Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in ℝ 2 /ℤ 2
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 /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.

@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 = {http://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. http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.015/

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