### Abstract

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by u ^{λ} the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family (Figure presented.) to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Royal Society of Edinburgh - Proceedings A |

DOIs | |

Publication status | Accepted/In press - 2016 Mar 3 |

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### Keywords

- asymptotic analysis
- ergodic problems
- Hamilton–Jacobi equations
- Mather measures
- weak Kolmogorov–Arnold–Moser theory

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Royal Society of Edinburgh - Proceedings A*, 1-18. https://doi.org/10.1017/S0308210515000517