In this paper, we study first-order stationary monotone meanfield games (MFGs) with Dirichlet boundary conditions. Whereas Dirichlet conditions may not be satisfied for Hamilton-Jacobi equations, here we establish the existence of solutions to 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.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): OSR-CRG2017-3452
Acknowledgements: The authors were partially supported by baseline and start-up funds from King Abdullah University of Science and Technology (KAUST) and by KAUST project OSR-CRG2017-3452.