Obstacle mean-field game problem
- Gomes, Diogo A.
- Patrizi, Stefania
2010 Mathematics Subject Classification
- 35J87 49L99
- mean-field games, obstacle problem, penalization method
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.
- Interfaces Free Bound., 17 (2015) pp. 55--68.