Aubry-Mather measures in the nonconvex setting

F. Cagnetti; D. Gomes; H. V. Tran

doi:10.1137/100817656

Aubry-Mather measures in the nonconvex setting

F. Cagnetti^*, D. Gomes, H. V. Tran

^*Corresponding author for this work

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

17 Scopus citations

Abstract

The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053-1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301-319], is used to construct analogues to the Aubry-Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.

Original language	English (US)
Pages (from-to)	2601-2629
Number of pages	29
Journal	SIAM Journal on Mathematical Analysis
Volume	43
Issue number	6
DOIs	https://doi.org/10.1137/100817656
State	Published - 2011
Externally published	Yes

Keywords

Adjoint method
Aubry-Mather theory
Nonconvex Hamiltonians
Weak KAM

ASJC Scopus subject areas

Analysis
Computational Mathematics
Applied Mathematics

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10.1137/100817656

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title = "Aubry-Mather measures in the nonconvex setting",

abstract = "The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053-1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301-319], is used to construct analogues to the Aubry-Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.",

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T1 - Aubry-Mather measures in the nonconvex setting

AU - Cagnetti, F.

AU - Gomes, D.

AU - Tran, H. V.

PY - 2011

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N2 - The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053-1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301-319], is used to construct analogues to the Aubry-Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.

AB - The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053-1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301-319], is used to construct analogues to the Aubry-Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.

KW - Adjoint method

KW - Aubry-Mather theory

KW - Nonconvex Hamiltonians

KW - Weak KAM

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JF - SIAM Journal on Mathematical Analysis

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Aubry-Mather measures in the nonconvex setting

Abstract

Keywords

ASJC Scopus subject areas

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