Homogenization of quadratic convolution energies in periodically perforated domains

TL;DR

This paper establishes a homogenization result for quadratic convolution energies in perforated domains, showing that the limit energy is a Dirichlet-type quadratic form characterized by a non-local cell problem.

Contribution

It introduces a homogenization theorem for quadratic convolution energies in perforated domains, including a novel extension theorem for such domains with periodic perforations.

Findings

Limit energy is a Dirichlet-type quadratic form.

The integrand is characterized by a non-local cell problem.

Extension theorem applies to a broad class of perforated domains.

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.