Improved approximations of resolvents in homogenization of fourth-order operators with periodic coefficients

TL;DR

This paper develops improved approximations for the resolvent operators of fourth-order elliptic operators with periodic coefficients in homogenization, achieving higher accuracy with remainder terms of order O(ε^2) and O(ε^3).

Contribution

It introduces new homogenization approximations for resolvents of fourth-order operators with periodic coefficients, with enhanced error estimates of order O(ε^2) and O(ε^3).

Findings

Resolvent approximation with O(ε^2) remainder in H^2 norm.

Refined resolvent approximation with O(ε^3) remainder in L^2 norm.

Use of two-scale expansions with smoothing to achieve higher accuracy.

Abstract

In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic fourth-order operator $A_\varepsilon$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent $(A_\varepsilon+1)^{-1}$, acting as an operator from $L^2$ to $H^2$, we find an approximation with remainder term of order $O(\varepsilon^2)$ as $\varepsilon$ tends to $0$. Relying on this result, we construct the resolvent approximation with remainder of order $O(\varepsilon^3)$ in the operator $L^2$-norm. We employ two-scale expansions that involve smoothing.