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

Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions