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

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

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.

Citation: Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068
References:
[1]

O. BondarenkoI. 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.

[2]

F. CakoniH. 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.

[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.

[4]

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

[5]

F. CakoniD. 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.

[6]

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

[7]

F. CakoniA. 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.

[8]

F. CakoniD. 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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[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. 
[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.

[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.

[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.

[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.

[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.
[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.

[28]

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

[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.

show all references

References:
[1]

O. BondarenkoI. 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.

[2]

F. CakoniH. 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.

[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.

[4]

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

[5]

F. CakoniD. 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.

[6]

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

[7]

F. CakoniA. 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.

[8]

F. CakoniD. 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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[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. 
[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.

[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.

[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.

[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.

[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.
[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.

[28]

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

[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.

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