Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T14:00:18.615Z Has data issue: false hasContentIssue false
  • English
  • Français

H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations*

Published online by Cambridge University Press:  14 November 2011

Luc Tartar
Luc Tartar
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213–3890, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

New mathematical objects, called H-measures, are introduced for studying oscillations and concentration effects in partial differential equations. Applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 193 - 230
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Avellaneda, M.. Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. 40 (1987), 527–554.Google Scholar
2Avellaneda, M.. Optimal bounds and microgeometries for elastic two-phase composites..S1AM J. Appl. Math. 47 (1987), 1216–1228.Google Scholar
3Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
4Calderon, A. P.. Commutators of singular integral operators Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099.Google Scholar
5Coifman, R. and Meyer, Y.. Au dela des op6rateurs pseudo-diffe'rentiels Astirisque 57 (1978) 1–185.Google Scholar
6David, G. and Journé, J. L.. Une caracté;risation des opérateurs intègraux singuliers bornésur L2(RN). C. R. Acad. Sci. Paris, Sir. I. Math.. 296 (1983), 761–764.Google Scholar
7DiPerna, R. J.. Convergence of approximate solutions to conservation law. Arch. Rational Mech. Anal. 82 (1983), 27–70.Google Scholar
8DiPerna, R. J.. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1–30.Google Scholar
9DiPerna, R. J.. Oscillations and concentrations in solutions to the equations of mechanics. In Directions in Partial Differential Equations, eds. Crandall, M. G., Rabinowitz, P. H. and Turner, R. E. L., pp. 43–53 (New York: Academic Press, 1987).Google Scholar
10DiPerna, R. J. and Lions, P. L.. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math., 130 (1989), 321–366.Google Scholar
11DiPerna, R. J. and Lions, P. L.. Global weak solutions of Vlasov Maxwell systems. Comm. Pure Appl. Math., 42 (1989), 729–757.CrossRefGoogle Scholar
12DiPerna, R. J. and Majda, A. J.. Oscillations and concentration in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108 (1987) 667–689.Google Scholar
13DiPerna, R. J. and Majda, A. J.. Concentration in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40 (1987), 301–345.Google Scholar
14Fefferman, C.. Recent progress in classical Fourier analysis. Proc. I.CM. Vancouver, 1974, pp. 95–118.Google Scholar
15Francfort, G. and Murat, F.. Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94d (1986), 307–334.Google Scholar
16Gerard, P.. Compacite’ par compensation et regularity 2-microlocale. Séminaire Equations mix Dirivees Partielles 1988–1989 (Ecole Polytechnique, Palaiseau, exp. VI).Google Scholar
17Höormander., L..The Analysis of Linear Partial Differenital Operators I-IV (Berlin: Springer, 1983–1985).Google Scholar
18Kohn, R. V. and Milton, G. W.. On bounding the effective conductivity of anisotropic composites. In Homogenization and Effective Moduli of Materials and Media, eds. Ericksen, J. L., Kinderlehrer, D., Kohn, R. V. and Lions, J. L., pp. 97–125. IMA Volumes in Mathematics and its Applications 1, (Berlin: Springer, 1986).Google Scholar
19Landau, L. D. and Lifschitz, E. M.. Electrodynamics of Continuous Media, (Oxford: Pergamon Press, 1984).Google Scholar
20Lions, P. L.. The concentration-compactness principle in the calculus of variations: the locally compact case, part 1 and 2. Ann. Inst. H. Poincari Anal. Non Linéaire (1984), 109–145; 223–283.Google Scholar
21Lions, P. L.. The concentration-compactness principle in the calculus of variations: the limit case, part 1 and 2. Rev. Mat. Iberoamericana, 1 (1985) (1), 145–201; (2), 45–121.Google Scholar
22Milton, G. W.. Modelling the properties of composites by laminates. In Homogenization and Effective Moduli of Materials and Media, eds. Ericksen, J. L., Kinderlehrer, D., Kohn, R. V. and Lions, J. L., pp. 150–174. IMA Volumes in Mathematics and its Applications 1 (Berlin: Springer, 1986).Google Scholar
23Murat, F.. Compacite par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 5 (1978), 489–507.Google Scholar
24Murat, F.. H-convergence (Seminaire d'analyse fonctionnelle et numerique de:I'Université d' Alger, 1977–1978.Google Scholar
25Murat, F. and Tartar, L.. In preparation.Google Scholar
26Tartar, L.. Homogeneisation en hydrodynamique In Singular Perturbations and Boundary Layer Theory, Lyon 1976. Lecture Notes in Mathematics 594 pp. 474–481 (Berlin: Springer, 1977).Google Scholar
27Tartar, L.. Cours Peccot, College de France, Paris, 1977. Unpublished. Some material used in [24].Google Scholar
28Tartar, L.. Estimations de coefficients homogénésés. In Computing Methods in Applied Sciences and Engineering, I. Lecture Notes in Mathematics 704 pp. 364–373 (Berlin: Springer, 1979).Google Scholar
29Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, ed. Knops, R. J, Research Notes in Mathematics 39 pp. 136–212. (London: Pitman 1979).Google Scholar
30Tartar, L.. The compensated compactness method applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations ed. Ball, J. M., NATO ASI Series C 111 pp. 263–285. (New York: Reidel, 1983).Google Scholar
31Tartar, L.. Estimations fines de coefficients homogónéisés. In Ennio de Giorgi Colloquium, ed. Krée, P., Research Notes in Mathematics 125, pp. 168–187. (London: Pitman, 1985).Google Scholar
32Tartar, L. Oscillations in nonlinear partial differential equations: compensated compactness and homogenization. In Nonlinear Systems of Partial Differential Equations in Applied Mathematics Part 1, pp. 243–266. Lectures in Applied Mathematics 23, (Providence, R.I.: Americal Mathematical Society, 1986).Google Scholar
33Tartar, L.. Remarks on homogenization. In Homogenization and Effective Moduli of Materials and Media, eds. Ericksen, J. L.Kinderlehrez, D., Kohn, R. V., and Lions, J. L., IMA Volumes in Mathematics and its Applications 1, pp. 228–246. (Berlin: Springer-Verlag, 1986).Google Scholar
34Young, L. C.. Lectures on the Calculus of Variation and Optimal Control Theory. (Philadelphia: W. B. Saunders, 1969).Google Scholar