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{\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/} }
TY - JOUR AU - Cardaliaguet, Pierre TI - Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 837 EP - 856 VL - 27 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.015/ DO - 10.1016/j.anihpc.2009.11.015 LA - en ID - AIHPC_2010__27_3_837_0 ER -
%0 Journal Article %A Cardaliaguet, Pierre %T Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 837-856 %V 27 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2009.11.015/ %R 10.1016/j.anihpc.2009.11.015 %G en %F 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) 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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