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

TL;DR

This paper investigates the regularity and support propagation in first-order mean field games with local coupling, revealing how different coupling types influence free boundary behavior and solution smoothness.

Contribution

It introduces new regularity results and free boundary analysis for mean field games with various couplings, including entropic and power-type, and links these to optimal transport problems.

Findings

Support propagates with infinite speed for entropic coupling.

Finite support propagation and free boundary formation for power-type coupling.

Free boundary is strictly convex and $C^{1,1}$ regular under non-degeneracy.

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\"older 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.