On some sharp bounds for the off-diagonal elements of the homogenized tensor (Q1907215)

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.