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arXiv · 1510.08209

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

Masaru Ikehata
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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 $\mbox{\boldmath $E$}$ and magnetic field $\mbox{\boldmath $H$}$ 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 impedance-or the Leontovich boundary condition $\mbox{\boldmath $ν$}\times\mbox{\boldmath $H$} -λ\,\mbox{\boldmath $ν$}\times(\mbox{\boldmath $E$}\times\mbox{\boldmath $ν$})=\mbox{\boldmath $0$}$ with $λ$ an unknown positive function and $\mbox{\boldmath $ν$}$ 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 impedance $λ$ is greater or less than the special value $\sqrt{ε/μ}$.

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Masaru Ikehata. 2015-11-16. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. https://doi.org/10.3934/ipi.2017006

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