Operator estimates for homogenization of the Robin Laplacian in a perforated domain

TL;DR

This paper provides quantitative operator estimates for the homogenization of the Robin Laplacian in a perforated domain, improving understanding of the convergence rate and spectral distance in the homogenization process.

Contribution

The paper derives explicit convergence rate estimates for the Robin Laplacian in perforated domains, extending previous qualitative results to quantitative operator norm bounds.

Findings

Established $L^2$ operator norm convergence rates.

Derived estimates on spectral distance between operators.

Extended homogenization results to include Robin boundary conditions.

Abstract

Let $\varepsilon>0$ be a small parameter. We consider the domain $\Omega_\varepsilon:=\Omega\setminus D_\varepsilon$, where $\Omega$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $\Delta_\varepsilon$ be the Laplace operator in $\Omega_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+\gamma_\varepsilon u = 0$ with $\gamma_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on $d_\varepsilon$ and $\gamma_\varepsilon$, the operator $\Delta_\varepsilon$ converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in $\Omega$ and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.