AUBRY-MATHER MEASURES IN THE NONCONVEX SETTING

作者：Cagnetti F^*; Gomes D; Tran H V

来源：SIAM Journal on Mathematical Analysis, 2011, 43(6): 2601-2629.

DOI：10.1137/100817656

• Aubry-Mather theory
• weak KAM
• nonconvex Hamiltonians
• adjoint method

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. 301319], 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.

• 出版日期2011