# American Institute of Mathematical Sciences

February  2017, 11(1): 99-123. doi: 10.3934/ipi.2017006

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

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

Received  November 2015 Revised  August 2016 Published  January 2017

Fund Project: 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.

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 $\sqrt{ε/μ}$.

Citation: Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006
##### References:
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##### References:
 [1] N. G. Alexopoulos and G. A. Tadler, Accuracy of the Leontovich boundary condition for continuous and discontinuous surface impedances, J. Appl. Phys., 46 (2008), 3326-3332.  doi: 10.1063/1.322058.  Google Scholar [2] C. A. Balanis, Antenna Theory, Analysis and Design 3$^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005. Google Scholar [3] E. B. Bykovskii, A solution of the mixed problem for the Maxwell's equations in the case of an ideal conducting boundary, Vestnik Leningrad Univ., 12 (1957), 50-66.   Google Scholar [4] M. Cheney and R. Borden, Fundamentals of Radar Imaging CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291.  Google Scholar [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory 3rd edn, New York, Springer, 2013.  Google Scholar [6] R. Courantt and D. Hilbert, Methoden der Mathematischen Physik Vol. 2, Berlin, Springer, 1937. Google Scholar [7] R. Dautray and J. -L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications Vol. 3, Springer-Verlag, Berlin, 1990.  Google Scholar [8] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308.  Google Scholar [9] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data Inverse Problems 28 (2012), 045010, 29pp. doi: 10.1088/0266-5611/28/4/045010.  Google Scholar [10] M. Ikehata, On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain Inverse Problems 31 (2015), 085011, 21pp. doi: 10.1088/0266-5611/31/8/085011.  Google Scholar [11] M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Problems and Imaging, 10 (2016), 131-163.  doi: 10.3934/ipi.2016.10.131.  Google Scholar [12] M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. Marin, L. , Munteanu, L. , Chiroiu, V. ), 2,123-147, Editura Academiei, Bucharest, Romania, in press. Google Scholar [13] M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain Math. Meth. Appl. Sci. 2016. doi: 10.1002/mma.4021.  Google Scholar [14] B. V. Kapitonov, On exponential decay as $t\longrightarrow∞$ of solutions of an exterior boundary value problem for the Maxwell system, Math. USSR Sbornik, 66 (1990), 475-498.  doi: 10.1070/SM1990v066n02ABEH001318.  Google Scholar [15] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-harmononic Maxwell's Equations, Expansion-, Integral-, and Variational Methods Springer, 2015.  Google Scholar [16] S. G. Krein and I. M. Kulikov, The Maxwell-Leontovich operator, (Russian)Differentsial'nye Uravneniya, 5 (1969), 1275-1282; English transl. in Differential Equations, 5(1969), 937-943. Google Scholar [17] V. A. Solonnikov, Overdetermined elliptic boundary value problems, Zap. Nauchn. Sem. LOMI, 21(1971), 112-158; English transl. in J. Soviet Math., 1 (1973), 477-512.  Google Scholar [18] M. V. Urev, On the Maxwell system under impedance boundary conditions with memory, Siberian Math. J., 55 (2014), 548-563.  doi: 10.1134/S0037446614030161.  Google Scholar [19] K. Yosida, Functional Analysis Third Edtition, Springer, New York, 1971. Google Scholar
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