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 |
Keywords:
contact Hamilton-Jacobi equations; nonlinear adjoint method; nonuniqueness of solutions; uniqueness structuresReferences:
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