On homogenization for locally periodic elliptic problems on a domain

Let $\Omega$ be a Lipschitz domain, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $\Omega$ with fairly general boundary conditions, including, in particular, the Dirichlet and Neumann boundary conditions, as well as mixed ones. We suppose that the parameter $\varepsilon$ is small and the function $A$ in the operator $\mathcal A^\varepsilon$ is Lipschitz in the first variable and periodic in the second, so its coefficients are locally periodic and rapidly oscillating. It is a classical result in homogenization theory that the resolvent $(\mathcal A^\varepsilon-\mu)^{-1}$ converges (in a certain sense) as $\varepsilon\to0$. We are interested in approximations for $(\mathcal A^\varepsilon-\mu)^{-1}$ and $\nabla(\mathcal A^\varepsilon-\mu)^{-1}$ in the operator norm on $L_p$ for a suitable $p$. The rates of the approximations depend on regularity of the effective operator $\mathcal A^0$. We prove that if $(\mathcal A^0-\mu)^{-1}$ is continuous from $L_p$ to the Besov space $B_{p,\infty}^{1+s}$ with $0<s\le1$, then the rates are, respectively, $\varepsilon^s$ and $\varepsilon^{s/p}$.