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Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in \(\mathbb R^2/\mathbb Z^2\). (English) Zbl 1201.35089
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 M. Arisawa and 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.

MSC:
35F21 Hamilton-Jacobi equations
35F20 Nonlinear first-order PDEs
49J15 Existence theories for optimal control problems involving ordinary differential equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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