# zbMATH — the first resource for mathematics

Microlocal defect measures. (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 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
Full Text:
##### References:
 [1] Bachelot A., C. R. A. S. Paris 299 pp 543– (1984) [2] DOI: 10.1016/0022-1236(82)90051-9 · Zbl 0487.58028 [3] DOI: 10.1002/cpa.3160420603 · Zbl 0698.35128 [4] Gérard P., Ann. Sci. Ecole Norm. Sup. 23 pp 89– (1990) [5] Gérard P., Exposé n{$$\deg$$} VI, Ecole Polyerchique 23 (1988) [6] Golse F., C. R. A. S. Paris 305 pp 801– (1987) [7] DOI: 10.1016/0022-1236(88)90051-1 · Zbl 0652.47031 [8] DOI: 10.1080/03605308508820384 · Zbl 0572.35032 [9] Hanouzet B., Research Notes in Mathematics 89 pp 208– (1983) [10] Hörmander L., The analysis of linear partial differential operators (1983) [11] Lions P. –L., Rev. Mat. Iberoamericana 1 pp 145– (1983) [12] Murat F., Ann. Scuola Norm. Sup. Pisa 5 pp 489– (1978) [13] Murat F., in Proceedings of the international meeting on recent methods in non–linear analysis (1979) [14] Murat F., Ann. Scuola Norm. Sup. Pisa 8 pp 69– (1981) [15] Tartar L., Res. Notes in Math. pp 136– (1979) [16] Tartar L., to appear in Proceedings of the Royal Society of Edinburgh 114 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.