In this paper we study master equations arising from mean field game
problems, under the Lasry-Lions monotonicity condition. Classical solutions of
such equations typically require very strong technical conditions. Moreover,
unlike the equations arising from mean field control problems, the mean field
game master equations are non-local and even classical solutions often do not
satisfy the comparison principle, so the standard viscosity solution approach
seems infeasible. We shall propose two notions of weak solutions for such
equations: one is in the spirit of vanishing viscosity solutions, relying on
the stability result; and the other is in the spirit of Sobolev solutions,
based on the integration by parts formula. We shall prove existence and
uniqueness of weak solutions in both senses. For the crucial regularity in
terms of the measures, we construct a smooth mollifer for functions on
Wasserstein space, which is new in the literature and is interesting in its own
right. In order to focus on the main ideas, in this paper we consider only a
very special case, and the more general cases will be studied in an
accompanying paper.