| Issue |
ESAIM: COCV
Volume 25, 2019
|
|
|---|---|---|
| Article Number | 44 | |
| Number of page(s) | 33 | |
| DOI | https://doi.org/10.1051/cocv/2018026 | |
| Published online | 20 September 2019 | |
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games*
Department of Mathematics, University of Padova,
Via Trieste 63,
35121
Padova, Italy.
** Corresponding author: bardi@math.unipd.it
Received:
10
July
2017
Accepted:
10
April
2018
This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T, is proved. Some results are also extended to MFGs with two populations.
Mathematics Subject Classification: 49L20 / 60H10
Key words: Mean Field Games / finite horizon / non-uniqueness of solutions / uniqueness of solutions / multipopulation MFG
The authors are partially supported by the research projects “Mean-Field Games and Nonlinear PDEs” of the University of Padova, and “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games” of the Fondazione CaRiPaRo. They are also members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
© EDP Sciences, SMAI 2019
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