Summary: In order to study weak continuity of quadratic forms on spaces of 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.