an:00028826
Zbl 0770.35001
GĂ©rard, Patrick
Microlocal defect measures
EN
Commun. Partial Differ. Equations 16, No. 11, 1761-1794 (1991).
0360-5302 1532-4133
1991
j
35A27 35S05 35B27 47F05
compensated compactness theorem; homogenization for differential operators; oscillating coefficients
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 [\textit{F. Murat}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 69-102 (1981; Zbl 0464.46034); \textit{L. Tartar}, Res. Notes Math. 39, 136-212 (1979; Zbl 0437.35004)] to variable coefficients and Golse-Lions-Perthme- Sentis's averaging lemma [\textit{F. Golse}, \textit{P.-L. Lions}, \textit{B. Perthame} and \textit{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.
0464.46034; 0437.35004; 0652.47031