"Microlocal defect measures and wave-front sets"Vojnović, IvanaJoint work with Ivan Ivec. Following the unpublished ideas of Folkmar Bornemann [3], we introduce notion of microlocal compactness wave front sets. They describe how far is a bounded sequence in a Sobolev space from being relatively compact. We distinguish local and nonlocal variants. Important property is a connection between microlocal defect functionals and wave front sets, that is connection with H-measures and H-distributions. H-measures are introduced independently by Luc Tartar (in [5]) and Patrick Gerard (in [4]). H-distributions as their generalization are introduced in [2] and further developed in [1]. Microlocal defect functionals act on spaces of symbols of appropriate pseudodifferential operators and they are associated to given weakly convergent sequence in some Sobolev space. We will give existence theorems and main properties of these functionals. Finally, we will show that support of H-measure for a given weakly con- vergent sequence (un) is equal to microlocal wave front set of that sequence. References [1] Aleksic, J.; Pilipovic, S.; Vojnovic, I. H - distributions via Sobolev spaces, Mediterranean Journal of Mathematics, 2016., pp 1-14. [2] Antonic, N.; Mitrovic, D., H-distributions: an extension of H-measures to an Lp - Lq setting, Abstr. Appl. Anal. 2011, Art. ID 901084, 12 pp. [3] Bornemann F., A note on microlocal obstructions to the compactness of sequences, unpublished paper. [4] Gerard, P. Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), no. 11. [5] Tartar, L., H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115(1990), no. 3-4, 193–230. |
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