Aubry-Mather measures in the non convex setting

Details Aubry-Mather measures in the non convex setting Aubry-Mather measures in the non convex setting

2010-05-08

arxiv.org/abs/1005.1317v2

SIAM J. Math. Anal. 43, pp. 2601-2629, 2011

Mathematics

Analysis of PDEs

final version

Scientific paper

10.1137/100817656

The adjoint method introduced in [Eva] and [Tra] 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 semidefinite matrix of Borel measures. However, in the important case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.

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