English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper, we focus on the homogenization process of the non-localelliptic 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 typecondition outside of the bounded domain $\mathcal O\subseteq \mathbb{R}^n$. Wefind the homogenized problem by using the $H$-convergence method, as$\varepsilon\to 0$, under standard uniform ellipticity, boundedness andsymmetry assumptions on coefficients $A_\varepsilon(x)$, with the homogenizedcoefficients as the standard $H$-limit (cf. \cite{MT1}) of the sequence$\{A_\varepsilon\}_{\varepsilon>0}$. We also prove that the commonly referredto as \textit{the strange term} in the literature (see \cite[Chapter 4]{MT})does not appear in the homogenized problem associated with the fractionalLaplace operator $(-\Delta)^s$ in a perforated domain. Both of these resultshave been obtained in the class of general microstructures. Consequently, wecould certify that the homogenization process, as $\varepsilon\to 0$, is stableunder $s\to 1^{-}$ in the non-perforated domains, but not necessarily in thecase of perforated domains.

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