American Institute of Mathematical Sciences

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August  2012, 6(3): 487-521. doi: 10.3934/ipi.2012.6.487

The Green function of the interior transmission problem and its applications

Kyoungsun Kim 1, , Gen Nakamura 2, and  Mourad Sini 3,

1. 

Department of Mathematics of Inha University, Incheon 402-751, South Korea
2. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
3. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz A-4040, Australia

Received  February 2012 Revised  May 2012 Published  September 2012

The interior transmission problem appears naturally in the scattering theory. In this paper, we construct the Green function associated to this problem. In addition, we provide point-wise estimates of this Green function similar to those known for the Green function related to the classical transmission problems. These estimates are, in particular, useful to the study of various inverse scattering problems. Here, we apply them to justify some asymptotic formulas already used for detecting partially coated dielectric mediums from far field measurements.
Keywords: interior transmission problem, pseudodifferential operators., Green function.
Mathematics Subject Classification: Primary: 35J08, 35J25; Secondary: 35S1.
Citation: Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
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show all references

References:
[1]

Inverse Problems, 26 (2010), 29pp.  Google Scholar

[2]

Interaction of Mechanics and Mathematics, Springer, 2006.  Google Scholar

[3]

J. Comput. Appl. Math., 146 (2002), 285-299. doi: 10.1016/S0377-0427(02)00361-8.  Google Scholar

[4]

Appl. Anal., 88 (2009), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[5]

Applicable Analysis, 89 (2010), 67-86. doi: 10.1080/00036810903437820.  Google Scholar

[6]

Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.  Google Scholar

[7]

Inverse Problems and Imaging, 1 (2007), 13-28.  Google Scholar

[8]

North-Holland, Amsterdam, 1982.  Google Scholar

[9]

Math. Res. Lett., 18 (2011), 279-293.  Google Scholar

[10]

Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[11]

MIT Press, Cambridge, 1981.  Google Scholar

[12]

SIAM J. Math. Anal., 44 (2012), 1165-1174.  Google Scholar

[13]

SIAM J. Appl. Math., 67 (2007), 1124-1146. doi: 10.1137/060654220.  Google Scholar

[14]

SIAM J. Scient. Comp., 31 (2009), 2665-2687. doi: 10.1137/080718024.  Google Scholar

[15]

Cambridge University Press, Cambridge, 2000.  Google Scholar

[16]

SIAM J. Math. Anal., 39 (2007), 819-837. doi: 10.1137/060658667.  Google Scholar

[17]

SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[18]

Amer. J. of Math., 91 (1969), 889-920. doi: 10.2307/2373309.  Google Scholar

[19]

AMS, Providence, 1992.  Google Scholar

[20]

Applied Mathematical Sciences, 116, Springer-Verlag, 1996.  Google Scholar

[21]

Inverse Problems, 26 (2010), 29 pp. doi: 10.1088/0266-5611/26/1/015010.  Google Scholar

[22]

Inverse Problems, 26 (2010), 24 pp.  Google Scholar

[23]

1, Plenum, New York, 1980.  Google Scholar

[24]

Appl. Anal., 83 (2004), 825-851. doi: 10.1080/00036810410001689283.  Google Scholar

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