Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms

TL;DR

This paper introduces microlocal compactness forms (MCFs), a new analytical tool that captures detailed oscillation and concentration behaviors in sequences of functions, extending existing theories like Young measures and H-measures.

Contribution

The paper develops the theory of MCFs, enabling precise analysis of oscillations, concentrations, and differential constraints, with applications to compensated compactness and PDE singularities.

Findings

MCFs extend Young measures and H-measures by including directional and location information.

A new weak-to-strong compactness theorem is established using MCFs.

Applications include analysis of microstructures, anisotropic functionals, and PDE singularities.

Abstract

This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily - a distinct advantage over Young measure theory. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. After developing several aspects of the abstract theory, we consider three applications: For lamination microstructures, the hierarchy of oscillations is reflected in the MCF. The directional information retained in an MCF is harnessed in the relaxation theory for anisotropic integral functionals. Finally, we indicate how the theory pertains to the study of propagation of singularities in certain systems of PDEs. The proofs combine measure theory, Young measures, and harmonic analysis.