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

Cardaliaguet, Pierre

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

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 ${ℝ}^2/{\mathbb{Z}}^2$. 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 ${ℝ}^2/{\mathbb{Z}}^2$. 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.

[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