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Interfaces and Free Boundaries


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Volume 17, Issue 1, 2015, pp. 55–68
DOI: 10.4171/IFB/333

Published online: 2015-05-26

Obstacle mean-field game problem

Diogo A. Gomes[1] and Stefania Patrizi[2]

(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