  This paper presents a novel first-order mean-field game model that includes a
prescribed incoming flow of agents in part of the boundary (Neumann boundary
condition) and exit costs in the remaining portion (Dirichlet boundary
condition). Our model is described by a system of a Hamilton-Jacobi equation
and a stationary transport (Fokker-Planck) equation equipped with mixed and
contact-set boundary conditions. We provide a rigorous variational formulation
for the system, allowing us to prove the existence of solutions using
variational techniques. Moreover, we establish the uniqueness of the gradient
of the value function, which is a key result for the analysis of the model. In
addition to the theoretical results, we present several examples that
illustrate the presence of regions with vanishing density.
