Homogenization of high order elliptic operators with periodic coefficients

Abstract

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, $b({\mathbf D})=\sum_{|α|=p}^d b_α{\mathbf D}^α$ is a differential operator of order $p$ with constant coefficients; $b_α$ are constant $(m\times n)$-matrices. It is assumed that $m\ge n$ and that the symbol $b({\boldsymbol ξ})$ has maximal rank. For the resolvent $(A_\varepsilon - ζI)^{-1}$ with $ζ\in {\mathbb C} \setminus [0,\infty)$, 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 $ζ$.