Abstract
Weak KAM theory for discounted Hamilton–Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of \(\alpha \)-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of \(\alpha \)-limit points is effectively exploited with properties of the corresponding dynamical systems.
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Acknowledgements
The authors would like to thank Professor Albert Fathi for valuable comments on this work. The authors are grateful to Professors Diogo A. Gomes, Yifeng Yu and one of the referees for valuable comments for the previous version of the manuscript. The first author is partially supported by JSPS Grants: KAKENHI #15K17574, #26287024, #16H03948. The second author is partially supported by JSPS Grant-in-aid for Young Scientists (B) #15K21369.
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Communicated by Y. Giga.
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Mitake, H., Soga, K. Weak KAM theory for discounted Hamilton–Jacobi equations and its application. Calc. Var. 57, 78 (2018). https://doi.org/10.1007/s00526-018-1359-1
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DOI: https://doi.org/10.1007/s00526-018-1359-1