Improved homogenization estimates for high order elliptic systems

TL;DR

This paper develops improved homogenization estimates for high-order elliptic systems with periodic coefficients, providing an approximation of the resolvent with an error of order  in the operator norm without regularity assumptions.

Contribution

It introduces a novel approximation method for the resolvent of high-order elliptic operators with periodic coefficients, achieving an -order error estimate in the operator norm.

Findings

Resolvent approximation with -order accuracy

No regularity assumptions beyond ellipticity and boundedness

Use of two-scale expansions with Steklov smoothing

Abstract

In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even order larger than 2 with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. We constuct an approximation for the resolvent of $L_\varepsilon$ with the remainder term of order $\varepsilon^2$ in the operator $L^2$-norm. We impose no regularity conditions on the operator beyond ellipticity and boundedness of coefficients. We use two scale expansions with correctors regularized by the Steklov smoothing.