Optimal regularity at the free boundary in one-dimensional first-order mean field games

Sebastian Munoz

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arXiv · 2605.06871

Optimal regularity at the free boundary in one-dimensional first-order mean field games

Sebastian Munoz

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Abstract

We establish sharp regularity for the value function, the pressure, and the free boundary in one-dimensional first-order mean field games with power coupling and compactly supported density. Under a standard nondegeneracy assumption on the initial datum, the pressure $p=m^\theta$ is Lipschitz continuous, the value function $u$ is $C^{1,1/2}$, and the two free boundary curves are smooth in time. If the initial pressure is smooth, then both $p$ and $u$ are smooth up to the free boundary from inside the positive phase. The proof works in Lagrangian coordinates and, through a singular change of variables, recasts the boundary degeneracy as a removable radial axis in effective dimension $N=4+2/\theta$, allowing the application of recent estimates for even solutions to elliptic problems with degenerate weights.

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Sebastian Munoz. 2026-05-07. Optimal regularity at the free boundary in one-dimensional first-order mean field games. https://arxiv.org/abs/2605.06871

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