American Institute of Mathematical Sciences

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March  2020, 19(3): 1387-1397. doi: 10.3934/cpaa.2020068

Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary

Yuebin Hao

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R.China

Received  March 2019 Revised  September 2019 Published  November 2019

The interior transmission eigenvalue problem plays a basic role in the study of inverse scattering problems for an inhomogeneous medium. In this paper, we consider the electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with conductive boundary. Our main focus is to understand the associated eigenvalue problem, more specifically to prove the transmission eigenvalues form a discrete set and show that they exist by employing a variety of variational techniques under various assumptions on the index of refraction.

Keywords: Transmission eigenvalues, conductive boundary condition, existence, discreteness.
Mathematics Subject Classification: Primary: 35Q61, 35P25; Secondary: 35J15.
Citation: Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068
References:
[1]

O. Bondarenko, I. Harris and A. Kleefeld, The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary, Appl. Anal., 96 (2017), 2-22.  doi: 10.1080/00036811.2016.1204440.  Google Scholar

[2]

F. Cakoni, H. Haddar and S. Meng, Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations, J. Integral Equations Appl., 27 (2015), 375-406.  doi: 10.1216/JIE-2015-27-3-375.  Google Scholar

[3]

L. Chesnel, Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach, Inverse Probl., 28 (2012), 065005, 14. doi: 10.1088/0266-5611/28/6/065005.  Google Scholar

[4]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.  Google Scholar

[5]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Paris, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[6]

F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2017), 145-162.  doi: 10.1137/090754637.  Google Scholar

[7]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[8]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[9]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[10]

F. Cakoni and H. Haddar, Transmission eigenvalues[Editorial], Inverse Probl., 29 (2013), 100201, 3. doi: 10.1088/0266-5611/29/10/100201.  Google Scholar

[11]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, In Inverse problems and applications: inside out. Ⅱ, Sci. Res. Inst. Publ., 60 (2013), 529–580, Cambridge Univ. Press, Cambridge.  Google Scholar

[12]

A. Cossonnière and H. Haddar, Surface integral formulation of the interior transmission problem, J. Integral Equations Appl., 25 (2013), 341-376.  doi: 10.1216/JIE-2013-25-3-341.  Google Scholar

[13]

A. Cossonnière and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal., 43 (2011), 1698-1715.  doi: 10.1137/100813890.  Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences, third edition. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[15]

C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM J. Math. Anal., 27 (1996), 1597-1630.  doi: 10.1137/S0036141094271259.  Google Scholar

[16]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[17]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Probl., 28 (2012), 055009, 23. doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[18]

H. Haddar, The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem, Math. Methods Appl. Sci., 27 (2004), 2111-2129.  doi: 10.1002/mma.465.  Google Scholar

[19]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Probl., 30 (2014), 035016, 21. doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[20]

H. Haddar and S. Meng, The spectral analysis of the interior transmission eigenvalue problem for maxwells equations, arXiv: 1707.04815v2. Google Scholar

[21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36 of Oxford Lecture Series in Mathematics and its ApplicationsOxford University Press, Oxford, 2008.   Google Scholar
[22]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Probl., 29 (2013), 104011, 21. doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[23]

J. Li, X. Li, H. Liu and Y. Wang, Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking, arXiv: 1701.05301v1. Google Scholar

[24]

E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174.  doi: 10.1137/11084738X.  Google Scholar

[25]

E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Probl., 29 (2013), 104003, 19. doi: 10.1088/0266-5611/29/10/104003.  Google Scholar

[26] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[27]

L.Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Probl., 29 (2013), 104001, 28. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[28]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.  doi: 10.1137/110836420.  Google Scholar

[29]

F. Yang and P. Monk, The interior transmission problem for regions on a conducting surface, Inverse Probl., 30 (2014), 015007, 34. doi: 10.1088/0266-5611/30/1/015007.  Google Scholar

show all references

References:
[1]

O. Bondarenko, I. Harris and A. Kleefeld, The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary, Appl. Anal., 96 (2017), 2-22.  doi: 10.1080/00036811.2016.1204440.  Google Scholar

[2]

F. Cakoni, H. Haddar and S. Meng, Boundary integral equations for the transmission eigenvalue problem for Maxwell's equations, J. Integral Equations Appl., 27 (2015), 375-406.  doi: 10.1216/JIE-2015-27-3-375.  Google Scholar

[3]

L. Chesnel, Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach, Inverse Probl., 28 (2012), 065005, 14. doi: 10.1088/0266-5611/28/6/065005.  Google Scholar

[4]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.  Google Scholar

[5]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C. R. Math. Acad. Sci. Paris, 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[6]

F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2017), 145-162.  doi: 10.1137/090754637.  Google Scholar

[7]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[8]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[9]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[10]

F. Cakoni and H. Haddar, Transmission eigenvalues[Editorial], Inverse Probl., 29 (2013), 100201, 3. doi: 10.1088/0266-5611/29/10/100201.  Google Scholar

[11]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, In Inverse problems and applications: inside out. Ⅱ, Sci. Res. Inst. Publ., 60 (2013), 529–580, Cambridge Univ. Press, Cambridge.  Google Scholar

[12]

A. Cossonnière and H. Haddar, Surface integral formulation of the interior transmission problem, J. Integral Equations Appl., 25 (2013), 341-376.  doi: 10.1216/JIE-2013-25-3-341.  Google Scholar

[13]

A. Cossonnière and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal., 43 (2011), 1698-1715.  doi: 10.1137/100813890.  Google Scholar

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences, third edition. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[15]

C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM J. Math. Anal., 27 (1996), 1597-1630.  doi: 10.1137/S0036141094271259.  Google Scholar

[16]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[17]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Probl., 28 (2012), 055009, 23. doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[18]

H. Haddar, The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem, Math. Methods Appl. Sci., 27 (2004), 2111-2129.  doi: 10.1002/mma.465.  Google Scholar

[19]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Probl., 30 (2014), 035016, 21. doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[20]

H. Haddar and S. Meng, The spectral analysis of the interior transmission eigenvalue problem for maxwells equations, arXiv: 1707.04815v2. Google Scholar

[21] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36 of Oxford Lecture Series in Mathematics and its ApplicationsOxford University Press, Oxford, 2008.   Google Scholar
[22]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Probl., 29 (2013), 104011, 21. doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[23]

J. Li, X. Li, H. Liu and Y. Wang, Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking, arXiv: 1701.05301v1. Google Scholar

[24]

E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44 (2012), 1165-1174.  doi: 10.1137/11084738X.  Google Scholar

[25]

E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Probl., 29 (2013), 104003, 19. doi: 10.1088/0266-5611/29/10/104003.  Google Scholar

[26] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[27]

L.Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Probl., 29 (2013), 104001, 28. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[28]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.  doi: 10.1137/110836420.  Google Scholar

[29]

F. Yang and P. Monk, The interior transmission problem for regions on a conducting surface, Inverse Probl., 30 (2014), 015007, 34. doi: 10.1088/0266-5611/30/1/015007.  Google Scholar

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