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 $A_{\epsilon }$ of order $2p$ given by the expression $b(\mathbf{D}{)}^{\ast }g(\mathbf{x}/\epsilon )b(\mathbf{D})$, $\epsilon >0$. 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 $b(\mathbf{\xi })$ has maximal rank. For the resolvent $(A_{\epsilon }-\zeta I{)}^{-1}$ 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 $\zeta$.

53 pages