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

Part of: General first-order equations and systems Representations of solutions Partial differential equations Hamilton-Jacobi theories, including dynamic programming Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  03 March 2016

Eman S. Al-Aidarous ,
Ebraheem O. Alzahrani ,
Hitoshi Ishii  and
Arshad M. M. Younas
Eman S. Al-Aidarous
Affiliation:
Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia (ealaidarous@kau.edu.sa)
Ebraheem O. Alzahrani
Affiliation:
Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia (eoalzahrani@kau.edu.sa)
Hitoshi Ishii
Affiliation:
Waseda University, Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan (hitoshi.ishii@waseda.jp)
Arshad M. M. Younas
Affiliation:
Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia (arshadm@kau.edu.sa)
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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 uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family 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.

Keywords

Hamilton–Jacobi equationsergodic problemsasymptotic analysisweak Kolmogorov–Arnold–Moser theoryMather measures

MSC classification

Primary: 35F21: Hamilton-Jacobi equations
Secondary: 35B40: Asymptotic behavior of solutions 35D40: Viscosity solutions 35F30: Boundary value problems for nonlinear first-order equations 37J50: Action-minimizing orbits and measures 49L25: Viscosity solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 2 , April 2016 , pp. 225 - 242
Copyright
Copyright © Royal Society of Edinburgh 2016 

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