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Archive for Rational Mechanics and Analysis

, Volume 215, Issue 1, pp 1–63 | Cite as

Directional Oscillations, Concentrations, and Compensated Compactness via Microlocal Compactness Forms

  • Filip Rindler
Article

Abstract

This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in 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 L p -spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow one to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby differential constraints on the functions in the sequence can also 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.

Keywords

Pseudodifferential Operator Young Measure Strong Compactness Differential Constraint Gradient Young Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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