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Generalized ergodic problems: existence and uniqueness structures of solutions. (English) Zbl 1430.35052
Summary: We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $$n$$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

##### MSC:
 35F21 Hamilton-Jacobi equations 35B10 Periodic solutions to PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B40 Asymptotic behavior of solutions to PDEs 35D40 Viscosity solutions to PDEs 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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