Homogenization of high order elliptic operators with periodic coefficients

Abstract:

A selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $\varepsilon >0$, is studied in $L_2(\mathbb {R}^d;\mathbb {C}^n)$. Here $g(\mathbf {x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function on $\mathbb {R}^d$; it is assumed that $g(\mathbf {x})$ is periodic with respect to some lattice. Next, $b(\mathbf {D})=\sum _{|\alpha |=p} b_\alpha \mathbf {D}^\alpha$ is a differential operator of order $p$ with constant coefficients; the $b_\alpha$ 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 $\zeta \in \mathbb {C} \setminus [0,\infty )$, approximations are obtained 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 $\zeta$.