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. © 2011 Society for Industrial and Applied Mathematics.