engleski

Localisation principle for 1-scale H-measures

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. In contrast to the semiclassical measures, H-measures are not suitable to treat problems with a characteristic length (e.g.~thickness of a plate). Luc Tartar overcame the mentioned restriction by introducing 1-scale H-measures, a generalisation of H-measures with a characteristic length. Moreover, these objects are also an extension of semiclassical measures, being functionals on continuous functions on a compactification of $R^d\setminus\{; ; ; 0\}; ; ; $. We improve and generalise Tartar's localisation principle for 1-scale H-measures from which we are able to derive known localisation principles for H-measures and semiclassical measures. The localisation principle for H-measures has already been successfully applied in many fields (compactness by compensation, small amplitude homogenisation, velocity averaging, averaged control etc.), and the new results expected to have an even wider class of possible applications.

H-measures ; semiclassical ; 1-scale ; localisation principle

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