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MATHER measures for space–time periodic nonconvex Hamiltonians

  • Eddaly Guerra-VelascoEmail author
Original Article
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Abstract

In Gomes (Nonlinearity 15(3):581–603, 2002) developed techniques and tools with the purpose of extending the Aubry–Mather theory in a stochastic setting, namely he proved the existence of stochastic Mather measures and their properties. These results were also extended in the time-dependent setting in the doctoral thesis of the Guerra-Velasco (http://132.248.9.195/ptd2015/abril/506015252/Index.html, 2015). However, to construct analogs to the Aubry–Mather measures for nonconvex Hamiltonians, it is necessary to use the adjoint method introduced by Evans (Arch Ration Mech Anal 197:1053–1088, 2010) and Tran (Calc Var Partial Differ Equ 41:301–319, 2011); the construction of the measures is in Cagnetti et al. (SIAM J Math Anal 43(6):2601–262, citelink2011CGT). The main goal of this paper is to construct Mather measures for space–time periodical nonconvex Hamiltonians using the techniques in [10, 21] and [7]. Moreover, we also will prove that there is only one value, such that the viscous Hamilton–Jacobi equation has a smooth periodic solution unique up to an additive constant.

Keywords

Hamilton–Jacobi Non-convex Periodic Hamiltonians 

Mathematics Subject Classification

37J50 49L25 

Notes

Acknowledgements

The author acknowledges the referees for their valuable suggestions and remarks that substantially improved this article. The author is grateful to Héctor Sánchez Morgado for his suggestions and to Boris Percino Figueroa for his support in the process of this research.

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Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  1. 1.Facultad de Ciencias en Física y MatemáticasCONACYT-Universidad Autónoma de ChiapasMéxicoMexico

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