Comptes Rendus
no. 9
Partial differential equations
On periodic subsolutions to steady second-order systems and applications
[Sur des sous-solutions périodiques de systèmes d'équations du deuxième ordre stables et applications]

Présenté par : the Editorial Board

Vinh Duc Nguyen1
1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 721-728.

Bostan et Namah (Remarks on bounded solutions of steady Hamilton–Jacobi equations, C. R. Acad. Sci. Paris, Ser. I 347(15–16) (2009) 873–878) ont montré que les solutions bornées de l'équation H(Du)=H(0), où H est superlinéaire et strictement convexe, sont les constantes. Dans cette note, on présente une autre preuve qui peut être appliquée facilement à des systèmes d'équations paraboliques dégénérées. Notre preuve s'applique à des sous-solutions périodiques au lieu des solutions bornées examinées par Bostan et Namah. Comme nous n'utilisons pas la formule d'Hopf–Lax dans la preuve, nous pouvons affaiblir un peu certaines régularités des hamiltoniens. Finalement, nous présentons une application au comportement asymptotique des solutions pour des systèmes d'équations paraboliques dégénérées.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.09.002
Vinh Duc Nguyen 1

1 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam 
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Vinh Duc Nguyen. On periodic subsolutions to steady second-order systems and applications. Comptes Rendus. Mathématique, Volume 357 (2019) no. 9, pp. 721-728. doi : 10.1016/j.crma.2019.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.09.002/

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