The Green function of the interior transmission problem and its applications

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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

Abstract
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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

References:

[1]	M. F. Ben Hassen, O. Ivanyshyn and M. Sini, Three-dimensional acoustic scattering by complex obstacles. The accuracy issue,, Inverse Problems, 26 (2010).
[2]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,'', Interaction of Mechanics and Mathematics, (2006).
[3]	F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media,, J. Comput. Appl. Math., 146 (2002), 285. doi: 10.1016/S0377-0427(02)00361-8.
[4]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Appl. Anal., 88 (2009), 475. doi: 10.1080/00036810802713966.
[5]	F. Cakoni, G. Nakamura, M. Sini and N. Zeev, The identification of a partially coated dielectric from far field measurements,, Applicable Analysis, 89 (2010), 67. doi: 10.1080/00036810903437820.
[6]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003.
[7]	D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.
[8]	J. Chazarain and A. Piriou, "Introduction to the Theory of Linear Partial Differential Equations,'', North-Holland, (1982).
[9]	M. Hitrik, K. Krupchyk, P. Ola and L. Paivarinta, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279.
[10]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,'', Oxford Lecture Series in Mathematics and its Applications, (2008).
[11]	H. Kumanogo, "Pseudodifferential Operators,'', MIT Press, (1981).
[12]	E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165.
[13]	J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder,, SIAM J. Appl. Math., 67 (2007), 1124. doi: 10.1137/060654220.
[14]	J. Liu and M. Sini, On the accuracy of the numerical detection of complex obstacles from far field data using the probe method,, SIAM J. Scient. Comp., 31 (2009), 2665. doi: 10.1137/080718024.
[15]	W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000).
[16]	G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data,, SIAM J. Math. Anal., 39 (2007), 819. doi: 10.1137/060658667.
[17]	L. Paivarinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525.
[18]	R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. of Math., 91 (1969), 889. doi: 10.2307/2373309.
[19]	N. Shimakura, "Partial Differential Operators of Elliptic Type,'', AMS, (1992).
[20]	M. E. Taylor, "Partial Differential Equations II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).
[21]	N. T. Thanh and M. Sini, An analysis of the accuracy of the linear sampling method for an acoustic inverse obstacle scattering problem,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/1/015010.
[22]	N. T. Thanh and M. Sini, Accuracy of the linear sampling method for inverse obstacle scattering: effect of geometrical and physical parameters,, Inverse Problems, 26 (2010).
[23]	F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators,'', 1, 1 (1980).
[24]	N. Valdivia, Uniqueness in inverse obstacle scattering with conductive boundary conditions,, Appl. Anal., 83 (2004), 825. doi: 10.1080/00036810410001689283.

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References:

[1]	M. F. Ben Hassen, O. Ivanyshyn and M. Sini, Three-dimensional acoustic scattering by complex obstacles. The accuracy issue,, Inverse Problems, 26 (2010).
[2]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,'', Interaction of Mechanics and Mathematics, (2006).
[3]	F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media,, J. Comput. Appl. Math., 146 (2002), 285. doi: 10.1016/S0377-0427(02)00361-8.
[4]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Appl. Anal., 88 (2009), 475. doi: 10.1080/00036810802713966.
[5]	F. Cakoni, G. Nakamura, M. Sini and N. Zeev, The identification of a partially coated dielectric from far field measurements,, Applicable Analysis, 89 (2010), 67. doi: 10.1080/00036810903437820.
[6]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inverse Problems, 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003.
[7]	D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.
[8]	J. Chazarain and A. Piriou, "Introduction to the Theory of Linear Partial Differential Equations,'', North-Holland, (1982).
[9]	M. Hitrik, K. Krupchyk, P. Ola and L. Paivarinta, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279.
[10]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,'', Oxford Lecture Series in Mathematics and its Applications, (2008).
[11]	H. Kumanogo, "Pseudodifferential Operators,'', MIT Press, (1981).
[12]	E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165.
[13]	J. J. Liu, G. Nakamura and M. Sini, Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder,, SIAM J. Appl. Math., 67 (2007), 1124. doi: 10.1137/060654220.
[14]	J. Liu and M. Sini, On the accuracy of the numerical detection of complex obstacles from far field data using the probe method,, SIAM J. Scient. Comp., 31 (2009), 2665. doi: 10.1137/080718024.
[15]	W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000).
[16]	G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data,, SIAM J. Math. Anal., 39 (2007), 819. doi: 10.1137/060658667.
[17]	L. Paivarinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525.
[18]	R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. of Math., 91 (1969), 889. doi: 10.2307/2373309.
[19]	N. Shimakura, "Partial Differential Operators of Elliptic Type,'', AMS, (1992).
[20]	M. E. Taylor, "Partial Differential Equations II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996).
[21]	N. T. Thanh and M. Sini, An analysis of the accuracy of the linear sampling method for an acoustic inverse obstacle scattering problem,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/1/015010.
[22]	N. T. Thanh and M. Sini, Accuracy of the linear sampling method for inverse obstacle scattering: effect of geometrical and physical parameters,, Inverse Problems, 26 (2010).
[23]	F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators,'', 1, 1 (1980).
[24]	N. Valdivia, Uniqueness in inverse obstacle scattering with conductive boundary conditions,, Appl. Anal., 83 (2004), 825. doi: 10.1080/00036810410001689283.

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