Composites with invisible inclusions: eigenvalues of R-linear problem

TL;DR

This paper investigates an R-linear eigenvalue problem related to metamaterials with small circular inclusions, deriving asymptotic formulas and exploring eigenfunction properties to enable neutral coatings.

Contribution

It introduces a new R-linear eigenvalue problem for metamaterials, derives asymptotic eigenvalue formulas, and discusses eigenfunction properties for neutral inclusion design.

Findings

Asymptotic eigenvalue formula for small inclusions

Validation of the nodal domains conjecture for univalent eigenfunctions

Potential for designing neutral coatings around inclusions

Abstract

An new eigenvalue $\mathbb R$-linear problem arisen in the theory of metamaterials is stated and constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of inclusions tend to zero. The nodal domains conjecture related to univalent eigenfunctions is posed. Demonstration of the conjecture allows to justify that a set of inclusions can be made neutral by surrounding it with an appropriate coating.