Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-17T21:26:53.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourier approach to homogenization problems

Published online by Cambridge University Press:  15 August 2002

Carlos Conca  and
M. Vanninathan
Carlos Conca
Affiliation:
Departamento de Ingeniería Matemática, and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170/3, Correo-3, Santiago, Chile; [email protected].
M. Vanninathan
Affiliation:
IISc-TIFR Mathematics Programme, TIFR Centre, P.O. Box 1234, Bangalore 560 012, India; [email protected].
Get access

Abstract

This article is divided into two chapters. Theclassical problem of homogenization of elliptic operators withperiodically oscillating coefficients is revisited in thefirst chapter. Following a Fourier approach, we discuss someof the basic issues of the subject: main convergence theorem,Bloch approximation, estimates on second order derivatives,correctors for the medium, and so on. The second chapter isdevoted to the discussion of some non-classical behaviour ofvibration problems of periodic structures.

Keywords

HomogenizationBloch wavescorrectorsregularityspectral problemsvibration problems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 489 - 511
Copyright
© EDP Sciences, SMAI, 2002

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

Aguirre, F. and Conca, C., Eigenfrequencies of a tube bundle immersed in a fluid. Appl. Math. Optim. 18 (1988) 1-38. CrossRef
Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. CrossRef
Allaire, G. and Conca, C., Bloch-wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. CrossRef
Allaire, G. and Conca, C., Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal. 29 (1997) 343-379. CrossRef
Allaire, G. and Conca, C., Bloch wave homogenization for a spectral problem in fluid-solid structures. Arch. Rational Mech. Anal. 135 (1996) 197-257. CrossRef
Allaire, G. and Conca, C., Analyse asymptotique spectrale de l'équation des ondes. Homogénéisation par ondes de Bloch. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 293-298.
Allaire, G. and Conca, C., Analyse asymptotique spectrale de l'équation des ondes. Complétude du spectre de Bloch. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 557-562.
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis in Periodic Structures. North-Holland, Amsterdam (1978).
Bloch, F., Über die Quantenmechanik der Electronen in Kristallgittern. Z. Phys. 52 (1928) 555-600. CrossRef
Boccardo, L. and Marcellini, P., Sulla convergenza delle soluzioni di disequazioni variazionali. Ann. Mat. Pura Appl. 4 (1977) 137-159.
Castro, C. and Zuazua, E., Une remarque sur l'analyse asymptotique spectrale en homogénéisation. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 1043-1048.
A. Cherkaev and R. Kohn, Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997).
C. Conca, S. Natesan and M. Vanninathan, Numerical experiments with the Bloch-Floquet approach in homogenization (to appear).
C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in Homogenization and Applications. SIAM J. Math. Anal. (in press).
C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in bounded domains. Preprint (2002).
C. Conca, R. Orive and M. Vanninathan, Application of Bloch decomposition in wave propagation problems (in preparation).
C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. J. Wiley and Sons/Masson, New York/Paris, Collection RAM 38 (1995).
Conca, C., Planchard, J. and Vanninathan, M., Limiting behaviour of a spectral problem in fluid-solid structures. Asymp. Anal. 6 (1993) 365-389.
C. Conca, J. Planchard, B. Thomas and M. Vanninathan, Problèmes Mathématiques en Couplage Fluide-Structure. Applications aux Faisceaux Tubulaires. Eyrolles, Paris (1994).
Conca, C. and Vanninathan, M., Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997) 1639-1659. CrossRef
Conca, C. and Vanninathan, M., On uniform H 2-estimates in periodic homogenization. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 499-517. CrossRef
Conca, C. and Vanninathan, M., A spectral problem arising in fluid-solid structures. Comput. Methods Appl. Mech. Engrg. 69 (1988) 215-242. CrossRef
G. Dal Maso, An Introduction to $\Gamma-$ Convergence. Birkhäuser, Boston (1993).
Figotin, A. and Kuchment, P., Band-gap structure of spectra of periodic dielectric and accoustic media. I, scalar model. SIAM J. Appl. Math. 56 (1996) 68-88. CrossRef
Floquet, G., Sur les équations différentielles linéaires à coefficients périodiques. Ann. École Norm. Sér. 2 12 (1883) 47-89. CrossRef
Gelfand, I.M., Expansion in series of eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk SSSR 73 (1950) 1117-1120.
P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire Equations aux Dérivées Partielles, Vol. 16, 1990-1991. École Polytechnique, Palaiseau (1991).
Gérard, P., Microlocal defect measures. Comm. Partial Differential Equation 16 (1991) 1761-1794. CrossRef
Gérard, P., Markowich, P.A., Mauser, N.J. and Poupaud, F., Homogenization limits and Wigner transforms. Comm. Pure. Appl. Math. 50 (1997) 321-377. 3.0.CO;2-C>CrossRef
L. Hörmander, Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin (1985).
S. Kesavan, Homogenization of elliptic eigenvalue problems, I and II. Appl. Math. Optim. 5 (1979) 153-167, 197-216. CrossRef
Lions, P.L. and Paul, T., Sur les mesures de Wigner. Revista Math. Iberoamer. 9 (1993) 553-618. CrossRef
Markowich, P.A., Mauser, N.J. and Poupaud, F., Wigner, A function approach to semiclassical limits: electrons in a periodic potential. J. Math. Phys. 35 (1994) 1066-1094. CrossRef
R. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media I and II. SIAM J. Math. Anal. 2 (1991) 1-15, 16-33.
F. Murat, (1977-78) H -Convergence, Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, mimeographed notes. English translation: Murat and L. Tartar, H -Convergence, in F. Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn. Birkhäuser Verlag, Boston. Series Progress in Nonlinear Differential Equations and their Applications 31 (1977).
F. Murat, A survey on compensated compactness, in Contributions to Modern Calculus of Variations, edited by L. Cesari, Pitman Res. Notes in Math. Ser. 148 (1987) 145-183.
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. CrossRef
Odeh, F. and Keller, J.B., Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys. 5 (1964) 1499-1504. CrossRef
Oleinik, O.A., Shamaev, A.S. and Yosifian, G.A., On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces. Upsekhi Math. Nauk. 44 (1989) 157-158.
Planchard, J., Global behaviour of large elastic tube-bundles immersed in a fluid. Comput. Mech. 2 (1987) 105-118. CrossRef
Planchard, J., Eigenfrequencies of a tube-bundle placed in a confined fluid. Comput. Methods Appl. Mech. Engrg. 30 (1982) 75-93. CrossRef
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, II. Fourier Analysis and Self-Adjointness, III. Scattering Theory, IV. Analysis of Operators. Academic Press, New York (1972-78).
E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer-Verlag, Berlin. Lecture Notes in Phys. 127 (1980).
J. Sánchez-Hubert and E. Sánchez-Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer-Verlag, Berlin (1989).
Santosa, F. and Symes, W.W., A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51 (1991) 984-1005. CrossRef
Tartar, L., H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 193-230. CrossRef
L. Tartar, Problèmes d'Homogénéisation dans les Equations aux Dérivées Partielles, Cours Peccot au Collège de France (1977). Partially written in F. Murat [].
Vanninathan, M., Homogenization and eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 239-271. CrossRef
Wilcox, C., Theory of Bloch waves. J. Anal. Math. 33 (1978) 146-167. CrossRef