Interfaces and Free Boundaries
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Published online: 2015-05-26
Obstacle mean-field game problemDiogo A. Gomes and Stefania Patrizi (1) King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
(2) Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.
Keywords: Mean-field games, obstacle problem, penalization method
Gomes Diogo, Patrizi Stefania: Obstacle mean-field game problem. Interfaces Free Bound. 17 (2015), 55-68. doi: 10.4171/IFB/333