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New estimates on the regularity of the pressure in density-constrained Mean Field Games

Abstract : We consider variational Mean Field Games endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player is augmented by a pressure effect, i.e. a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optimal transport techniques after time-discretization, the results obtained in a paper of the second author with Cardaliaguet and Mészáros. We prove H1 and L infinity regularity under very mild assumptions on the data, and explain the consequences for the MFG, in terms of the value function and of the Lagrangian equilibrium formulation.
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https://hal.archives-ouvertes.fr/hal-01930480
Contributor : Hugo Lavenant <>
Submitted on : Monday, May 13, 2019 - 12:56:43 PM
Last modification on : Monday, December 14, 2020 - 5:25:15 PM
Long-term archiving on: : Tuesday, October 1, 2019 - 4:03:17 PM

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  • HAL Id : hal-01930480, version 2
  • ARXIV : 1811.09147

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Hugo Lavenant, Filippo Santambrogio. New estimates on the regularity of the pressure in density-constrained Mean Field Games. Journal of the London Mathematical Society, London Mathematical Society, inPress. ⟨hal-01930480v2⟩

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