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 $$-Δu+|Du|^γ=f$$ in $\mathbb T^d$, where $d\geq 2$, $γ>2$, and $f\in L^{q_c}(\mathbb T^d)$ with $q_c=d(γ-1)