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

Fioralba  Cakoni; Houssem  Haddar; Isaac  Harris

doi:10.3934/ipi.2015.9.1025

Article Contents

2015, Volume 9, Issue 4: 1025-1049. Doi: 10.3934/ipi.2015.9.1025

This issue Previous Article Stabilized BFGS approximate Kalman filter Next Article A new Kohn-Vogelius type formulation for inverse source problems

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

1.
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019
2.
INRIA, CMAP, Ecole polytechnique, Université Paris Saclay, Route de Saclay, 91128 Palaiseau
3.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received: September 2014

Revised: March 2015

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A25.

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