Homogenisation and spectral convergence of high-contrast convolution type operators

TL;DR

This paper studies the homogenisation and spectral properties of high-contrast convolution operators in periodic media, adapting two-scale convergence to nonlocal operators and analyzing the spectrum's limit behavior as microstructure scales to zero.

Contribution

It introduces a novel adaptation of two-scale convergence for nonlocal convolution operators and characterizes the spectral convergence in high-contrast periodic media.

Findings

Spectrum of the limit operator is a subset of the limit of the original spectrum.

The spectra of the limit and original operators need not coincide.

Homogenisation results are obtained for both whole space and bounded domains.

Abstract

The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition. Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset the limit of the spectrum of the original operator, and that they need not coincide.