On periodic subsolutions to steady second-order systems and applications

Bostan and Namah (Remarks on bounded solutions of steady Hamilton–Jacobi equations, C. R. Acad. Sci. Paris, Ser. I 347(15–16) (2009) 873–878) proved that constant functions are the only bounded solutions to $H(Du)=H(0)$ when H is superlinear and strictly convex. In this short note, we present a proof other than that of Bostan and Namah for equations that can be easily applied to some types of possibly degenerate parabolic systems. Our proof applies for periodic subsolutions instead of bounded solutions like that of Bostan and Namah; however, we need periodic subsolutions, which is quite restrictive. We do not consider Hopf–Lax's formula in our proof, so we can relax some restrictions on H. We also present an application to the large-time behavior of solutions to degenerate parabolic systems.