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

Fioralba Cakoni (1) , Houssem Haddar (2) , Isaac Harris (1)

1 Department of Mathematical Sciences
2 DeFI - Shape reconstruction and identification

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.

Mots clés: Interior transmission problem; homogenization; transmission eigenvalues; periodic in-homogeneous medium; inverse scattering problem

Domaines: Analyse numérique [cs.NA]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging , 2015, 9 (4), pp.1025 - 1049. ⟨10.3934/ipi.2015.9.1025⟩. ⟨hal-01110294⟩