# Homogenization in Perforated Domains and Interior Lipschitz Estimates

## TL;DR

This paper proves interior Lipschitz estimates for linear elasticity systems with rapidly oscillating periodic coefficients in perforated domains, using direct methods to establish convergence rates and Liouville estimates.

## Contribution

It introduces a direct approach to obtain interior Lipschitz estimates and convergence rates for elasticity systems in perforated domains, avoiding compactness arguments.

## Key findings

- Established $H^1$-convergence rates for solutions.
- Derived interior Lipschitz estimates at the macroscopic scale.
- Proved Liouville type estimates in unbounded perforated domains.

## Abstract

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale $\varepsilon$ by establishing $H^1$-convergence rates for such solutions. The interior estimates are derived directly without the use of compactness via an argument presented in [3] that was adapted for elliptic equations in [2] and [11]. As a consequence, we derive a Liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains.