Inserted: 24 may 2010
Last Updated: 13 nov 2013
Journal: SIAM J. Math. Anal.
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