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 . 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 . 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--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}, volume = {27}, year = {2010}, pages = {837-856}, doi = {10.1016/j.anihpc.2009.11.015}, mrnumber = {2629882}, zbl = {1201.35089}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_3_837_0} }
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) pp. 837-856. doi : 10.1016/j.anihpc.2009.11.015. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_3_837_0/
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