Homogenization of high order elliptic operators with periodic coefficients

TL;DR

This paper develops operator norm approximations for the resolvent of high order elliptic operators with periodic coefficients, advancing homogenization theory for complex differential operators in mathematical physics.

Contribution

It provides new norm-resolvent approximation results for high order elliptic operators with periodic coefficients, including explicit error estimates depending on the small parameter and spectral parameter.

Findings

Operator norm approximations for resolvents are established.

Error estimates depend explicitly on the homogenization parameter and spectral shift.

Results extend homogenization techniques to higher order elliptic operators.

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, $b({\mathbf D})=\sum_{|\alpha|=p}^d b_\alpha {\mathbf D}^\alpha$ 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 $\zeta$.