Homogenization of high order elliptic operators with periodic coefficients

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we study a selfadjoint strongly elliptic operator $A_{\epsilon }$ of order $2p$ given by the expression $b(\mathbf{D}{)}^{\ast }g(\mathbf{x}\mathrm{/}\epsilon )b(\mathbf{D})$, $\epsilon >0$. Here $g(\mathbf{x})$ is a bounded and positive definite $(m\times m)$-matrix-valuedfunctionin${\mathbb R}^d$;itisassumedthat$g({\mathbf x})$isperiodicwithrespecttosomelattice\mathrm{.}Next,$b({\mathbf D})=\sum_{|\alpha|=p}^d b_\alpha {\mathbf D}^\alpha$ is a differential operator of order $p$ with constant coefficients; $b_{\alpha }$ are constant $(m\times n)$-matrices\mathrm{.}Itisassumedthat$m\ge n$andthatthesymbol$b({\boldsymbol \xi})$hasmaximalrank\mathrm{.}Fortheresolvent$(A_\varepsilon - \zeta I)^{-1}$with$\zeta \in {\mathbb C} \setminus [0,\infty)$, we obtain approximations in the norm of operators in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ and in the norm of operators acting from $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ to the Sobolev space $H^p({\mathbb{R}}^d;{\mathbb{C}}^n)$, with error estimates depending on $\epsilon$ and $\zeta$.