Operator estimates for Neumann sieve problem

TL;DR

This paper provides quantitative estimates on the convergence rate of Neumann Laplacians in perforated domains with many small holes, advancing understanding of the sieve problem in spectral theory.

Contribution

The authors derive explicit operator norm estimates for the convergence rate of Neumann Laplacians in perforated domains, improving previous qualitative results under general conditions.

Findings

Established $L^2\to L^2$ operator norm convergence rates.

Derived $L^2\to H^1$ estimates with a corrector.

Extended the class of shapes and arrangements of holes for convergence analysis.

Abstract

Let $\Omega$ be a domain in $\mathbb{R}^n$, $\Gamma$ be a hyperplane intersecting $\Omega$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $\Gamma\cap\Omega$; when $\varepsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\mathscr{A}_\varepsilon$ be the Neumann Laplacian in the perforated domain $\Omega_\varepsilon=\Omega\setminus\Gamma_\varepsilon$, where $\Gamma_\varepsilon=\Gamma\setminus (\cup_k D_{k,\varepsilon})$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, $\mathscr{A}_\varepsilon$ converges in the strong resolvent sense to the Laplacian on $\Omega\setminus\Gamma$ subject to the so-called $\delta'$-conditions on $\Gamma$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms; in the latter case a special corrector is required.