Aubry-Mather Measures in the Nonconvex Setting
Aubry-Mather Measures in the Nonconvex Setting
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非凸环境中的奥布里-马瑟测度
DOI：
10.1137/100817656
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
H. Tran
中科院分区：
文献类型：
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作者：
F. Cagnetti;Diogo Gomes;H. Tran
在[L. C.埃文斯，阿奇。定量。机械肛门，197（2010），pp. 1053-1088]和[H. V. Tran，Calc.变种偏微分方程，41（2011），pp。301-319]，被用来构建类似的Aubry-Mather措施的非凸哈密尔顿。更确切地说，概率测度的一般结构，这在凸设置同意Mather措施，提供。这些措施可能无法在哈密顿流下是不变的，并出现耗散，这是由一个半正定矩阵的Borel措施。然而，在一致拟凸哈密顿的情况下，耗散消失，因此不变性是有保证的。
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.