American Institute of Mathematical Sciences

Advanced
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
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). Google Scholar

[2]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,'', Interaction of Mechanics and Mathematics, (2006). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. Google Scholar

[8]

J. Chazarain and A. Piriou, "Introduction to the Theory of Linear Partial Differential Equations,'', North-Holland, (1982). Google Scholar

[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. Google Scholar

[10]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,'', Oxford Lecture Series in Mathematics and its Applications, (2008). Google Scholar

[11]

H. Kumanogo, "Pseudodifferential Operators,'', MIT Press, (1981). Google Scholar

[12]

E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000). Google Scholar

[16]

G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data,, SIAM J. Math. Anal., 39 (2007), 819. doi: 10.1137/060658667. Google Scholar

[17]

L. Paivarinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar

[18]

R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. of Math., 91 (1969), 889. doi: 10.2307/2373309. Google Scholar

[19]

N. Shimakura, "Partial Differential Operators of Elliptic Type,'', AMS, (1992). Google Scholar

[20]

M. E. Taylor, "Partial Differential Equations II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996). Google Scholar

[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. Google Scholar

[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). Google Scholar

[23]

F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators,'', 1, 1 (1980). Google Scholar

[24]

N. Valdivia, Uniqueness in inverse obstacle scattering with conductive boundary conditions,, Appl. Anal., 83 (2004), 825. doi: 10.1080/00036810410001689283. Google Scholar

show all references

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). Google Scholar

[2]

F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,'', Interaction of Mechanics and Mathematics, (2006). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

D. Colton, L. Paivarinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. Google Scholar

[8]

J. Chazarain and A. Piriou, "Introduction to the Theory of Linear Partial Differential Equations,'', North-Holland, (1982). Google Scholar

[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. Google Scholar

[10]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,'', Oxford Lecture Series in Mathematics and its Applications, (2008). Google Scholar

[11]

H. Kumanogo, "Pseudodifferential Operators,'', MIT Press, (1981). Google Scholar

[12]

E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,'', Cambridge University Press, (2000). Google Scholar

[16]

G. Nakamura and M. Sini, Obstacle and boundary determination from scattering data,, SIAM J. Math. Anal., 39 (2007), 819. doi: 10.1137/060658667. Google Scholar

[17]

L. Paivarinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar

[18]

R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. of Math., 91 (1969), 889. doi: 10.2307/2373309. Google Scholar

[19]

N. Shimakura, "Partial Differential Operators of Elliptic Type,'', AMS, (1992). Google Scholar

[20]

M. E. Taylor, "Partial Differential Equations II. Qualitative Studies of Linear Equations,'', Applied Mathematical Sciences, (1996). Google Scholar

[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. Google Scholar

[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). Google Scholar

[23]

F. Treves, "Introduction to Pseudodifferential and Fourier Integral Operators,'', 1, 1 (1980). Google Scholar

[24]

N. Valdivia, Uniqueness in inverse obstacle scattering with conductive boundary conditions,, Appl. Anal., 83 (2004), 825. doi: 10.1080/00036810410001689283. Google Scholar

[1]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[2]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[3]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[4]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2014, 13 (1) : 175-202. doi: 10.3934/cpaa.2014.13.175
[5]

Bernd Ammann, Robert Lauter and Victor Nistor. Algebras of pseudodifferential operators on complete manifolds. Electronic Research Announcements, 2003, 9: 80-87.

[6]

Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443
[7]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[8]

Catarina Carvalho, Victor Nistor, Yu Qiao. Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electronic Research Announcements, 2017, 24: 68-77. doi: 10.3934/era.2017.24.008
[9]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
[10]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[11]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[12]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
[13]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[14]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[15]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
[16]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[17]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[18]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[19]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[20]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences