*(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.

##### 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 |