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Microlocal defect measures. (English) Zbl 0770.35001
Summary: In order to study weak continuity of quadratic forms on spaces of $L\sp 2$ 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 [{\it F. Murat}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 69-102 (1981; Zbl 0464.46034); {\it L. Tartar}, Res. Notes Math. 39, 136-212 (1979; Zbl 0437.35004)] to variable coefficients and Golse-Lions-Perthme- Sentis’s averaging lemma [{\it F. Golse}, {\it P.-L. Lions}, {\it B. Perthame} and {\it 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.

##### MSC:
 35A27 Microlocal methods; sheaf-theoretic methods (PDE) 35S05 General theory of pseudodifferential operators 35B27 Homogenization; equations in media with periodic structure (PDE) 47F05 Partial differential operators
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