Improved $L^2$-approximation of resolvents in homogenization of fourth order operators

A 4th order elliptic operator $A_\varepsilon$ in the diverdence form acting in the entire space $\mathbb{R}^d$ and having $\varepsilon$-periodic coefficients is studied ($\varepsilon$ is a small parameter). An approximation for the resolvent $(A_\varepsilon+1)^{-1}$ is found with error estimate of order $\varepsilon^3$ in the operator $(L^2{\to}L^2)$-norm. The method of double-scale approximation with a generalised shift in the form of smoothing is used.