Spectral homogenization for a Robin-Neumann problem

TL;DR

This paper investigates the asymptotic behavior of eigenpairs in a perforated domain with Robin-Neumann boundary conditions, deriving a homogenized problem and convergence rates using advanced mathematical techniques.

Contribution

It introduces a homogenization approach for Robin-Neumann spectral problems in perforated domains, including eigenpair convergence rates and eigenspace analysis.

Findings

Eigenpairs converge to the homogenized problem with rate √ε.

Homogenized problem derived using Višik lemma and Mosco convergence.

Asymptotic analysis of eigenpairs as perforation size tends to zero.

Abstract

We consider a Neumann-Robin spectral problem in a perforated domain $\Omeps$. By homogenization techniques we find the suitable homogenized problem and we discuss the asymptotics of eigenpairs, as the size of the perforation tends to zero. Our results involve an approach based on Vi\v{s}\'ik lemma and the Mosco convergence of eigenspaces. We prove that eigenpairs of our problem converge to eigenpairs of the homogenized problem with rate $\sqrt{\eps}$.