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

Rita Ferreira; Diogo Gomes; Teruo Tada

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arXiv · 1804.07175

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

Rita Ferreira·Diogo Gomes·Teruo Tada

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Abstract

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.

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Rita Ferreira, Diogo Gomes, Teruo Tada. 2018-04-19. Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions. https://arxiv.org/abs/1804.07175

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