| Abstract: |
| We extend the weak--strong uniqueness principle for mean-field game (MFG) systems to a broad class of second-order stationary and time-dependent problems. Under standard monotonicity, growth, and coercivity assumptions on the Hamiltonian and relying strictly on the integrability exponents guaranteed by the existing theory for monotone MFG systems, we show that any weak solution must coincide with a given strong solution. Our analysis covers models with spatially dependent scalar diffusion coefficients, using monotonicity arguments and a coefficient-adapted mollification strategy to manage the variable diffusion terms. We extend this strategy to establish weak--strong uniqueness in the corresponding second-order, initial--terminal, time-dependent setting. Finally, to address the critical quadratic growth regime, we derive a new a priori second-order estimate for a stationary MFG system with logarithmic coupling, quantifying the control of second spatial derivatives of $u$ weighted by $m$ and of $Dm$ in terms of the data and thereby establishing weak--strong uniqueness in this setting. Our results provide a unified framework for uniqueness and regularity in a wide array of MFG models. |
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