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

Ferreira, Rita; Gomes, Diogo; Tada, Teruo

doi:10.1090/proc/14475

Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions
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by Rita Ferreira, Diogo Gomes and Teruo Tada

Proc. Amer. Math. Soc. 147 (2019), 4713-4731

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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