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Homogenization and operator estimates for Steklov problems in perforated domains

Andrii Khrabustovskyi, Jari Taskinen

Abstract

Let the set $Ω_\varepsilon$ be obtained from the bounded domain $Ω$ by removing a family of $\varepsilon$-periodically distributed identical balls. In $Ω_\varepsilon$ one considers the standard Steklov spectral problem. It is known from [Girouard-Henrot-Lagacé, 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 $-Δu=λQ u$ on $Ω$ 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 $Ω$ is not assumed to be bounded.

Homogenization and operator estimates for Steklov problems in perforated domains

Abstract

Let the set be obtained from the bounded domain by removing a family of -periodically distributed identical balls. In one considers the standard Steklov spectral problem. It is known from [Girouard-Henrot-Lagacé, 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 , the Steklov spectrum converges to the spectrum of the problem on with some weight . 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 is not assumed to be bounded.

Paper Structure

This paper contains 11 sections, 14 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Geometry
  4. Assumption on the 'holes' $D_{i,\varepsilon}$
  5. The operator ${\mathscr{R}}_{\varepsilon}$
  6. Main results
  7. Remarks on holes shape assumptions
  8. Auxiliary estimates
  9. Proof of the main results
  10. Proof of Theorem \ref{['th1']}
  11. Proof of Theorem \ref{['th2']}

Key Result

Theorem 2.1

Under the geometric assumptions imposed above the following estimates hold: and

Figures (2)

  • Figure 1: Examples of space tessellations satisfying \ref{['Gamma:cond:1']}. Left: a Voronoi tessellation generated by a countable set of points. Right: a rectangular tessellation.
  • Figure 2: The perforated domain $\Omega_{\varepsilon}$ defined by \ref{['Omega:e']}. The dotted lines correspond to the boundaries of the cells $\Gamma_{i,\varepsilon}$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1: MK06
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 15 more