Homogenization approach for the transmission eigenvalue problem for periodic media and application to the inverse problem

TL;DR

This paper studies how the transmission eigenvalues in periodic media converge to those of a homogenized problem, enabling the extraction of material properties from scattering data.

Contribution

It establishes the convergence of solutions and eigenvalues in periodic media to their homogenized counterparts, and proposes a method to infer material properties from measurable eigenvalues.

Findings

Transmission eigenvalues converge to homogenized eigenvalues

Solution of the transmission problem converges to the homogenized solution

First transmission eigenvalue can reveal effective material properties

Abstract

We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The convergence results presented here are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper.