An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity

TL;DR

This paper introduces an operator-asymptotic method for periodic homogenization in 3D linearized elasticity, providing uniform resolvent estimates and natural derivation of correctors, advancing theoretical understanding of elastic media.

Contribution

It develops a novel asymptotic approach combined with a uniform Korn inequality, yielding new norm-resolvent estimates and natural correctors in elasticity homogenization.

Findings

Established $L^2\to L^2$, $L^2\to H^1$ estimates for the resolvent.

Derived higher-order correctors consistent with classical formulas.

Provided a uniform approximation framework for solutions in elastic media.

Abstract

We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength $|\chi|$ for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (``quasimomentum") $\chi$. We develop an asymptotic procedure in powers of $|\chi|$, combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order $L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ and higher-order $L^2 \to L^2$ correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae.