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

TL;DR

This paper proves sharp regularity results for the value function, pressure, and free boundary in one-dimensional first-order mean field games with power coupling, using a novel change of variables and elliptic estimates.

Contribution

It introduces a new approach using a singular change of variables to analyze boundary regularity in mean field games, achieving optimal regularity results.

Findings

Pressure is Lipschitz continuous under standard assumptions.

Value function is C^{1,1/2} regular.

Free boundary curves are smooth in time.

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.