Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems
A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton--Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of $$-\Delta u+|Du|^\gamma=f$$ in $\mathbb T^d$, where $d\geq 2$, $\gamma>2$, and $f\in L^{q_c}(\mathbb T^d)$ with $q_c=d(\gamma-1)/\gamma$. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of $L^{q_c}$, we rule out this concentration and prove maximal $L^{q_c}$ regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local $L^{\gamma q_c}$-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton--Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.