Erceg, Marko (2016) Onescale Hmeasures and variants. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Abstract
Microlocal defect functionals (Hmeasures, Hdistributions, 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 onescale Hmeasures, a generalisation of Hmeasures with a characteristic length, comprehending the notion of semiclassical measures. In order to better understand the use of onescale Hmeasures, we start by studying more deeply the relation between Hmeasures and semiclassical measures. The new condition, ($\omega_n$)concentrating property is introduced and we show that Hmeasures 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 onescale Hmeasures, 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 Hmeasures 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 onescale Hdistributions, which are also generalisations of Hdistributions, and derive a corresponding localization principle. Finally, we address some variants with and without characteristic length suitable for problems with different scaling among variables.
Item Type:  Thesis (Doctoral thesis) 

Keywords:  Hmeasures, Wigner measures, Hdistributions, semiclassical limit, compactness by compensation 
Supervisor:  Antonić, Nenad 
Date:  2016 
Number of Pages:  122 
Subjects:  NATURAL SCIENCES > Mathematics NATURAL SCIENCES > Mathematics > Mathematical Analysis NATURAL SCIENCES > Mathematics > Other Mathematical Disciplines 
Additional Information:  MSC 2010: 35B27, 35K10, 35S05, 46F05, 46G10, 54D35 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  01 Jul 2016 10:44 
Last Modified:  01 Jul 2016 10:44 
URI:  http://digre.pmf.unizg.hr/id/eprint/4941 
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