Homogenization and operator estimates for Steklov problems in perforated domains

TL;DR

This paper studies the asymptotic behavior of Steklov spectral problems in perforated domains with periodically distributed holes, proving operator norm convergence and spectral estimates as the perforation scale tends to zero.

Contribution

It extends previous results by establishing resolvent operator convergence and quantitative spectral estimates under general conditions on holes, even in unbounded domains.

Findings

Resolvent operators converge in operator norm as perforation size shrinks.

Spectral Hausdorff distance estimates are provided.

Results apply to unbounded domains with general hole configurations.

Abstract

Let the set $\Omega_\varepsilon$ be obtained from the bounded domain $\Omega$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $\Omega_\varepsilon$ one considers the standard Steklov spectral problem. It is known from [Girouard-Henrot-Lagac\'e, ARMA (2021)] that, if the radii of the holes shrink at a critical rate such that the surface area of a single hole is comparable to the volume of a periodicity cell, then, in the limit $\varepsilon \to 0$, the Steklov spectrum converges to the spectrum of the problem $-\Delta u=\lambda Q u$ on $\Omega$ with some weight $Q>0$. In the present work, we extend this result by proving, under fairly general assumptions on the locations and shapes of the holes, convergence of the associated resolvent operators in the operator norm topology, together with quantitative estimates for the Hausdorff distance between the spectra. The underlying domain $\Omega$ is not assumed to be bounded.