KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
KAUST Grant NumberOSR-CRG2017-3452
Embargo End Date2022-03-11
Permanent link to this recordhttp://hdl.handle.net/10754/662281
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AbstractIn this paper, we study a priori estimates for a first-order mean-field planning problem with a potential. In the theory of mean-field games (MFGs), a priori estimates play a crucial role to prove the existence of classical solutions. In particular, uniform bounds for the density of players’ distribution and its inverse are of utmost importance. Here, we investigate a priori bounds for those quantities for a planning problem with a non-vanishing potential. The presence of a potential raises non-trivial difficulties, which we overcome by exploring a displacement-convexity property for the mean-field planning problem with a potential together with Moser’s iteration method. We show that if the potential satisfies a certain smallness condition, then a displacement-convexity property holds. This property enables Lq bounds for the density. In the one-dimensional case, the displacement-convexity property also gives Lq bounds for the inverse of the density. Finally, using these Lq estimates and Moser’s iteration method, we obtain L∞ estimates for the density of the distribution of the players and its inverse. We conclude with an application of our estimates to prove existence and uniqueness of solutions for a particular first-order mean-field planning problem with a potential.
CitationBakaryan, T., Ferreira, R., & Gomes, D. (2021). Some estimates for the planning problem with potential. Nonlinear Differential Equations and Applications NoDEA, 28(2). doi:10.1007/s00030-021-00681-z
SponsorsT. Bakaryan, R. Ferreira, and D. Gomes were partially supported by baseline and start-up funds from King Abdullah University of Science and Technology (KAUST) OSR-CRG2017-3452.