Layer potentials for Lam\'e systems and homogenization of perforated elastic medium with clamped holes

TL;DR

This paper develops a layer potential approach to analyze the homogenization of Lamé systems in perforated elastic media with clamped holes, providing quantitative error estimates across various hole-to-cell ratios.

Contribution

It introduces a unified layer potential method for homogenization of perforated elastic media, including natural correctors and asymptotic analysis of cell problems.

Findings

Established quantitative homogenization results for perforated Lamé systems.

Developed a unified layer potential framework applicable to different hole-cell ratios.

Provided error estimates and asymptotic analysis for the cell problems.

Abstract

We investigate Lam\'e systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lam\'e system to those in the whole space when the period tends to infinity.