engleski

Characteristic length of sequences via one-scale H-measures

By Vitali's theorem, the lack of strong convergence of L^2 bounded sequences is caused by oscillations and/or concentration effects, which motivates the study of these effects. Although the direction of oscillations and the point of concentrations can be detected using H-measures (also called microlocal defect measures), sometimes this is not satisfactory since we are also interested in the scale of this phenomena (e.q. frequency of oscillations). We introduce (\omega_n)-concentrating property which, together with the known (\omega_n)-oscillating property, can give a full picture of the scale of observed sequences. These conditions also illustrate when H-measures and semiclassical measures (also called Wigner measures) coincide. Furthermore, we would like to present a more recent variant of microlocal defect funcionals, one-scale H-measures, which were introduced as a generalisation of H-measures with a characteristic length, being also an extension of semiclassical measures. Using the framework of one-scale H-measures, we develop a variant of compensated compactness suitable for partial differential equations with a characteristic length.

oscillations ; semiclassical ; one-scale H-measures ; compensated compactness

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