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On some sharp bounds for the off-diagonal elements of the homogenized tensor. (English) Zbl 0847.35011

The author investigates bounds for the off-diagonal elements of the homogenized tensor \(q\) for the stationary heat equation, assuming that the original conductivity is generated by a \(Y\)-periodic scalar function \(\lambda(x)\); \(Y\) denotes the unit cell in \(\mathbb{R}^N\). He finds out the estimate \(|q_{rs}|\leq q_a- (q_{rr}+ q_{ss})/2\), \(r\neq s\), where \(q_a= (1/ |Y|) \int_Y h(y) dy\). In order to show that the obtained bounds are sharp, an explicit formula for the homogenized tensor in the case of a laminate structure is presented.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35Q99 Partial differential equations of mathematical physics and other areas of application
74E05 Inhomogeneity in solid mechanics
74E30 Composite and mixture properties
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References:

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