Calculus of Variations and Geometric Measure Theory
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F. Cagnetti - D. Gomes - H. V. Tran

Aubry-Mather Measures in the Non Convex setting

created by cagnetti on 24 May 2010
modified on 13 Nov 2013


Published Paper

Inserted: 24 may 2010
Last Updated: 13 nov 2013

Journal: SIAM J. Math. Anal.
Volume: 43
Pages: 2601-2629
Year: 2011


The adjoint method, introduced in Evans and Tran, is used to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that 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 semi-definite matrix of Borel measures. However, in the important case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed

Keywords: Aubry-Mather theory, weak KAM, non convex Hamiltonians, adjoint method


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