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H-measures applied to symmetric systems*

Published online by Cambridge University Press:  14 November 2011

Nenad Antonić
Nenad Antonić
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, Zagreb, Croatia e-mail: nantonic@srce.hr
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Abstract

H-measures were recently introduced by Luc Tartar as a tool which might provide better understanding of propagating oscillations. Independently, Patrick Gerard introduced the same objects under the name of microlocal defect measures. Partial differential equations of mathematical physics can often be written in the form of a symmetric system:

where Ak and B are matrix functions, while u is an unknown vector function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems. This result, combined with the localisation property, is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation, Maxwell's and Dirac's systems, and second-order equations in two variables.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1133 - 1155
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Antonić, N.. Memory effects in homogenisation: linear second-order equations. Arch. Rational Mech. Anal. 125 (1993), 1–24.CrossRefGoogle Scholar
2Evans, L. C.. Weak convergence methods for Nonlinear Partial Differential Equations, CBMS 74 (Providence, RI: American Mathematical Society, 1990).CrossRefGoogle Scholar
3Francfort, G. and Murat, F.. Oscillations and energy densities in the wave equation. Comm. Partial Differential Equations 17 (1992), 1785–865.CrossRefGoogle Scholar
4Friedrichs, K. O.. Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7 (1954), 345–92.CrossRefGoogle Scholar
5Friedrichs, K. O.. Symmetric positive linear differential equations. Comm. Pure Appl. Math. 11 (1958), 333–418.Google Scholar
6Gérard, P.. Compacité par compensation et régularité 2-microlocale. In Séminaire Equations aux Dérivées Partielles 1988–89 (Ecole Polytechnic, Palaiseau, exp. VI).Google Scholar
7Gérard, P.. Microlocal defect measures. Comm. Partial Differential Equations 16 (1991), 1761–94.Google Scholar
8Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1983).Google Scholar
9Hörmander, L.. The Analysis of Linear Partial Differential Operators I–IV (Berlin: Springer, 1983–1985).Google Scholar
10Morawetz, C. S.. A uniqueness theorem for Frankl's problem. Comm. Pure Appl. Math. 11 (1958), 697–703.Google Scholar
11Tartar, L.. Remarks on homogenization. In Homogenization and effective moduli of materials and media, eds Eriksen, J. L. et al. , IMA 1, 228–46 (New York: Springer, 1986).CrossRefGoogle Scholar
12Tartar, L.. H-measures, a new approach for studying homogenisation, oscillations, and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 193–230.Google Scholar
13Tartar, L.. H-measures and applications. Proceedings of the International Congress of Mathematicians, Kyoto, 1990, Vol. 2, 1215–23 (Tokyo: Mathematical Society of Japan and Springer-Verlag, 1991).Google Scholar
14Trèves, F.. Introduction to pseudodifferential and Fourier integral operators (New York: Plenum Press, 1980).CrossRefGoogle Scholar
15Tricomi, F.. Sulle equazioni alle derivate parziali di 20 ordine, di tipo misto. Mem. R. Accad. Nazionale Linzei, ser. V 14 (1923), fasc. VII.Google Scholar
16Weinstein, A.. The singular solutions and the Cauchy problem for generalised Tricomi equations. Comm. Pure Appl. Math. 7 (1954), 105–16.Google Scholar