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

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.