  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. At 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 the planning problem both for space and time derivatives.
