MATHER measures for space–time periodic nonconvex Hamiltonians

Abstract

In Gomes (Nonlinearity 15(3):581–603, 2002) developed techniques and tools with the purpose of extending the Aubry–Mather theory in a stochastic setting, namely he proved the existence of stochastic Mather measures and their properties. These results were also extended in the time-dependent setting in the doctoral thesis of the Guerra-Velasco (http://132.248.9.195/ptd2015/abril/506015252/Index.html, 2015). However, to construct analogs to the Aubry–Mather measures for nonconvex Hamiltonians, it is necessary to use the adjoint method introduced by Evans (Arch Ration Mech Anal 197:1053–1088, 2010) and Tran (Calc Var Partial Differ Equ 41:301–319, 2011); the construction of the measures is in Cagnetti et al. (SIAM J Math Anal 43(6):2601–262, citelink2011CGT). The main goal of this paper is to construct Mather measures for space–time periodical nonconvex Hamiltonians using the techniques in [10, 21] and [7]. Moreover, we also will prove that there is only one value, such that the viscous Hamilton–Jacobi equation has a smooth periodic solution unique up to an additive constant.