# Generalized multi-scale Young measures

## TL;DR

This paper develops generalized multi-scale Young measures to analyze oscillation and concentration effects in variational problems, extending previous measures to a multi-scale setting and applying them to homogenization of PDE-constrained measures.

## Contribution

It introduces a new framework of generalized multi-scale Young measures, extending Pedregal's measures to handle multiple scales and PDE constraints, with comprehensive properties and applications.

## Key findings

- Established compactness and representation properties of the measures
- Extended the framework to include nonlinear compositions and localization
- Applied to characterize the Gamma-limit of homogenized convex integrals

## Abstract

This paper is devoted to the construction of generalized multi-scale Young measures, which are the extension of Pedregal's multi-scale 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 multi-scale Young measures such as compactness, representation of non-linear 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.