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Mathematical Tools for Studying Oscillations and Concentrations: from Young Measures to H-Measures and Their Variants

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Multiscale Problems in Science and Technology

Abstract

The following set of notes was taken during the six lectures that I gave in Dubrovnik in September 2000. My purpose was to retrace the evolution of the subject over the last twenty years. A few years ago, I have thought that it should be helpful for young researchers to be told how and when new ideas were introduced, and what was known at the time when they appeared, so that they learn about the discovery process; in contrast, the traditional way of writing mathematical papers tends to suppress the human part of the research activity and only describes the results, which unfortunately are not always attributed to their real authors.

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References

  1. Grégoire Allaire: Homogenization and two-scale convergence, SIAM Journal of Mathematical Analysis 23 (1992) 1482–1518

    Article  MATH  Google Scholar 

  2. Grégoire Allaire, François murat: Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis 7 (1993), no. 2, 81–95.

    MathSciNet  MATH  Google Scholar 

  3. Youcef Amirat, Kamel Hamdache, Abdelhamid ziani: Étude d’une equation de transport à mémoire, Comptes Rendus de l Académie des sciences Paris Sér. I Math. 311 (1990), no. 11, 685–688.

    MATH  Google Scholar 

  4. Nenad Antonić: Memory effects in homogenisation: Linear second-order equations, Archive for Rational Mechanics and Analysis 125 (1993) 1–24

    Article  MathSciNet  MATH  Google Scholar 

  5. Nenad Antonić: H-measures applied to symmetric systems, Proceedings of the Royal Society of Edinburgh 126A (1996) 1133–1155

    Google Scholar 

  6. Ivo Babuška: Homogenization and its applications. Mathematical and computational problems, in Numerical solutions of partial differential equations — III, pp. 89–116, Academic Press, 1976

    Google Scholar 

  7. Alain Bensoussan, Jacques-Louis Lions, George Papanicolaou: Asymptotic analysis for periodic structures, North-Holland, 1978

    Google Scholar 

  8. David J. Bergman: Bulk physical properties of composite media, in Les méthodes de l’homogénéisation: théorie et applications en physique, (Bréausans-Nappe, 1983), pp. 1–128, Collect. Dir. Etudes Rech. Elec. France, 57, Eyrolles, Paris, 1985

    Google Scholar 

  9. Phillipe Braidy, Didier Pouilloux: unpublished memoir

    Google Scholar 

  10. Alberto Calderón: Commutators of singular integral operators, Proceedings of the National Academy of Sciences of USA 53 (1965) 1092–1099

    Article  MATH  Google Scholar 

  11. Yves Colin de Verdière: Ergodicité et fonctions propres du laplacien, Communications in Mathematical Physics 102 (1985), no. 3, 497–502.

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantine M. Dafermos: Hyperbolic conservation laws in continuum physics, Springer, 2000

    Google Scholar 

  13. Ronald J. DiPerna, Andrew J. Majda: Oscillations and concentration in weak solutions of the incompressible fluid equations, Communications in Mathematical Physics 108 (1987) 667–689

    Article  MathSciNet  MATH  Google Scholar 

  14. Paul Adrien Maurice Dirac: The principles of Quantum Mechanics, Clarendon Press, 1956

    Google Scholar 

  15. Gilles Francfort, François Murat: Oscillations and energy densities in the wave equation, Communications in Partial Differential Equations 17 (1992) 1785–1865

    Article  MathSciNet  Google Scholar 

  16. Gilles Francfort, François Murat, Luc Tartar: Fourth order moments of nonnegative measures on S 2 and applications, Archive for Rational Mechanics and Analysis 131 (1995) 305–333

    Article  MathSciNet  MATH  Google Scholar 

  17. Patrick Gérard: Mesures semi-classiques et ondes de Bloch, in Séminaire Equations aux Dérivées Partielles, École Polytechnique, Palaiseau, exp. XVI (1990)

    Google Scholar 

  18. Patrick Gérard: Microlocal defect measures, Communications in Partial Differential Equations 16 (1991) 1761–1794

    Article  MathSciNet  MATH  Google Scholar 

  19. Patrick Gérard: Oscillations and concentration effects in semilinear disper-sive wave equations, Journal of Functional Analysis 141 (1996) 60–98

    Article  MathSciNet  MATH  Google Scholar 

  20. Patrick Gérard, Erik Leichtnam: Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Mathematical Journal 71 (1993) 559–607

    Article  MathSciNet  MATH  Google Scholar 

  21. Zvi Hashin, Shmuel Shtrikman: A variational approach to the theory of effective magnetic permeability of multiphase materials, Journal of Applied Physics 35 (1962) 3125–3131

    Google Scholar 

  22. Lars Hörmander: The analysis of linear partial differential operators I–IV, Springer, 1983–1985

    Google Scholar 

  23. John K. Hunter, Joseph B. Keller: Weakly nonlinear high frequency waves, Communications in Pure and Applied Mathematics 36 (1983) 547–569

    Article  MathSciNet  MATH  Google Scholar 

  24. Richard D. James, David Kinderlehrer: Frustration in ferromagnetic materials, Continuum Mechanics and Thermodynamics 2 (1990) 215–239

    Article  MathSciNet  Google Scholar 

  25. Joseph B. Keller, George C. Papanicolaou, Leonid V. Ryzhik: Transport equations for elastic and other waves in random medium, Wave motion 24 (1996) 327–330

    Article  MathSciNet  MATH  Google Scholar 

  26. Joseph B. Keller, George C. Papanicolaou, Leonid V. Ryzhik: Stability of the P-to-S energy ratio in the diffusive regime, Bull Seismological Soc. Amer. 86 (1996) 1107–1115

    Google Scholar 

  27. Lev Landau, Evgenij Lifschitz: Électrodynamique des milieux continus, Mir, 1969

    Google Scholar 

  28. Pierre-Louis Lions: The concentration-compactness principle in the calculus of variations: The locally compact case I & II, Annales de l’nstitute Henri Poincaré 1 C (1984) 109–145, 223–283; The limit case I & II, Revista Matemática Iberoamericana 1 (1985) No. 145-201, No. 45-121

    Google Scholar 

  29. Pierre-Louis Lions, Thierry Paul: Sur les mesures de Wigner, Revista Matemática Iberoamericana 9 (1993) 553–618

    Google Scholar 

  30. Luisa Mascarenhas: A linear homogenisation problem with time dependent coefficient, Transactions of the American Mathematical Society 281 (1984) 179–195

    Article  MathSciNet  MATH  Google Scholar 

  31. Graeme W. Milton: Theory of composites, Cambridge, 2001

    Google Scholar 

  32. Stefan Müller: Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations, Problems involving change of type (Stuttgart, 1988), 31–44, Lecture Notes in Phys., 359, Springer, Berlin, 1990.

    Chapter  Google Scholar 

  33. François Murat, Luc Tartar: Calcul des variations et homogénéisation, in Les méthodes de l’homogénéisation: théorie et applications en physique, (Bréau-sans-Nappe, 1983), pp. 319–369, Collect. Dir. Etudes Rech. Elec. France, 57, Eyrolles, Paris, 1985

    Google Scholar 

  34. François Murat, Luc Tartar: H-convergence, translation of lecture notes from 1978, in Topics in the mathematical modeling of composite materials, Cherkaev-Kohn (eds.), Vol. 31 of Progress in nonlinear differential equations and their applications, 21–43, Birkhaüser, Boston, 1997

    Google Scholar 

  35. François Murat, Luc Tartar: work still in preparation

    Google Scholar 

  36. G. Nguetseng: A general convergence result for a functional related to the theory of homogenization, SIAM Journal of Mathematical Analysis 20 (1989) 608–623

    Article  Google Scholar 

  37. Henri Sanchez-Palencia: Non-homogeneous media and vibration theory, Springer, 1980

    Google Scholar 

  38. Alexander Shnirelman: Ergodic properties of eigentunctions, Uspehi matematičeskih nauk 29 (1974), no. 6(180), 181–182.

    Google Scholar 

  39. Laurent Schwartz: Théorie des distributions, Hermann, 1957

    Google Scholar 

  40. Sergej L. Sobolev: Applications of functional analysis in mathematical physics, American Mathematical Society, 1963

    Google Scholar 

  41. Sergio Spagnolo: Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore, Annali délia Scuola Normale Superiore Pisa 21 (1967) 657–699

    MATH  Google Scholar 

  42. Sergio Spagnolo: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Annali della Scuola Normale Superiore Pisa 22 (1968) 571–597

    MATH  Google Scholar 

  43. Sergio Spagnolo, Antonio Marino: Un tipo di approssimazione dell’operatore \( \sum\nolimits_{ij} {D_i (a_{ij} (x)D_j )} \) con operatori \( \sum\nolimits_j {D_j (\beta (x)D_j )} \), Annali della Scuola Normale Superiore Pisa 23 (1969) 657–673

    Google Scholar 

  44. Luc Tartar: Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136–212, Pitman Res. Notes Math. Ser. 39, Longman, 1979

    Google Scholar 

  45. Luc Tartar: Estimation unes des coefficients homogénéisés, in E. De Giorgi colloquium, pp. 168–187, Pitman, 1985

    Google Scholar 

  46. Luc Tartar: Remarks on homogenization, in Homogenization and effective moduli of materials and media, pp. 228–246, IMA Vol. Math. Appl. 1, Springer, 1986

    Google Scholar 

  47. Luc Tartar: Remarks on oscillations and Stokes’ equation, in Macroscopic modelling of turbulent flows, pp. 24–31, Lecture Notes in Physics 230, Springer, 1985

    Google Scholar 

  48. Luc Tartar: Nonlocal effects induced by homogenization, in Partial differential equations and the calculus of variations, Vol. II, pp. 925–938, Progr. Nonlinear Differential Equations Appl. 2, Birkhaüser, 1989

    Google Scholar 

  49. Luc Tartar: H-measures and small amplitude homogenization, in Random Media and Composites, (Leesburg, VA, 1988) 89–99, SIAM, Philadelphia, PA 1989

    Google Scholar 

  50. Luc Tartar: Propagation of oscillations and concentration effects in partial differential equations, in Ordinary and Partial Differential Equations, Vol. III, (Dundee, 1990), pp. 208–219, Pitman Res. Notes Math. Ser. 254, Longman, 1991

    Google Scholar 

  51. Luc Tartar: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh 115A (1990) 193–230

    Google Scholar 

  52. Luc Tartar: H-measures and applications, in Proceedings of the International Congress of Mathematicians, pp. 1215–1223, Mathematical Society of Japan, Tokyo, 1991

    Google Scholar 

  53. Luc Tartar: Memory effects and homogenization, Archive for Rational Mechanics and Analysis 111 (1990) 121–133

    Article  MathSciNet  MATH  Google Scholar 

  54. Luc Tartar: Beyond YOUNG measures, Meccanica 30 (1995) 505–526

    Article  MathSciNet  MATH  Google Scholar 

  55. Luc Tartar: Homogenization and Hyperbolicity, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 25 (1997) 785–805

    Google Scholar 

  56. Luc Tartar: How to describe oscillations of solutions of nonlinear partial differential equations, in Transactions of the Sixth Army Conference on Applied Mathematics and Computing (Boulder, CO. 1988), 1133–1141, ARO Rep. 89-1, U.S. Army Res. Office, Research Triangle Park, NC, 1989

    Google Scholar 

  57. Luc Tartar: Approximation of H-measures, Conference in honour of Roland Glowinski (Tours, May 1997); text finished too late to be included in the proceedings. Research Report 97-204, Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

    Google Scholar 

  58. Luc Tartar: H-measures and applications, in Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol XI, pp. 282–290, Pitman Res. Notes Math. Ser. 299, Longman, 1994

    Google Scholar 

  59. Luc Tartar: Homogenization, Compensated Compactness, and H-measures, lecture notes in preparation

    Google Scholar 

  60. Wolf Von Wahl: The equations of NAVIER-STOKES and abstract parabolic equations, Friedrich Vieweg & Sohn, Braunschweig, 1985

    MATH  Google Scholar 

  61. Jack Warga: Optimal control of differential and functional equations, Academic press, 1972

    Google Scholar 

  62. Vasilij V. Žikov, Sergej M. Kozlov, Olga A. Oleinik: Homogenization of differential operators and integral functionals, Springer, 1994

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213-3890, USA

    Luc Tartar

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  1. Luc Tartar
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Nenad Antonić

  2. Dept. of Applied Analysis and Scientific Computing, CWI Amsterdam, P.O.Box 94079, Kruislaan 413, 1090 GB, Amsterdam, The Netherlands

    C. J. van Duijn

  3. IWR, University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    Willi Jäger

  4. Analyse Numérique, Bât 101 U.F.R. Mathématiques, Université Lyon 1, 43, Bd. du 11 Novembre 1918, 69622, Villeurbanne Cedex, France

    Andro Mikelić

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Tartar, L. (2002). Mathematical Tools for Studying Oscillations and Concentrations: from Young Measures to H-Measures and Their Variants. In: Antonić, N., van Duijn, C.J., Jäger, W., Mikelić, A. (eds) Multiscale Problems in Science and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56200-6_1

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  • DOI: https://doi.org/10.1007/978-3-642-56200-6_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43584-6

  • Online ISBN: 978-3-642-56200-6

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