  The primary objective of this paper is to understand first-order,
time-dependent mean-field games with Neumann boundary conditions, a question
that remains under-explored in the literature. This matter is particularly
relevant given the importance of boundary conditions in crowd models. In our
model, the Neumann conditions result from players entering the domain according
to a prescribed current, for instance, in a crowd entry scenario into an
open-air concert or stadium. We formulate the model as a standard mean-field
game coupling a Hamilton-Jacobi equation with a Fokker-Planck equation. Then,
we introduce a relaxed variational problem and use Fenchel-Rockafellar duality
to study the relation between these problems. Finally, we prove the existence
and uniqueness of solutions for the system using variational methods.
