In 1D first-order MFGs with power coupling and compactly supported density, under nondegeneracy the pressure is Lipschitz, the value function is C^{1,1/2}, and free boundaries are smooth in time.
Zamboni,H¨ older continuity for solutions of linear degenerate elliptic equations under minimal assump- tions, J
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Optimal regularity at the free boundary in one-dimensional first-order mean field games
In 1D first-order MFGs with power coupling and compactly supported density, under nondegeneracy the pressure is Lipschitz, the value function is C^{1,1/2}, and free boundaries are smooth in time.