Minimax probabilities for Aubry-Mather problems

Diogo A. Gomes; Nara Jung; Artur O. Lopes

doi:10.1142/S0219199710004019

Minimax probabilities for Aubry-Mather problems

Diogo A. Gomes, Nara Jung, Artur O. Lopes

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem, we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First, we give some examples showing that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are T-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.

Original language	English (US)
Pages (from-to)	789-813
Number of pages	25
Journal	Communications in Contemporary Mathematics
Volume	12
Issue number	5
DOIs	https://doi.org/10.1142/S0219199710004019
State	Published - Oct 2010
Externally published	Yes

Bibliographical note

Funding Information:
D. Gomes was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems through FCT Program POCTI/FEDER and also by grant DENO/FCT-PT (PTDC/EEA-ACR/67020/2006). A. O. Lopes was partially supported by CNPq, PRONEX — Sistemas Dinâmicos, Instituto do Milênio, and is beneficiary of CAPES financial support.

Keywords

Aubry-Mather measures
Lagrangian cost
Minimax measures
discrete Aubry-Mather problem
duality
holonomic measures

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1142/S0219199710004019

Cite this

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title = "Minimax probabilities for Aubry-Mather problems",

abstract = "In this paper, we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem, we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First, we give some examples showing that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are T-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.",

keywords = "Aubry-Mather measures, Lagrangian cost, Minimax measures, discrete Aubry-Mather problem, duality, holonomic measures",

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N2 - In this paper, we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem, we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First, we give some examples showing that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are T-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.

AB - In this paper, we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem, we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and non-existence results. First, we give some examples showing that in continuous time stationary minimax Mather measures are either trivial or fail to exist. A more natural definition in continuous time are T-periodic minimax Mather measures. We give a complete characterization of these measures and discuss several examples.

KW - Aubry-Mather measures

KW - Lagrangian cost

KW - Minimax measures

KW - discrete Aubry-Mather problem

KW - duality

KW - holonomic measures

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JO - Communications in Contemporary Mathematics

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IS - 5

ER -

Minimax probabilities for Aubry-Mather problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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