On the Problem of Maximal $$L^q$$-regularity for Viscous Hamilton–Jacobi Equations

Abstract

AbstractIn this paper we prove a conjecture by P.-L. Lions on maximal regularity of $$L^q$$ L q -type for periodic solutions to $$-\Delta u + |Du|^\gamma = f$$ - Δ u + | D u | γ = f in $$\mathbb {R}^d$$ R d , under the (sharp) assumption that $$q > d \frac{\gamma -1}{\gamma }$$ q > d γ - 1 γ .