| Abstract: |
| In this joint work with Tigran Bakaryan and Giuseppe Di Fazio, we present recent findings on the regularity properties of stationary mean-field games (MFGs) on the torus, focusing on systems with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The goal is to bridge the gap between known low-regularity results for bounded diffusions and the smooth solutions typically associated with the Laplacian. By employing the Hopf-Cole transformation, we reformulate the system into a scalar elliptic equation, enabling us to establish the existence of $C^{1,\alpha}$ solutions. These results have significant implications for understanding the fine structure of equilibria in MFG models, especially in applications with non-linear and non-smooth dynamics. |
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