Fourier approach to homogenization problems
ESAIM: Control, Optimisation and Calculus of Variations (2002)
- Volume: 8, page 489-511
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topConca, Carlos, and Vanninathan, M.. "Fourier approach to homogenization problems." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 489-511. <http://eudml.org/doc/245915>.
@article{Conca2002,
abstract = {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.},
author = {Conca, Carlos, Vanninathan, M.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; Bloch waves; correctors; regularity; spectral problems; vibration problems; elliptic operators},
language = {eng},
pages = {489-511},
publisher = {EDP-Sciences},
title = {Fourier approach to homogenization problems},
url = {http://eudml.org/doc/245915},
volume = {8},
year = {2002},
}
TY - JOUR
AU - Conca, Carlos
AU - Vanninathan, M.
TI - Fourier approach to homogenization problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 489
EP - 511
AB - 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.
LA - eng
KW - homogenization; Bloch waves; correctors; regularity; spectral problems; vibration problems; elliptic operators
UR - http://eudml.org/doc/245915
ER -
References
top- [1] F. Aguirre and C. Conca, Eigenfrequencies of a tube bundle immersed in a fluid. Appl. Math. Optim. 18 (1988) 1-38. Zbl0663.76003MR928208
- [2] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. Zbl0770.35005MR1185639
- [3] G. Allaire and C. Conca, Bloch-wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. Zbl0901.35005MR1614641
- [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. Zbl0918.35018MR1616495
- [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. Zbl0857.73008MR1418465
- [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. Zbl0844.35075
- [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. Zbl0844.35076
- [8] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis in Periodic Structures. North-Holland, Amsterdam (1978). Zbl0404.35001MR503330
- [9] F. Bloch, Über die Quantenmechanik der Electronen in Kristallgittern. Z. Phys. 52 (1928) 555-600. Zbl54.0990.01JFM54.0990.01
- [10] L. Boccardo and P. Marcellini, Sulla convergenza delle soluzioni di disequazioni variazionali. Ann. Mat. Pura Appl. 4 (1977) 137-159. Zbl0333.35030MR425344
- [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. Zbl0863.35011
- [12] A. Cherkaev and R. Kohn, Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997). Zbl0870.00018MR1493036
- [13] C. Conca, S. Natesan and M. Vanninathan, Numerical experiments with the Bloch-Floquet approach in homogenization (to appear). Zbl1121.65119MR2199541
- [14] C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in Homogenization and Applications. SIAM J. Math. Anal. (in press). Zbl1010.35004MR1897707
- [15] C. Conca, R. Orive and M. Vanninathan, Bloch Approximation in bounded domains. Preprint (2002). Zbl1077.35017MR1897707
- [16] C. Conca, R. Orive and M. Vanninathan, Application of Bloch decomposition in wave propagation problems (in preparation). Zbl1077.35017
- [17] C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. J. Wiley and Sons/Masson, New York/Paris, Collection RAM 38 (1995). Zbl0910.76002MR1652238
- [18] C. Conca, J. Planchard and M. Vanninathan, Limiting behaviour of a spectral problem in fluid-solid structures. Asymp. Anal. 6 (1993) 365-389. Zbl0806.35133MR1203625
- [19] C. Conca, J. Planchard, B. Thomas and M. Vanninathan, Problèmes Mathématiques en Couplage Fluide-Structure. Applications aux Faisceaux Tubulaires. Eyrolles, Paris (1994). Zbl0824.73002
- [20] C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57 (1997) 1639-1659. Zbl0990.35019MR1484944
- [21] C. Conca and M. Vanninathan, On uniform -estimates in periodic homogenization. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 499-517. Zbl1005.35014MR1838500
- [22] C. Conca and M. Vanninathan, A spectral problem arising in fluid-solid structures. Comput. Methods Appl. Mech. Engrg. 69 (1988) 215-242. Zbl0669.73071MR949516
- [23] G. Dal Maso, An Introduction to Convergence. Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
- [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. Zbl0852.35014MR1372891
- [25] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. École Norm. Sér. 2 12 (1883) 47-89. MR1508722JFM15.0279.01
- [26] I.M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients. Dokl. Akad. Nauk SSSR 73 (1950) 1117-1120. MR39154
- [27] P. Gérard, Mesures semi-classiques et ondes de Bloch, in Séminaire Equations aux Dérivées Partielles, Vol. 16, 1990–1991. École Polytechnique, Palaiseau (1991). Zbl0739.35096
- [28] P. Gérard, Microlocal defect measures. Comm. Partial Differential Equation 16 (1991) 1761-1794. Zbl0770.35001MR1135919
- [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. Zbl0881.35099MR1438151
- [30] L. Hörmander, Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin (1985). Zbl0601.35001MR781536
- [31] S. Kesavan, Homogenization of elliptic eigenvalue problems, I and II. Appl. Math. Optim. 5 (1979) 153-167, 197-216. Zbl0428.35062MR533617
- [32] P.L. Lions and T. Paul, Sur les mesures de Wigner. Revista Math. Iberoamer. 9 (1993) 553-618. Zbl0801.35117MR1251718
- [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. Zbl0805.35106MR1262733
- [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. Zbl0729.35010
- [35] F. Murat, (1977-78) -Convergence, Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger, mimeographed notes. English translation: Murat and L. Tartar, -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). Zbl0920.35019
- [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. MR894077
- [37] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. Zbl0688.35007MR990867
- [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. Zbl0129.46004MR168924
- [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. Zbl0711.47003
- [40] J. Planchard, Global behaviour of large elastic tube-bundles immersed in a fluid. Comput. Mech. 2 (1987) 105-118. Zbl0635.73070
- [41] J. Planchard, Eigenfrequencies of a tube-bundle placed in a confined fluid. Comput. Methods Appl. Mech. Engrg. 30 (1982) 75-93. Zbl0483.70016MR659568
- [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). Zbl0242.46001MR493419
- [43] E. Sánchez–Palencia, Non-Homogeneous Media and Vibration Theory. Springer-Verlag, Berlin. Lecture Notes in Phys. 127 (1980). Zbl0432.70002
- [44] J. Sánchez–Hubert and E. Sánchez–Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer-Verlag, Berlin (1989). Zbl0698.70003
- [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. Zbl0741.73017MR1117428
- [46] L. Tartar, -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. Zbl0774.35008MR1069518
- [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. Zbl0486.35063MR635561
- [49] C. Wilcox, Theory of Bloch waves. J. Anal. Math. 33 (1978) 146-167. Zbl0408.35067MR516045
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.