Localisation principle for one-scale H-measures

TL;DR

This paper introduces and analyzes one-scale H-measures, a generalization of microlocal defect functionals with a characteristic length, providing new proofs, strengthening results, and extending localization principles.

Contribution

It presents a comprehensive introduction to one-scale H-measures, improves localization principles, and develops a variant of compactness by compensation for equations with a characteristic length.

Findings

Strengthened localization principle for one-scale H-measures.

Generalized results to include semiclassical measures.

Developed a variant of compactness by compensation.

Abstract

Microlocal defect functionals (H-measures, H-distributions, semiclassical measures, etc.) are objects which determine, in some sense, the lack of strong compactness for weakly convergent ${\rm L}^p$ sequences. Recently, Luc Tartar introduced one-scale H-measures, a generalisation of H-measures with a characteristic length, which also comprehend the notion of semiclassical measures. We present a self-contained introduction to one-scale H-measures, carrying out some alternative proofs, and strengthening some results, comparing these objects to known microlocal defect functionals. Furthermore, we improve and generalise Tartar's localisation principle for these objects from which we are able to derive the known localisation principles for both H-measures and semiclassical measures. Moreover, we develop a variant of compactness by compensation suitable for equations with a characteristic length.