Bloch wave homogenization and spectral asymptotic analysis

Journal des Mathematiques Pures et Appliquees

Volumn 77, Issue 2, 1998, Pages 153-208

  Allaire, Grégoire ^a,b   Conca, Carlos ^c  

^a CEA SACLAY   (France)

^b SORBONNE UNIVERSITÉ   (France)

^c UNIVERSITY OF CHILE   (Chile)

Author keywords

Bloch waves; Boundary layers; Homogenization; Spectral analysis

Indexed keywords

EID: 0031994541     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-7824(98)80068-8     Document Type: Review

References (50)

1. [1] F. AGUIRRE and C. CONCA, Eigenfrequencies of a tube bundle immersed in a fluid, Appl. Math. Optim., 18, 1988, pp. 1-38.
2. [2] G. ALLAIRE, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, (6), 1992, pp. 1482-1518.
3. [3] G. ALLAIRE and M. BRIANE, Multiscale convergence and reiterated homogenization, Proc. Roy. Soc. Edinburgh, 126A, 1996, pp. 297-342.
4. [4] G. ALLAIRE and C. CONCA, Analyse asymptotique spectrale de l'équation des ondes. Homogénéisation par ondes de Bloch, C. R. Acad. Sci. Paris Série I Math., t.321, 1995, pp. 293-298.
5. [5] G. ALLAIRE and C. CONCA, Analyse asymptotique spectrale de l'équation des ondes. Complétude du spectre de Bloch, C. R. Acad. Sci. Paris Série I Math., t.321, 1995, pp. 557-562.
6. [6] G. ALLAIRE and C. CONCA, Bloch wave homogenization for a spectral problem in fluid-solid structures, Arch. Rational. Mech. Anal., 135, (3), 1996, pp. 117-257.
7. [7] G. ALLAIRE and C. CONCA, Boundary layers in the homogenization of a spectral problem in fluid-solid structures, to appear in SIAM J. Math. Anal.
8. [8] I. BABUŠKA, Solution of interface problems by homogenization I, II, III, SIAM J. Math. Anal., 7, 1976, pp. 603-634, pp. 635-645 and 8, 1977, pp. 923-937.
9. [9] N. BAKHVALOV and G. PANASENKO, Homogenization: averaging processes in periodic media, Mathematics and its applications, volume 36, Kluwer Academic Publishers, Dordrecht, 1990.
10. [10] A. BENSOUSSAN, J.-L. LIONS and G. PAPANICOLAOU, Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978.
11. [11] A. BENSOUSSAN, J.-L. LIONS and G. PAPANICOLAOU, Boundary layer analysis in homogenization of diffusion equations with Dirichlet conditions in the half space, in Proc. of Intern. Symp. SDE, Kyoto 1976, pp. 21-40, K. Ito ed., J. Wiley, New York, 1978.
12. [12] F. BLOCH, Uber die Quantenmechanik der Electronen in Kristallgittern, Z. Phys. 52, 1928, pp. 555-600.
13. [13] L. BOCCARDO and P. MARCELLINI, Sulla convergenza delle soluzioni di disequazioni variazionali, Ann. Mat. Pura Appl., 4, 1977, pp. 137-159.
14. [14] C. CASTRO and E. ZUAZUA, Une remarque sur l'analyse asymptotique spectrale en homogénéisation, C. R. Acad. Sci. Paris Série I Math., t.322, 1996, pp. 1043-1047.
15. [15] C. CASTRO and E. ZUAZUA, Une remarque sur les séries de Fourier non-harmoniques et son application à la contrôlabilité des cordes avec densité régulière, C. R. Acad. Sci. Paris Sér. I Math., t. 323, 1996, pp. 365-370.
16. [16] C. CONCA, J. PLANCHARD and M. VANNINATHAN, Limiting behaviour of a spectral problem in fluid-solid structures, Asymptotic Analysis, 6, 1993, pp. 365-389.
17. [17] C. CONCA, J. PLANCHARD and M. VANNINATHAN, Fluids and periodic structures, RMA, 38, J. Wiley & Masson, Paris 1995.
18. [18] C. CONCA and M. VANNINATHAN, A spectral problem arising in fluid-solid structures, Comput. Methods Appl. Mech. Engrg., 69, 1988, pp. 215-242.
19. [19] C. CONCA and M. VANNINATHAN, Homogenization of periodic structures via Bloch decomposition, it SIAM J. Appl. Math., 57 in press 1997.
20. [20] G. DAL MASO, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.
21. [21] A. FIGOTIN and P. KUCHMENT, Band-gap structure of spectra of periodic dielectric and acoustic media. I, scalar model, SIAM J. Appl. Math., 56, 1996, pp. 68-88.
22. [22] G. FLOQUET, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Ecole Norm. Sér., 12 2, 1883, pp. 47-89.
23. [23] I. M. GELFAND, Expansion in series of eigenfunctions of an equation with periodic coefficients, Dokl. Akad. Nauk. SSSR, 73, 1950, pp. 1117-1120.
24. [24] P. GÉRARD, Mesures semi-classiques et ondes de Bloch, in Séminaire Equations aux Dérivées Partielles 1990-1991, volume 16, Ecole Polytechnique, Palaiseau, 1991.
25. [25] C. GÉRARD and A. MARTÍNEZ, J. Sjostrand, A mathematical approach to the effective Hamiltonian in perturbed periodic problems, Comm. Math. Phys., 142, 1991, pp. 217-244.
26. [26] I. C. GOHBERG and M. G. KREǏN, Introduction to the theory of linear nonselfadjoint operators, AMS, Translations of Mathematical Monographs, Vol. 18, Providence, Rhode Island, 1969.
27. [27] V. JIKOV, S. KOZLOV and O. OLEINIK, Homogenization of Differential Operators, Springer-Verlag, Berlin, 1995.
28. [28] T. KATO, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966.
29. [29] S. KESAVAN, Homogenization of elliptic eigenvalue problems, I and II, Appl. Math. Optim., 5, 1979, pp. 153-167 and pp. 197-216.
30. [30] J.-L. LIONS, Some methods in the mathematical analysis of systems and their control, Science Press, Beijing, Gordon and Breach, New York, 1981.
31. [31] P.-L. LIONS and T. PAUL, Sur les mesures de Wigner, Revista Math. Iberoamer., 9, 1993, pp. 553-618.
32. [32] P. A. MARKOWICH, N. J. MAUSER and F. POUPAUD, A Wigner function approach to semiclassical limits: electrons in a periodic potential, J. Math. Phys., 35, 1994, pp. 1066-1094.
33. [33] R. MORGAN and I. BABUSKA, An approach for constructing families of equations for periodic media. I and II, SIAM J. Math. Anal. 22, 1991, pp. 1-15, pp. 16-33.
34. [34] S. MOSKOW and M. VOGELIUS, First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof, to appear in Proc. Roy. Soc., Edinburgh, 1996.
35. [35] F. MURAT and L. TARTAR, H-convergence, in Topics in the mathematical modeling of composite materials, A. CHERKAEV and R. V. KOHN eds., series: Progress in Nonlinear Differential Equations and their Applications, Birkhaüser, Boston 1997. French version: mimeographed notes, séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978.
36. [35] F. MURAT and L. TARTAR, H-convergence, in Topics in the mathematical modeling of composite materials, A. CHERKAEV and R. V. KOHN eds., series: Progress in Nonlinear Differential Equations and their Applications, Birkhaüser, Boston 1997. French version: mimeographed notes, séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1978.
37. [36] G. NGUETSENG, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (3), 1989, pp. 608-623.
38. [37] F. ODEH and J. KELLER, Partial differential equations with periodic coefficients and Bloch waves in crystals, J. Math. Phys. 5, 11, 1964, pp. 1499-1504.
39. [38] O. A. OLEINIK, A. S. SHAMAEV and G. A. YOSIFIAN, On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces, Upsekhi Math. Nauk., 44, 1989, pp. 157-158.
40. [39] J. PLANCHARD, Eigenfrequencies of a tube-bundle placed in a confined fluid, Comput. Methods Appl. Mech. Engrg., 30, 1982, pp. 75-93.
41. [40] J. PLANCHARD, Global behaviour of large elastic tube-bundles immersed in a fluid, Comput. Mech., 2, 1987, pp. 105-118.
42. [41] M. REED and B. SIMON, Methods of modern mathematical physics, IV Analysis of operators, Academic Press, New York, 1978.
43. [42] F. RELLICH, Storungtheorie der Spektralzerlegung, Math. Ann. 113, 1937, pp. 600-619.
44. [43] J. SÁNCHEZ-HUBERT and E. SÁNCHEZ-PALENCIA, Vibration and coupling of continuous systems. Asymptotic methods, Springer Verlag, Berlin, 1989.
45. [44] E. SÁNCHEZ-PALENCIA, Non homogeneous media and vibration theory, Lecture notes in physics, 127, Springer Verlag, 1980.
46. [45] F. SANTOSA and W. SYMES, A dispersive effective medium for wave propagation in periodic composites, SIAM J. Appl. Math. 51, 1991, pp. 984-1005.
47. [46] F. SANTOSA and M. VOGELIUS, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math. 53, 1993, pp. 1636-1668.
48. [47] L. TARTAR, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115A, 1990, pp. 193-230.
49. [48] M. VANNINATHAN, Homogenization and eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90, 1981, pp. 239-271.
50. [49] C. WILCOX, Theory of Bloch waves, J. Anal. Math., 33, 1978, pp. 146-167.