In $L_2({ \mathbb R}^d;{ \mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({ \mathbf D})^* g({ \mathbf x}/\varepsilon) b({ \mathbf D})$, $\varepsilon >0$. Here $g({ \mathbf x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${ \mathbb R}^d$; it is assumed that $g({ \mathbf x})$ is periodic with respect to some lattice. 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. It is assumed that $m\ge n$ and that the symbol $b({ \boldsymbol \xi})$ has maximal rank. For the resolvent $(A_\varepsilon - \zeta I)^{ -1}$ 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 $\varepsilon$ and $\zeta$.
Categories: math.AP
