Homogenization of quadratic convolution energies in periodically perforated domains

TL;DR

This work addresses the homogenization of quadratic convolution energies defined on periodically perforated domains. By combining an extension theorem for perforated sets with a blow-up/compactness framework, the authors prove that the nonlocal energy $F_\varepsilon$ Gamma-converges to a local Dirichlet-type energy $F_{\rm hom}(u)=\int_{\Omega} \langle A_{\rm hom}\nabla u,\nabla u\rangle dx$, where the homogenized tensor $A_{\rm hom}$ is given by a non-local cell problem that couples short- and long-range interactions through the kernel $a$. The analysis requires decay of $a$ at infinity and a nondegeneracy condition near the origin; an extension theorem for perforated domains is central to passing to the limit. The results rigorously connect nonlocal convolution models on perforated media to classical local homogenized models, with implications for macroscopic descriptions of systems with long-range interactions.

Abstract

We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

Theorems & Definitions (24)

• Theorem 2.1: compactness theorem
• Proposition 2.2: treatment of boundary values
• proof
• Remark 2.3
• Definition 3.1
• Theorem 3.2: extension theorem
• Lemma 3.3
• proof : Proof of Lemma \ref{['ene_locali']}
• proof : Proof of Theorem \ref{['t_ext']}
• Definition 4.1