engleski

One-scale H-measures, variants and applications

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 in his recent book overcame the mentioned restriction by introducing one-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\{; ; ; ; ; 0}; ; ; ; ; . We improve and generalise Tartar's localisation principle for one-scale H-measures 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. Since one-scale H-measures are adequate only for the L^2 framework, we introduce the generalisation, one-scale H-distributions, as a counterpart of H-distributions with a characteristic length, and address some important features.

H-measures ; localisation principle ; compensated compactness

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