  In this paper, we study first-order stationary monotone mean-field games
(MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations
Dirichlet conditions may not be satisfied, here, we establish the existence of
solutions of MFGs that satisfy those conditions. To construct these solutions,
we introduce a monotone regularized problem. Applying Schaefer's fixed-point
theorem and using the monotonicity of the MFG, we verify that there exists a
unique weak solution to the regularized problem. Finally, we take the limit of
the solutions of the regularized problem and using Minty's method, we show the
existence of weak solutions to the original MFG.
