Generalized Mather problem and selection principles for viscosity solutions and mather measures

Abstract In this paper we study the generalized Mather problem, whose applications range from classical mechanics to stochastic control and discrete dynamics, and present some applications to selection principles for viscosity solutions and Mather measures. In the first part of the paper we discuss the motivation of the problem, use duality theory to establish its main properties, and study the support of generalized Mather measures. Our framework unifies several well-known problems, such as the ones studied in [D. Gomes, A stochastic analogue of Aubry–Mather theory, Nonlinearity 15 (2002), pp. 581–603. MR 1901094 (2003b:37096)] or [D. Gomes, Viscosity Solution Methods and the Discrete Aubry–Mather Problem, Discrete and Continuous Dynamical Systems 13 (2005), pp. 103–116.], and includes new ones such as the discounted cost infinite horizon. This last problem, as well as other perturbation problems are studied in the second part of the paper in the setting of this unified framework. More precisely, in the second part, we study perturbation problems for generalized Mather measures and corresponding singular limits of Hamilton–Jacobi equations. In particular, we establish a selection criterion for limits of sequences of Mather measures and viscosity solutions, which allows to determine possible limiting viscosity solutions and corresponding Mather measures. We apply our results to the vanishing discount cost infinite horizon problem, and also present an application to the vanishing viscosity problem.