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Homogenization limits on diffusion equations in thin domains. (English) Zbl 0672.73009
The problem deals with a composite material in a plate-like domain in the limit as the plate thickness tends to zero. the related mathematical study is to derive a consistent two-dimensional model for this structure as the limit of some three-dimensional problem. The paper studies a simple problem: a linear diffusion equation for a composite medium in a thin plate-like domain of \(R^ n\), \(\Omega_{\epsilon}=\omega X(- \epsilon /2,\epsilon /2)\), where \(\omega \subset R^{n-1}\) is smooth and bounded. The diffusivity matrices are spatially varying.
Two main results are proved. The first theorem shows that, as the thickness tends to zero, the limit solution must satisfy a corresponding effective diffusion equation in the mid section \(\omega\). This analysis does not need any information about the geometry of the inhomogenities. In the particular case of a mixture of two isotropic components distributed with a horizontal periodicity (in the first n-1 variables), optimal bounds on the effective conductivity of the composite, as the thickness tends to zero, are given by the second theorem. These bounds depend only on the diffusivity coefficients of the two components and on their volume fraction, they are independent of the geometry.
This study extends results originally obtained by D. Caillerie [RAIRO, Anal. Numer. 15, 295-319 (1981; Zbl 0483.35003)].
Reviewer: Th.Lévy

74E05 Inhomogeneity in solid mechanics
74E30 Composite and mixture properties
35A35 Theoretical approximation in context of PDEs
74K20 Plates
74S30 Other numerical methods in solid mechanics (MSC2010)
46S30 Constructive functional analysis
Full Text: DOI EuDML
[1] D CAILLERIE, Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis R A I R O Analyse Numérique, 15 (1981), pp 295-319 Zbl0483.35003 MR642495 · Zbl 0483.35003 · eudml:193384
[2] D CAILLERIE, Thin elastic and periodic plates Math Methods Appl Sci , 6 (1984), pp 159-191 Zbl0543.73073 MR751739 · Zbl 0543.73073 · doi:10.1002/mma.1670060112
[3] A DAMLAMIAN, M VOGELIUS, Homogenization limits of the equations of elasticity in thin domains SIAM J Math Anal , 18 (2) (1987), pp 435-451 Zbl0614.73012 MR876283 · Zbl 0614.73012 · doi:10.1137/0518034
[4] E DE GIORGI, Quelques problèmes de \(\Gamma\)-convergence, in proceedings of the Conference Computing Methods in Applied Sciences and Engineering, Versailles 1979 (R Glowinski and J L Lions eds ), North Holland, 1980, pp 637-643 Zbl0445.49018 MR584059 · Zbl 0445.49018
[5] R KOHN - G W MILTON, On bounding the effective conductivity of anisotropic composites, in Homogenization and Effective Moduli of Materials and Media (J R Ericksen, D Kinderlehrer, R Kohn, J L Lions eds ) IMA Volumes in Math Appls, 1, Springer, 1986, pp 97-125 Zbl0631.73012 MR859413 · Zbl 0631.73012
[6] R KOHN, M VOGELIUS, A new model for thin plates with rapidly varying thickness II a convergence proof Quart Appl Math , 43 (1985) pp 1-22 Zbl0565.73046 MR782253 · Zbl 0565.73046
[7] K A LURIE, A V CHERKAEV, Exact estimates of the conductivity of composites formed by two isotropically conducting media taken in prescribed proportions Proc Royal Soc Edinburgh, 99A (1984), pp 71-87 Zbl0564.73079 MR781086 · Zbl 0564.73079 · doi:10.1017/S030821050002597X
[8] F MURAT, H-convergence, Séminaire d’Analyse Fonctionnelle et Numérique 1977/1978 Département de Mathématiques, Université d’Alger
[9] F MURAT, G FRANCFORT, Homogenization and optimal bounds in linear elasticity Arch Rat Mech Anal , 94 (1986), pp 307-334 Zbl0604.73013 MR846892 · Zbl 0604.73013 · doi:10.1007/BF00280908
[10] L TARTAR, Cours Peccot, Collège de France, Paris, 1977
[11] L TARTAR, Estimations fines des coefficients homogénéises, in Ennio De Giorgi’s Colloquium (P Kree ed), Pitman Research Notes in Math, Pitman, 1985 Zbl0586.35004 MR909716 · Zbl 0586.35004
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