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Microlocal defect measures. (English) Zbl 0770.35001
Summary: In order to study weak continuity of quadratic forms on spaces of \(L^ 2\) solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.
A systematic use of these measures leads in particular to a compensated compactness theorem, generalizing Murat-Tartar’s compensated compactness [F. Murat, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 69-102 (1981; Zbl 0464.46034); L. Tartar, Res. Notes Math. 39, 136-212 (1979; Zbl 0437.35004)] to variable coefficients and Golse-Lions-Perthme- Sentis’s averaging lemma [F. Golse, P.-L. Lions, B. Perthame and R. Sentis, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)].
We also obtain results on homogenization for differential operators of order 1 with oscillating coefficients.

35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text: DOI
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