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Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. (English) Zbl 1050.35012
The most important step in the analysis of the homogenization problem is the identification of the effective (averaged) Hamiltonian, which is usually accomplished by studying the cell problem, an auxiliary equation arising in the formal expansion. The cell problem can be thought as a nonlinear eigenvalue problem with the effective Hamiltonian and the solution or approximate solution playing, respectively, the roles of the eigenvalue and the eigenfunction or approximate eigenfunction. It is also expected that the fine properties of the corrector may help to obtain more precise information about the averaged (effective) Hamiltonian.
The existence of exact correctors is the topic of this note. More precisely, the authors show here that, in general, there do not exist correctors with the necessary behavior to guarantee the uniqueness of the effective Hamiltonian. They then state a necessary and sufficient condition for the existence of appropriate correctors and present several examples that shed light onto the problem. The authors adopt and extend the terminology of the Mather theory for periodic dynamical systems to the stationary ergodic setting.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F20 Nonlinear first-order PDEs
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