On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method

Masaru Ikehata

doi:10.3934/ipi.2017006

Article Contents

2017, Volume 11, Issue 1: 99-123. Doi: 10.3934/ipi.2017006 open access

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On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method

Masaru Ikehata

Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

Received: October 31, 2015

Revised: July 31, 2016

Published: March 2018

The author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 25400155) and (B) (No. 26287020) of Japan Society for the Promotion of Science

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Abstract

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field $\boldsymbol{E}$ and magnetic field $\boldsymbol{ H}$ which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity $ε$ and magnetic permeability $μ$. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition $\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$ with admittance $λ$ an unknown positive function and $\boldsymbol{ ν}$ the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the admittance $λ$ is greater or less than the special value $\sqrt{ε/μ}$.

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Mathematics Subject Classification: Primary: 35R30, 35L50, 35Q61; Secondary: 78A46, 78M35.

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