Fourier approach to homogenization problems

Conca, Carlos; Vanninathan, M.

doi:10.1051/cocv:2002048

Conca, Carlos ; Vanninathan, M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 489-511.

Résumé

This article is divided into two chapters. The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in the first chapter. Following a Fourier approach, we discuss some of 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 is devoted to the discussion of some non-classical behaviour of vibration problems of periodic structures.

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv:2002048

Classification : 35B27, 35A25, 42C30
Mots clés : homogenization, Bloch waves, correctors, regularity, spectral problems, vibration problems

@article{COCV_2002__8__489_0,
     author = {Conca, Carlos and Vanninathan, M.},
     title = {Fourier approach to homogenization problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {489--511},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002048},
     mrnumber = {1932961},
     zbl = {1065.35045},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2002048/}
}

TY  - JOUR
AU  - Conca, Carlos
AU  - Vanninathan, M.
TI  - Fourier approach to homogenization problems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 489
EP  - 511
VL  - 8
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2002048/
DO  - 10.1051/cocv:2002048
LA  - en
ID  - COCV_2002__8__489_0
ER  -

%0 Journal Article
%A Conca, Carlos
%A Vanninathan, M.
%T Fourier approach to homogenization problems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 489-511
%V 8
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2002048/
%R 10.1051/cocv:2002048
%G en
%F COCV_2002__8__489_0

Conca, Carlos; Vanninathan, M. Fourier approach to homogenization problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 489-511. doi : 10.1051/cocv:2002048. http://archive.numdam.org/articles/10.1051/cocv:2002048/

Bibliographie
Cité par

[1] F. Aguirre and C. Conca, Eigenfrequencies of a tube bundle immersed in a fluid. Appl. Math. Optim. 18 (1988) 1-38. | MR | Zbl

[2] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR | Zbl

[3] G. Allaire and C. Conca, Bloch-wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | MR | Zbl

[4] G. Allaire and C. Conca, Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal. 29 (1997) 343-379. | MR | Zbl

[5] G. Allaire and C. Conca, Bloch wave homogenization for a spectral problem in fluid-solid structures. Arch. Rational Mech. Anal. 135 (1996) 197-257. | MR | Zbl

[6] G. Allaire and C. Conca, 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. | Zbl

[7] G. Allaire and C. Conca, 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. | Zbl

[8] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis in Periodic Structures. North-Holland, Amsterdam (1978). | MR | Zbl

[9] F. Bloch, Über die Quantenmechanik der Electronen in Kristallgittern. Z. Phys. 52 (1928) 555-600. | JFM

[10] L. Boccardo and P. Marcellini, Sulla convergenza delle soluzioni di disequazioni variazionali. Ann. Mat. Pura Appl. 4 (1977) 137-159. | MR | Zbl

[11] C. Castro and E. Zuazua, Une remarque sur l'analyse asymptotique spectrale en homogénéisation. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 1043-1048. | Zbl

[12] A. Cherkaev and R. Kohn, Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997). | MR | Zbl

[13] C. Conca, S. Natesan and M. Vanninathan, Numerical experiments with the Bloch-Floquet approach in homogenization (to appear). | MR | Zbl

[14] C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in Homogenization and Applications. SIAM J. Math. Anal. (in press). | MR | Zbl

[15] C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in bounded domains. Preprint (2002). | MR

[16] C. Conca, R. Orive and M. Vanninathan, Application of Bloch decomposition in wave propagation problems (in preparation).

[17] C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. J. Wiley and Sons/Masson, New York/Paris, Collection RAM 38 (1995). | MR | Zbl

[18] C. Conca, J. Planchard and M. Vanninathan, Limiting behaviour of a spectral problem in fluid-solid structures. Asymp. Anal. 6 (1993) 365-389. | MR | Zbl

[19] C. Conca, J. Planchard, B. Thomas and M. Vanninathan, Problèmes Mathématiques en Couplage Fluide-Structure. Applications aux Faisceaux Tubulaires. Eyrolles, Paris (1994). | Zbl

[20] C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997) 1639-1659. | MR | Zbl

[21] C. Conca and M. Vanninathan, On uniform $H^{2}$ -estimates in periodic homogenization. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 499-517. | MR | Zbl

[22] C. Conca and M. Vanninathan, A spectral problem arising in fluid-solid structures. Comput. Methods Appl. Mech. Engrg. 69 (1988) 215-242. | MR | Zbl

[23] G. Dal Maso, An Introduction to $Γ -$ Convergence. Birkhäuser, Boston (1993). | MR | Zbl

[24] A. Figotin and P. Kuchment, Band-gap structure of spectra of periodic dielectric and accoustic media. I, scalar model. SIAM J. Appl. Math. 56 (1996) 68-88. | MR | Zbl

[25] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. École Norm. Sér. 2 12 (1883) 47-89. | JFM | Numdam | MR

[26] I.M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk SSSR 73 (1950) 1117-1120. | MR

[27] P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire Equations aux Dérivées Partielles, Vol. 16, 1990-1991. École Polytechnique, Palaiseau (1991). | Numdam | Zbl

[28] P. Gérard, Microlocal defect measures. Comm. Partial Differential Equation 16 (1991) 1761-1794. | MR | Zbl

[29] P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure. Appl. Math. 50 (1997) 321-377. | MR | Zbl

[30] L. Hörmander, Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin (1985). | MR | Zbl

[31] S. Kesavan, Homogenization of elliptic eigenvalue problems, I and II. Appl. Math. Optim. 5 (1979) 153-167, 197-216. | MR | Zbl

[32] P.L. Lions and T. Paul, Sur les mesures de Wigner. Revista Math. Iberoamer. 9 (1993) 553-618. | MR | Zbl

[33] 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) 1066-1094. | MR | Zbl

[34] R. Morgan and I. Babuška, An approach for constructing families of homogenized equations for periodic media I and II. SIAM J. Math. Anal. 2 (1991) 1-15, 16-33. | Zbl

[35] 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). | Zbl

[36] 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. | MR

[37] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR | Zbl

[38] F. Odeh and J.B. Keller, Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math. Phys. 5 (1964) 1499-1504. | MR | Zbl

[39] 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) 157-158. | Zbl

[40] J. Planchard, Global behaviour of large elastic tube-bundles immersed in a fluid. Comput. Mech. 2 (1987) 105-118. | Zbl

[41] J. Planchard, Eigenfrequencies of a tube-bundle placed in a confined fluid. Comput. Methods Appl. Mech. Engrg. 30 (1982) 75-93. | MR | Zbl

[42] 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). | MR | Zbl

[43] E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer-Verlag, Berlin. Lecture Notes in Phys. 127 (1980). | Zbl

[44] J. Sánchez-Hubert and E. Sánchez-Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer-Verlag, Berlin (1989). | Zbl

[45] F. Santosa and W.W. Symes, A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51 (1991) 984-1005. | MR | Zbl

[46] L. Tartar, $H$ -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. | MR | Zbl

[47] L. Tartar, Problèmes d'Homogénéisation dans les Equations aux Dérivées Partielles, Cours Peccot au Collège de France (1977). Partially written in F. Murat [25].

[48] M. Vanninathan, Homogenization and eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 239-271. | MR | Zbl

[49] C. Wilcox, Theory of Bloch waves. J. Anal. Math. 33 (1978) 146-167. | MR | Zbl

Cité par Sources :