Homogenization for non-local elliptic operators in both perforated and non-perforated domains

TL;DR

This paper studies the homogenization of non-local elliptic operators in perforated and non-perforated domains, showing stability of the process and clarifying the absence of the strange term in the homogenized limit.

Contribution

It introduces a homogenization approach for non-local elliptic operators using $H$-convergence, and demonstrates the non-appearance of the strange term in perforated domains.

Findings

Homogenization process is stable as $s o 1^{-}$ in non-perforated domains.

The strange term does not appear in the homogenized problem for perforated domains.

Results hold for general microstructures.

Abstract

In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with $0<s<1$, considering non-homogeneous Dirichlet type condition outside of the bounded domain $\mathcal O\subseteq \mathbb{R}^n$. We find the homogenized problem by using the $H$-convergence method, as $\varepsilon\to 0$, under standard uniform ellipticity, boundedness and symmetry assumptions on coefficients $A_\varepsilon(x)$, with the homogenized coefficients as the standard $H$-limit (cf. \cite{MT1}) of the sequence $\{A_\varepsilon\}_{\varepsilon>0}$. We also prove that the commonly referred to as \textit{the strange term} in the literature (see \cite[Chapter 4]{MT}) does not appear in the homogenized problem associated with the fractional Laplace operator $(-\Delta)^s$ in a perforated domain. Both of these results have been obtained in the class of general microstructures. Consequently, we could certify that the homogenization process, as $\varepsilon\to 0$, is stable under $s\to 1^{-}$ in the non-perforated domains, but not necessarily in the case of perforated domains.