# One-scale H-measures and variants

Erceg, Marko (2016) One-scale H-measures and variants. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

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 $L^p$ sequences. Recently, Luc Tartar introduced one-scale H-measures, a generalisation of H-measures with a characteristic length, comprehending the notion of semiclassical measures. In order to better understand the use of one-scale H-measures, we start by studying more deeply the relation between H-measures and semiclassical measures. The new condition, ($\omega_n$)-concentrating property is introduced and we show that H-measures can be reconstructed from the semiclassical measures if the corresponding sequence is both ($\omega_n$)-oscillatory and concentrating, but also that such ($\omega_n$) does not necessarily exist. Most applications of semiclassical measures are related to a suitable variant of homogenization limit of partial differential equations, which we illustrate on a second order linear parabolic equation with a detailed analysis of different regimes of corresponding characteristic lengths. Furthermore, we present a comprehensive analysis of 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. Obtained results then we generalise to the $L^p$ setting via one-scale H-distributions, which are also generalisations of H-distributions, and derive a corresponding localization principle. Finally, we address some variants with and without characteristic length suitable for problems with different scaling among variables.