Maximal Regularity of Superquadratic Hamilton-Jacobi Equations: The Endpoint Case in Lions'Conjecture
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 $-\Delta u+|Du|^\gamma=f$ on $\mathbb T^d$, 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. A critical local energy estimate excludes concentration of the gradient energy in $L^{\gamma q_c}$ at the first blow-up scale, while small-drift regularity upgrades weak compactness of the blow-up sequence to strong local compactness.