[Sur des sous-solutions périodiques de systèmes d'équations du deuxième ordre stables et applications]
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 , 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 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.
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@article{CRMATH_2019__357_9_721_0,
author = {Nguyen, Vinh Duc},
title = {On periodic subsolutions to steady second-order systems and applications},
journal = {Comptes Rendus. Math\'ematique},
pages = {721--728},
publisher = {Elsevier},
volume = {357},
number = {9},
year = {2019},
doi = {10.1016/j.crma.2019.09.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2019.09.002/}
}
TY - JOUR AU - Nguyen, Vinh Duc TI - On periodic subsolutions to steady second-order systems and applications JO - Comptes Rendus. Mathématique PY - 2019 SP - 721 EP - 728 VL - 357 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.09.002/ DO - 10.1016/j.crma.2019.09.002 LA - en ID - CRMATH_2019__357_9_721_0 ER -
%0 Journal Article %A Nguyen, Vinh Duc %T On periodic subsolutions to steady second-order systems and applications %J Comptes Rendus. Mathématique %D 2019 %P 721-728 %V 357 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.09.002/ %R 10.1016/j.crma.2019.09.002 %G en %F CRMATH_2019__357_9_721_0
Nguyen, Vinh Duc. On periodic subsolutions to steady second-order systems and applications. Comptes Rendus. Mathématique, Tome 357 (2019) no. 9, pp. 721-728. doi : 10.1016/j.crma.2019.09.002. http://www.numdam.org/articles/10.1016/j.crma.2019.09.002/
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