Aubry-Mather measures in the nonconvex setting

F. Cagnetti*, Diogo Gomes, H. V. Tran

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 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 languageEnglish (US)
Pages (from-to)2601-2629
Number of pages29
JournalSIAM Journal on Mathematical Analysis
Volume43
Issue number6
DOIs
StatePublished - Dec 1 2011

Keywords

  • Adjoint method
  • Aubry-Mather theory
  • Nonconvex Hamiltonians
  • Weak KAM

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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