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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

##### MSC:
 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
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##### References:
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