Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Alessio Figalli; Diogo A. Gomes; Diego Marcon

doi:10.1007/s00526-016-1016-5

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Alessio Figalli, Diogo A. Gomes, Diego Marcon

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Abstract

Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup. © 2016, Springer-Verlag Berlin Heidelberg.

Original language	English (US)
Journal	Calculus of Variations and Partial Differential Equations
Volume	55
Issue number	4
DOIs	https://doi.org/10.1007/s00526-016-1016-5
State	Published - Jun 22 2016

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: A. Figalli is partially supported by the NSF Grants DMS-1262411 and DMS-1361122. D. Gomes was partially supported by KAUST baseline and start-up funds. D. Marcon was partially supported by the UT Austin-Portugal partnership through the FCT doctoral fellowship SFRH/BD/33919/2009.

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title = "Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations",

abstract = "Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi{\textquoteright}s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup. {\textcopyright} 2016, Springer-Verlag Berlin Heidelberg.",

author = "Alessio Figalli and Gomes, {Diogo A.} and Diego Marcon",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: A. Figalli is partially supported by the NSF Grants DMS-1262411 and DMS-1361122. D. Gomes was partially supported by KAUST baseline and start-up funds. D. Marcon was partially supported by the UT Austin-Portugal partnership through the FCT doctoral fellowship SFRH/BD/33919/2009.",

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AU - Figalli, Alessio

AU - Gomes, Diogo A.

AU - Marcon, Diego

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: A. Figalli is partially supported by the NSF Grants DMS-1262411 and DMS-1361122. D. Gomes was partially supported by KAUST baseline and start-up funds. D. Marcon was partially supported by the UT Austin-Portugal partnership through the FCT doctoral fellowship SFRH/BD/33919/2009.

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N2 - Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup. © 2016, Springer-Verlag Berlin Heidelberg.

AB - Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup. © 2016, Springer-Verlag Berlin Heidelberg.

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JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 4

ER -

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

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