Twisted Lax–Oleinik formulas and weakly coupled systems of Hamilton–Jacobi equations
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 209-224.

Nous démontrons que les solutions de viscosité d’un système faiblement couplé d’équations d’Hamilton–Jacobi peuvent être approchées par des itérations d’opérateurs tordus à la Lax–Oleinik. On établit la convergence vers la solution du schéma itératif et mettons en exergue quelques propriétés supplémentaires des solutions approchées.

We show that viscosity solutions of evolutionary weakly coupled systems of Hamilton–Jacobi equations can be approximated by iterated twisted Lax–Oleinik like operators. We establish convergence to the solution of the iterated scheme and discuss further properties of the approximate solutions.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1598
Classification : 35F21,  49L25,  37J50
Mots clés : weakly coupled systems of Hamilton–Jacobi equations, viscosity solutions, weak KAM Theory
@article{AFST_2019_6_28_2_209_0,
     author = {Zavidovique, Maxime},
     title = {Twisted {Lax{\textendash}Oleinik} formulas and weakly coupled systems of {Hamilton{\textendash}Jacobi} equations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {209--224},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {2},
     year = {2019},
     doi = {10.5802/afst.1598},
     mrnumber = {3957680},
     zbl = {07095681},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1598/}
}
TY  - JOUR
AU  - Zavidovique, Maxime
TI  - Twisted Lax–Oleinik formulas and weakly coupled systems of Hamilton–Jacobi equations
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2019
DA  - 2019///
SP  - 209
EP  - 224
VL  - Ser. 6, 28
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1598/
UR  - https://www.ams.org/mathscinet-getitem?mr=3957680
UR  - https://zbmath.org/?q=an%3A07095681
UR  - https://doi.org/10.5802/afst.1598
DO  - 10.5802/afst.1598
LA  - en
ID  - AFST_2019_6_28_2_209_0
ER  - 
Zavidovique, Maxime. Twisted Lax–Oleinik formulas and weakly coupled systems of Hamilton–Jacobi equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 209-224. doi : 10.5802/afst.1598. http://archive.numdam.org/articles/10.5802/afst.1598/

[1] Barles, Guy Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications (Berlin), 17, Springer, 1994, x+194 pages | MR 1613876 | Zbl 0819.35002

[2] Barles, Guy; Ishii, Hitoshi; Mitake, Hiroyoshi A new PDE approach to the large time asymptotics of solutions of Hamilton–Jacobi equations, Bull. Math. Sci., Volume 3 (2013) no. 3, pp. 363-388 | Article | MR 3128036 | Zbl 1284.35065

[3] Barles, Guy; Souganidis, Panagiotis E. Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., Volume 4 (1991) no. 3, pp. 271-283 | MR 1115933 | Zbl 0729.65077

[4] Barles, Guy; Souganidis, Panagiotis E. On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., Volume 31 (2000) no. 4, pp. 925-939 | Article | MR 1752423

[5] Cagnetti, Filippo; Gomes, Diogo; Mitake, Hiroyoshi; Tran, Hung V. A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015) no. 1, pp. 183-200 | Article | MR 3303946 | Zbl 1312.35020

[6] Camilli, Fabio; Ley, Olivier; Loreti, Paola Homogenization of monotone systems of Hamilton–Jacobi equations, ESAIM, Control Optim. Calc. Var., Volume 16 (2010) no. 1, pp. 58-76 | Article | MR 2598088 | Zbl 1187.35008

[7] Cannarsa, Piermarco; Sinestrari, Carlo Semiconcave functions, Hamilton–Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser, 2004, xiv+304 pages | MR 2041617 | Zbl 1095.49003

[8] Cannarsa, Piermarco; Soner, Halil Mete Generalized one-sided estimates for solutions of Hamilton–Jacobi equations and applications, Nonlinear Anal., Theory Methods Appl., Volume 13 (1989) no. 3, pp. 305-323 | Article | MR 986450 | Zbl 0681.49030

[9] Clarke, Francis Functional analysis, calculus of variations and optimal control, Graduate Texts in Mathematics, 264, Springer, 2013, xiv+591 pages | Article | MR 3026831 | Zbl 1277.49001

[10] Clarke, Frank H.; Vinter, Richard B. Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Am. Math. Soc., Volume 289 (1985) no. 1, pp. 73-98 | Article | MR 779053 | Zbl 0563.49009

[11] Davini, Andrea; Fathi, Albert; Iturriaga, Renato; Zavidovique, Maxime Convergence of the solutions of the discounted Hamilton–Jacobi equation: convergence of the discounted solutions, Invent. Math., Volume 206 (2016) no. 1, pp. 29-55 | Article | MR 3556524 | Zbl 1362.35094

[12] Davini, Andrea; Siconolfi, Antonio A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations, SIAM J. Math. Anal., Volume 38 (2006) no. 2, pp. 478-502 | Article | MR 2237158 | Zbl 1109.49034

[13] Davini, Andrea; Siconolfi, Antonio; Zavidovique, Maxime Random Lax–Oleinik semigroups for Hamilton–Jacobi systems, J. Math. Pures Appl., Volume 120 (2018), pp. 294-333 | Article | MR 3906162 | Zbl 1415.35088

[14] Davini, Andrea; Zavidovique, Maxime Aubry sets for weakly coupled systems of Hamilton–Jacobi equations, SIAM J. Math. Anal., Volume 46 (2014) no. 5, pp. 3361-3389 | Article | MR 3265180 | Zbl 1327.35063

[15] Engler, Hans; Lenhart, Suzanne M. Viscosity solutions for weakly coupled systems of Hamilton–Jacobi equations, Proc. Lond. Math. Soc., Volume 63 (1991) no. 1, pp. 212-240 | Article | MR 1105722 | Zbl 0704.35030

[16] Fathi, Albert Sur la convergence du semi-groupe de Lax–Oleinik, C. R. Math. Acad. Sci. Paris, Volume 327 (1998) no. 3, pp. 267-270 | Article | MR MR1650261 | Zbl 1052.37514

[17] Ibrahim, H.; Siconolfi, Antonio; Zabad, Sahar Cycle characterization of the Aubry set for weakly coupled Hamilton–Jacobi systems, Commun. Contemp. Math., Volume 20 (2018) no. 6, 1750095, 28 pages (Art. ID 1750095, 28O pages) | Article | MR 3848071 | Zbl 06950442

[18] Ishii, Hitoshi; Mitake, Hiroyoshi; Tran, Hung V. The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, J. Math. Pures Appl., Volume 108 (2017) no. 2, pp. 125-149 | Article | MR 3670619 | Zbl 1375.35182

[19] Iturriaga, Renato; Sánchez-Morgado, Héctor Limit of the infinite horizon discounted Hamilton–Jacobi equation, Discrete Contin. Dyn. Syst., Volume 15 (2011) no. 3, pp. 623-635 | Article | MR 2774130 | Zbl 1217.37053

[20] Mitake, Hiroyoshi; Siconolfi, Antonio; Tran, Hung V.; Yamada, N. A Lagrangian approach to weakly coupled Hamilton–Jacobi systems, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 821-846 | Article | MR 3466199 | Zbl 1343.35065

[21] Roos, Valentine, 2013 (personal communication)

[22] Wei, Qiaoling Viscosity solution of the Hamilton–Jacobi equation by a limiting minimax method, Nonlinearity, Volume 27 (2014) no. 1, pp. 17-41 | Article | MR 3151090 | Zbl 1284.35120

Cité par Sources :