Microlocal defect measures. (English) Zbl 0770.35001

Gérard, Patrick

Commun. Partial Differ. Equations 16, No. 11, 1761-1794 (1991).

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 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

Keywords:

compensated compactness theorem; homogenization for differential operators; oscillating coefficients

Citations:

Zbl 0464.46034; Zbl 0437.35004; Zbl 0652.47031

References:

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