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Perturbation estimates of weak KAM solutions and minimal invariant sets for nearly integrable Hamiltonian systems


Authors: Qinbo Chen and Min Zhou
Journal: Proc. Amer. Math. Soc. 145 (2017), 201-214
MSC (2010): Primary 37Jxx, 70Hxx
DOI: https://doi.org/10.1090/proc/13193
Published electronically: June 30, 2016
MathSciNet review: 3565373
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Abstract: For nearly integrable and Tonelli system

$\displaystyle H_{\epsilon }=H_0(p)+\epsilon H_1(q,p,t). \quad (q,p,t)\in \mathbb{T}^n\times \mathbb{R}^n\times \mathbb{T},$

we give the perturbation estimates of weak KAM solution $ u_{\epsilon }$ with respect to parameter $ \epsilon $ and prove the stability of the Mather set $ \tilde {\mathcal {M}}_\epsilon $, Aubry set $ \tilde {\mathcal {A}}_\epsilon $, Mañé set $ \tilde {\mathcal {N}}_\epsilon $ and even the backward (forward) calibrated curves under the perturbation.

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Additional Information

Qinbo Chen
Affiliation: Department of Mathematics, Nanjing University, Nanjing, Jiangsu, People’s Republic of China 210093
Email: qinboChen1990@gmail.com

Min Zhou
Affiliation: School of Information Management, Nanjing University, Nanjing, Jiangsu, People’s Republic of China 210093
Email: minzhou@nju.edu.cn

DOI: https://doi.org/10.1090/proc/13193
Keywords: Weak KAM solutions, perturbation estimates, Mather theory, minimal invariant sets
Received by editor(s): December 7, 2015
Received by editor(s) in revised form: March 7, 2016
Published electronically: June 30, 2016
Additional Notes: The authors were supported by the National Basic Research Program of China (973 Program) (Grant No. 2013CB834100), the National Natural Science Foundation of China (Grant No. 11171146, Grant No. 11201222) and a program PAPD of Jiangsu Province, China.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society