Homogenization for non-self-adjoint locally periodic elliptic operators

We study the homogenization problem for matrix strongly elliptic operators on $L_2({\mathbb{R}}^d{)}^n$ of the form $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$\mathrm{.}Thefunction$A$ is Lipschitz in the first variable and periodic in the second. We do not require that $A^{\ast }=A$, so ${\mathcal{A}}^{\epsilon }$ need not be self-adjoint. In this paper, we provide, for small $\epsilon$, two terms in the uniform approximation for $({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$ and a first term in the uniform approximation for $\mathrm{\nabla }({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$. Primary attention is paid to proving sharp-order bounds on the errors of the approximations.