### 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 |

### Fingerprint

### 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

**A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions.** / Al-Aidarous, Eman S.; Alzahrani, Ebraheem O.; Ishii, Hitoshi; Younas, Arshad M M.

Research output: Contribution to journal › Article

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

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TY - JOUR

T1 - A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

AU - Al-Aidarous, Eman S.

AU - Alzahrani, Ebraheem O.

AU - Ishii, Hitoshi

AU - Younas, Arshad M M

PY - 2016/3/3

Y1 - 2016/3/3

N2 - 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.

AB - 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.

KW - asymptotic analysis

KW - ergodic problems

KW - Hamilton–Jacobi equations

KW - Mather measures

KW - weak Kolmogorov–Arnold–Moser theory

UR - http://www.scopus.com/inward/record.url?scp=84960115647&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960115647&partnerID=8YFLogxK

U2 - 10.1017/S0308210515000517

DO - 10.1017/S0308210515000517

M3 - Article

SP - 1

EP - 18

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

ER -