Improved resolvent 𝐿²-approximations in homogenization of fourth order operators

A divergent elliptic operator A ε A_\varepsilon of the fourth order with ε \varepsilon -periodic coefficients acting in the space R d \mathbb {R}^d is treated, where ε \varepsilon is a small parameter. For the resolvent ( A ε + 1 ) − 1 (A_\varepsilon +1)^{-1} , approximations are constructed in the operator ( L 2 → L 2 ) {(L^2\to L^2)} -norm with remainder of order ε 3 \varepsilon ^3 . The method of two-scale expansions with the use of smoothing is employed.