an:05718430
Zbl 1201.35089
Cardaliaguet, Pierre
Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in \(\mathbb R^2/\mathbb Z^2\)
EN
Ann. Inst. Henri Poincar??, Anal. Non Lin??aire 27, No. 3, 837-856 (2010).
00262173
2010
j
35F21 35F20 49J15 35R15
time average; resonance; non-resonance
The author investigates the long time average behavior of the solutions of the Hamilton-Jacobi equations with a noncoercive, nonconvex Hamiltonian on the torus \(\mathbb{R}^2/\mathbb{Z}^2\). Following an approach initiated by \textit{M. Arisawa} and \textit{P.-L. Lions} [Commun. Partial Differ. Equations 23, No. 11-12, 2187--2217 (1998; Zbl 1126.93434)], the author gives nonresonance conditions under which the long time average converges to a constant. The main idea in the proof consists of establishing some rigidity properties of the solutions. The author shows that in the resonant case the limit still exists, although it is not constant in general. He also computes the limit at points where it is not locally constant. The paper ends with some statements of open problems.
Athanase Papadopoulos (Strasbourg)
Zbl 1126.93434