Inverse Problems and Imaging (IPI)

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

Pages: 1025 - 1049, Volume 9, Issue 4, November 2015      doi:10.3934/ipi.2015.9.1025

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Fioralba Cakoni - Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, United States (email)
Houssem Haddar - INRIA, CMAP, Ecole polytechnique, Université Paris Saclay, Route de Saclay, 91128 Palaiseau, France (email)
Isaac Harris - Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States (email)

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.

Keywords:  Interior transmission problem, transmission eigenvalues, periodic inhomogeneous medium, inverse scattering problem, homogenization.
Mathematics Subject Classification:  Primary: 35R30, 35Q60, 35J40, 78A25.

Received: September 2014;      Revised: March 2015;      Available Online: October 2015.