  We analyze the Master Equation and the convergence problem within Mean Field
Games theory considering a bounded domain with homogeneous Dirichlet
conditions. This framework characterizes N-players differential games where
each player's dynamic ends when touching the boundary, leading to two Dirichlet
boundary conditions in the Master Equation. In the first part of the paper we
analyze both the well-posedness of the Master Equation and the regularity of
its solutions studying the global regularity at the boundary of solutions for a
suitable class of parabolic equations. Such results are then exploited to
consider the convergence problem of the Nash equilibria, towards the Mean Field
Games system strategies, in the N-players system, also proving optimal
trajectories convergence. Eventually, we apply our findings to solve a toy
model related to an optimal liquidation scenario.
