  We study the short-time existence and uniqueness of solutions to a coupled
system of partial differential equations arising in mean field game theory. It
has the generic form $$ \left\{ \begin{array}{c} -\partial_t u - \Delta u +
H(t,x,m,\nabla u) = f(t,x,m) \\ \partial_t m - \Delta m - \mathrm{div}
\left(m\nabla_p H(t,x,m,\nabla u)\right) = 0 \end{array}\right. $$ plus
initial-final and boundary conditions. The novelty of the problem is that the
Hamiltonian $H(t,x,m,p)$ may take such forms as $m^{-\alpha}|p|^r$ for some
$\alpha \geq 0$ and $r > 1$. Our main result is the existence of weak solutions
for small times $T$ so long as $r$ is not too large, and uniqueness under
additional constraints. The main ingredient in the proof is an a priori
estimate on solutions to the Fokker-Planck equation. We also briefly consider
existence and uniqueness of solutions to an optimal control problem related to
mean field games.
