Homogenization estimates for high order elliptic operators

TL;DR

This paper develops homogenization estimates for high-order elliptic operators with periodic coefficients, providing precise operator norm approximations of the resolvent with an error of order  in the whole space.

Contribution

It constructs an -order approximation of the resolvent for high-order elliptic operators with periodic coefficients, using a novel homogenized operator different from standard approaches.

Findings

Operator resolvent approximation with -order accuracy

Use of auxiliary periodic problems and smoothing operators

Homogenized operator differs from traditional models

Abstract

In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent $(A_\varepsilon+1)^{-1}$, we construct an approximation with the remainder term of order $\varepsilon^2$ in the operator $(L^2{\to}H^m)$-norm, using the resolvent of the homogenized operator, solutions of several auxiliary periodic problems on the unit cube, and smoothing operators. The homogenized operator here differs from the one commonly employed in homogenization.