Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$

Cardaliaguet, Pierre

doi:10.1016/j.anihpc.2009.11.015

Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in

ℝ^{2} / ℤ^{2}

Cardaliaguet, Pierre

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 837-856.

Résumé
Abstract

Nous considérons le comportement en temps grand de la moyenne temporelle de solutions d'équations de Hamilton–Jacobi pour un hamiltonien non convexe et non coercif dans le tore $ℝ^{2} / ℤ^{2}$ . Nous mettons en évidence des conditions de non-résonnance sous lesquelles cette moyenne converge vers une constante. Dans le cas où il y a résonnance, nous montrons que la limite existe, bien qu'étant non constante en général. Nous calculons la limite aux points où celle-ci est non localement constante.

The paper investigates the long time average of the solutions of Hamilton–Jacobi equations with a noncoercive, nonconvex Hamiltonian in the torus $ℝ^{2} / ℤ^{2}$ . We give nonresonance conditions under which the long-time average converges to a constant. In the resonant case, we show that the limit still exists, although it is nonconstant in general. We compute the limit at points where it is not locally constant.

MR Zbl

DOI : 10.1016/j.anihpc.2009.11.015

@article{AIHPC_2010__27_3_837_0,
     author = {Cardaliaguet, Pierre},
     title = {Ergodicity of {Hamilton{\textendash}Jacobi} equations with a noncoercive nonconvex {Hamiltonian} in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {837--856},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.11.015},
     mrnumber = {2629882},
     zbl = {1201.35089},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.11.015/}
}

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Cardaliaguet, Pierre. Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 837-856. doi : 10.1016/j.anihpc.2009.11.015. https://www.numdam.org/articles/10.1016/j.anihpc.2009.11.015/

Bibliographie
Cité par

[1] O. Alvarez, M. Bardi, Ergodic problems in differential games, S. Jorgensen, M. Quincampoix, T.L. Vincent (ed.), Advances in Dynamic Game Theory, Ann. Internat. Soc. Dynam. Games vol. 9, Birkhäuser, Boston (2007), 131-152

[2] O. Alvarez, M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman–Isaacs equations, Mem. Amer. Math. Soc., in press | MR

[3] O. Alvarez, H. Ishii, Hamilton–Jacobi equations with partial gradient and application to homogenization, Comm. Partial Differential Equations 26 no. 5–6 (2001), 983-1002 | MR | Zbl

[4] M. Arisawa, P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations 23 no. 11–12 (1998), 2187-2217 | MR | Zbl

[5] Z. Artstein, V. Gaitsgory, The value function of singularly perturbed control systems, Appl. Math. Optim. 41 no. 3 (2000), 425-445 | MR | Zbl

[6] M. Bardi, On differential games with long-time-average cost, Advances in Dynamic Games and Their Applications, Ann. Internat. Soc. Dynam. Games vol. 10, Birkhäuser Boston, Boston, MA (2009), 3-18 | MR | Zbl

[7] G. Barles, Some homogenization results for non-coercive Hamilton–Jacobi equations, Calc. Var. Partial Differential Equations 30 no. 4 (2007), 449-466 | MR | Zbl

[8] I. Birindelli, J. Wigniolle, Homogenization of Hamilton–Jacobi equations in the Heisenberg group, Commun. Pure Appl. Anal. 2 no. 4 (2003), 461-479 | MR | Zbl

[9] M.G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67 | Zbl

[10] D.A. Gomes, Hamilton–Jacobi methods for vakonomic mechanics, NoDEA Nonlinear Differential Equations Appl. 14 no. 3–4 (2007), 233-257 | MR | Zbl

[11] C. Imbert, M. Monneau, Homogenization of first-order equations with $u / ϵ$ -periodic Hamiltonians. Part I: Local equations, Arch. Ration. Mech. Anal. 187 no. 1 (2008), 49-89 | MR | Zbl

[12] P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton–Jacobi equations, unpublished work

[13] M. Quincampoix, J. Renault, On the existence of a limit value in some non-expansive optimal control problems, preprint hal-00377857 | MR | Zbl

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