Free boundary regularity and support propagation in mean field games and optimal transport
Cardaliaguet, Pierre; Munoz, Sebastian; Porretta, Alessio (2023), Free boundary regularity and support propagation in mean field games and optimal transport. https://basepub.dauphine.psl.eu/handle/123456789/25073
TypeDocument de travail / Working paper
External document linkhttps://hal.science/hal-04174138
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Dipartimento di Matematica [Roma II] [DIPMAT]
Abstract (EN)We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural nondegeneracy assumption, the free boundary is strictly convex and enjoys C 1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.
Subjects / Keywordsself-similar solutions; degenerate elliptic equations; intrinsic scaling; displacement convexity; Lagrangian coordinates; free boundary; mean field games; Hamilton-Jacobi equations; continuity equation
Showing items related by title and author.
Cardaliaguet, Pierre; Graber, Philip Jameson; Porretta, Alessio; Tonon, Daniela (2015) Article accepté pour publication ou publié