A local proof for the characterization of Young measures generated by sequences in BV

TL;DR

This paper provides a new, localized proof for the characterization of Young measures generated by BV sequences, utilizing advanced localization and Hahn-Banach techniques, and introduces homogeneous Young measures at different points.

Contribution

It offers a novel proof avoiding relaxation theorems, introduces homogeneous Young measures at regular and singular points, and demonstrates how to generate sequences respecting atomic parts.

Findings

New proof technique based on localization and Hahn-Banach

Introduction of homogeneous Young measures at different points

Ability to generate sequences respecting atomic parts in BV-Young measures

Abstract

This work presents a new proof of the recent characterization theorem for generalized Young measures generated by sequences in BV by Kristensen and the author [Arch. Ration. Mech. Anal. 197 (2010), 539-598]. The present argument is based on a localization technique together with a local Hahn-Banach argument in novel function spaces combined with an application of Alberti's Rank-One Theorem. This strategy avoids employing a relaxation theorem as in the previously known proof, and the new tools introduced in its course should prove useful in other contexts as well. In particular, we introduce homogeneous Young measures, separately at regular and singular points, which exhibit rather different behaviour than the classical homogeneous Young measures. As an application, we show how for BV-Young measures with an atomic part one can find a generating sequence respecting this structure.