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 non-degeneracy 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 H & ouml;lder 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. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Cardaliaguet, P., Munoz, S., Porretta, A. (2024). Free boundary regularity and support propagation in mean field games and optimal transport. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 190 [10.1016/j.matpur.2024.103599].

Free boundary regularity and support propagation in mean field games and optimal transport

Cardaliaguet, Pierre
;
Porretta, Alessio
2024-01-01

Abstract

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 non-degeneracy 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 H & ouml;lder 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. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
2024
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
Settore MATH-03/A - Analisi matematica
English
Con Impact Factor ISI
Self-similar solutions
Degenerate elliptic equations
Lagrangian coordinates
Free boundary
Mean field games
Cardaliaguet, P., Munoz, S., Porretta, A. (2024). Free boundary regularity and support propagation in mean field games and optimal transport. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 190 [10.1016/j.matpur.2024.103599].
Cardaliaguet, P; Munoz, S; Porretta, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/388111
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