2012/12/06 by Filip Rindler, Rindler, Filip
Computer Science · Engineering · Mathematics · #28B05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1212.1430
openalex publication_date 2012/12/06 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
This work introduces microlocal compactness forms (MCFs) as a new tool to\nstudy oscillations and concentrations in \Lp-bounded sequences of\nfunctions. Decisively, MCFs retain information about the location, value\ndistribution, and direction of oscillations and concentrations, thus extending\nat the same time the theories of (generalized) Young measures and H-measures.\nIn \Lp-spaces oscillations and concentrations precisely discriminate\nbetween weak and strong compactness, and thus MCFs allow to quantify the\ndifference in compactness. The definition of MCFs involves a Fourier variable,\nwhereby also differential constraints on the functions in the sequence can be\ninvestigated easily - a distinct advantage over Young measure theory.\nFurthermore, pointwise restrictions are reflected in the MCF as well, paving\nthe way for applications to Tartar's framework of compensated compactness;\nconsequently, we establish a new weak-to-strong compactness theorem in a\n"geometric" way. After developing several aspects of the abstract theory, we\nconsider three applications: For lamination microstructures, the hierarchy of\noscillations is reflected in the MCF. The directional information retained in\nan MCF is harnessed in the relaxation theory for anisotropic integral\nfunctionals. Finally, we indicate how the theory pertains to the study of\npropagation of singularities in certain systems of PDEs. The proofs combine\nmeasure theory, Young measures, and harmonic analysis.\n