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


Authors: Rita Ferreira, Diogo Gomes and Teruo Tada
Journal: Proc. Amer. Math. Soc. 147 (2019), 4713-4731
MSC (2010): Primary 35J56, 35A01
DOI: https://doi.org/10.1090/proc/14475
Published electronically: July 30, 2019
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Abstract: In this paper, we study first-order stationary monotone mean-field 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.


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Additional Information

Rita Ferreira
Affiliation: King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
Email: rita.ferreira@kaust.edu.sa

Diogo Gomes
Affiliation: King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
Email: diogo.gomes@kaust.edu.sa

Teruo Tada
Affiliation: King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia
Email: teruo.tada@kaust.edu.sa

DOI: https://doi.org/10.1090/proc/14475
Keywords: Mean-field games, stationary problems, weak solutions, Dirichlet boundary conditions
Received by editor(s): May 3, 2018
Received by editor(s) in revised form: November 14, 2018
Published electronically: July 30, 2019
Additional Notes: 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.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2019 American Mathematical Society