Transmission eigenvalues for inhomogeneous media containing obstacles

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August 2012, 6(3): 373-398. doi: 10.3934/ipi.2012.6.373

Transmission eigenvalues for inhomogeneous media containing obstacles

Fioralba Cakoni ^1, , Anne Cossonnière ^2, and Houssem Haddar ^3,

1.	Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, United States
2.	CERFACS, 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France
3.	INRIA Saclay Ile de France/CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Received November 2011 Revised June 2012 Published September 2012

Abstract
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We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.

Keywords: Interior transmission problem, inhomogeneous medium, transmission eigenvalues, inverse scattering problem..

Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A2.

Citation: Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems and Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373

References:

[1]	A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of t-coercivity to study the interior transmission eigenvalue problem, C. R. Acad. Sci., Ser. I, 340 (2011).
[2]	A. S. Bonnet-BenDhia, P. Ciarlet and C. Maria Zwölf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math, 234 (2010), 1912-1919. doi: 10.1016/j.cam.2009.08.041.
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.
[4]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal., 42 (2010), 2912-2921. doi: 10.1137/100793542.
[5]	F. Cakoni, D. Colton and H. Haddar, The computation of lower bounds for the norm of the index of refraction in an anisotropic media, J. Integral Equations and Applications, 21 (2009), 203-227. doi: 10.1216/JIE-2009-21-2-203.
[6]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162. doi: 10.1137/090754637.
[7]	F. Cakoni, D. Colton and H. Haddar, On the determination of dirichlet or transmission eigenvalues from far field data, C. R. Acad. Sci. Paris, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003.
[8]	F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data, Inverse Problems, 23 (2007), 507-522. doi: 10.1088/0266-5611/23/2/004.
[9]	F. Cakoni and D. Gintides, New results on transmission eigenvalues, Inverse Problems and Imaging, 4 (2010), 39-48. doi: 10.3934/ipi.2010.4.39.
[10]	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.
[11]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2008), 475-493. doi: 10.1080/00036810802713966.
[12]	F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. Jour. Comp. Sci. Math, 3 (2010), 142-167.
[13]	D. Colton and R. Kress, "Inverse Acoustic and Eletromagnetic Scattering Theory," Springer, New York, 2nd edition, 1998.
[14]	D. Colton, L. Päivärinta and J.Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.
[15]	A. Cossonniere and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal., 43 (2011), 1698-1715. doi: 10.1137/100813890.
[16]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Analysis, 42 (2010), 619-651. doi: 10.1137/100793748.
[17]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, Math Research Letters, 18 (2011), 279-293.
[18]	A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.
[19]	A. Kirsch and N. Grinberg, The factorization method for inverse problems, in "Mathematics and its Applications," Oxford Lecture Series, 36, Oxford University Press, Oxford, 2008.
[20]	A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 2965-2986. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.
[21]	L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.
[22]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.

show all references

References:

[1]	A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of t-coercivity to study the interior transmission eigenvalue problem, C. R. Acad. Sci., Ser. I, 340 (2011).
[2]	A. S. Bonnet-BenDhia, P. Ciarlet and C. Maria Zwölf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math, 234 (2010), 1912-1919. doi: 10.1016/j.cam.2009.08.041.
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.
[4]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal., 42 (2010), 2912-2921. doi: 10.1137/100793542.
[5]	F. Cakoni, D. Colton and H. Haddar, The computation of lower bounds for the norm of the index of refraction in an anisotropic media, J. Integral Equations and Applications, 21 (2009), 203-227. doi: 10.1216/JIE-2009-21-2-203.
[6]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162. doi: 10.1137/090754637.
[7]	F. Cakoni, D. Colton and H. Haddar, On the determination of dirichlet or transmission eigenvalues from far field data, C. R. Acad. Sci. Paris, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003.
[8]	F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data, Inverse Problems, 23 (2007), 507-522. doi: 10.1088/0266-5611/23/2/004.
[9]	F. Cakoni and D. Gintides, New results on transmission eigenvalues, Inverse Problems and Imaging, 4 (2010), 39-48. doi: 10.3934/ipi.2010.4.39.
[10]	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.
[11]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Applicable Analysis, 88 (2008), 475-493. doi: 10.1080/00036810802713966.
[12]	F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. Jour. Comp. Sci. Math, 3 (2010), 142-167.
[13]	D. Colton and R. Kress, "Inverse Acoustic and Eletromagnetic Scattering Theory," Springer, New York, 2nd edition, 1998.
[14]	D. Colton, L. Päivärinta and J.Sylvester, The interior transmission problem, Inverse Problems and Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.
[15]	A. Cossonniere and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal., 43 (2011), 1698-1715. doi: 10.1137/100813890.
[16]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Analysis, 42 (2010), 619-651. doi: 10.1137/100793748.
[17]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, Math Research Letters, 18 (2011), 279-293.
[18]	A. Kirsch, On the existence of transmission eigenvalues, Inverse Problems and Imaging, 3 (2009), 155-172.
[19]	A. Kirsch and N. Grinberg, The factorization method for inverse problems, in "Mathematics and its Applications," Oxford Lecture Series, 36, Oxford University Press, Oxford, 2008.
[20]	A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 2965-2986. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.
[21]	L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.
[22]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.

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