Generalized Multiscale Young Measures

This paper is devoted to the construction of generalized multiscale Young measures, which are the extension of Pedregal's multiscale Young measures [Trans. Amer. Math. Soc., 358 (2006), pp. 591-602] to the setting of generalized Young measures introduced by DiPerna and Majda [Comm. Math. Phys., 108 (1987), pp. 667-689]. As a tool for variational problems, these are well-suited objects for the study (at different length-scales) of oscillation and concentration effects of convergent sequences of measures. Important properties of multiscale Young measures such as compactness, representation of nonlinear compositions, localization principles, and differential constraints are extensively developed in the second part of this paper. As an application, we use this framework to address the Gamma-limit characterization of the homogenized limit of convex integrals defined on spaces of measures satisfying a general linear PDE-constraint.