Homogenization and norm resolvent convergence for elliptic operators in a strip perforated along a curve

TL;DR

This paper studies the homogenization of elliptic operators in a perforated strip along a curve, proving norm resolvent convergence and spectrum convergence under weak perforation assumptions.

Contribution

It establishes the norm resolvent convergence of elliptic operators in non-periodic perforated domains and characterizes all possible homogenized limits.

Findings

Proves norm resolvent convergence in various operator norms

Provides estimates for the rate of convergence

Shows convergence of the spectrum of the operators

Abstract

We consider an infinite planar straight strip perforated by small holes along a curve. In such domain, we consider a general second order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm resolvent convergence, we prove the convergence of the spectrum.