and uniqueness of weak solutions of the problem in the case of alarge class of Hamiltonians with arbitrary superlinear order of growth atinfinity and local coupling functions. We require the initial and finalmeasures to be merely summable. At the same time (relying on the techniquesdeveloped recently by Graber and M\'esz\'aros), under stronger monotonicity andconvexity conditions on the data, we obtain Sobolev estimates on the solutionsof the planning problem both for space and time derivatives.

The planning problem in Mean Field Games as regularized mass transport