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 & 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., 340 (2011). Google Scholar
[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. doi: 10.1016/j.cam.2009.08.041. Google Scholar
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics. Springer-Verlag, (2006). Google Scholar
[4]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar
[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. doi: 10.1216/JIE-2009-21-2-203. Google Scholar
[6]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145. doi: 10.1137/090754637. Google Scholar
[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. doi: 10.1016/j.crma.2010.02.003. Google Scholar
[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. doi: 10.1088/0266-5611/23/2/004. Google Scholar
[9]	F. Cakoni and D. Gintides, New results on transmission eigenvalues,, Inverse Problems and Imaging, 4 (2010), 39. doi: 10.3934/ipi.2010.4.39. Google Scholar
[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. doi: 10.1137/090769338. Google Scholar
[11]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Applicable Analysis, 88 (2008), 475. doi: 10.1080/00036810802713966. Google Scholar
[12]	F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. Jour. Comp. Sci. Math, 3 (2010), 142. Google Scholar
[13]	D. Colton and R. Kress, "Inverse Acoustic and Eletromagnetic Scattering Theory,", Springer, (1998). Google Scholar
[14]	D. Colton, L. Päivärinta and J.Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar
[15]	A. Cossonniere and H. Haddar, The electromagnetic interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 43 (2011), 1698. doi: 10.1137/100813890. Google Scholar
[16]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Analysis, 42 (2010), 619. doi: 10.1137/100793748. Google Scholar
[17]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, Math Research Letters, 18 (2011), 279. Google Scholar
[18]	A. Kirsch, On the existence of transmission eigenvalues,, Inverse Problems and Imaging, 3 (2009), 155. Google Scholar
[19]	A. Kirsch and N. Grinberg, The factorization method for inverse problems,, in, 36 (2008). Google Scholar
[20]	A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities,, Math. Meth. Appl. Sci., 21 (1998), 2965. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P. Google Scholar
[21]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar
[22]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar

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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., 340 (2011). Google Scholar
[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. doi: 10.1016/j.cam.2009.08.041. Google Scholar
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics. Springer-Verlag, (2006). Google Scholar
[4]	F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar
[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. doi: 10.1216/JIE-2009-21-2-203. Google Scholar
[6]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145. doi: 10.1137/090754637. Google Scholar
[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. doi: 10.1016/j.crma.2010.02.003. Google Scholar
[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. doi: 10.1088/0266-5611/23/2/004. Google Scholar
[9]	F. Cakoni and D. Gintides, New results on transmission eigenvalues,, Inverse Problems and Imaging, 4 (2010), 39. doi: 10.3934/ipi.2010.4.39. Google Scholar
[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. doi: 10.1137/090769338. Google Scholar
[11]	F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium,, Applicable Analysis, 88 (2008), 475. doi: 10.1080/00036810802713966. Google Scholar
[12]	F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. Jour. Comp. Sci. Math, 3 (2010), 142. Google Scholar
[13]	D. Colton and R. Kress, "Inverse Acoustic and Eletromagnetic Scattering Theory,", Springer, (1998). Google Scholar
[14]	D. Colton, L. Päivärinta and J.Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13. doi: 10.3934/ipi.2007.1.13. Google Scholar
[15]	A. Cossonniere and H. Haddar, The electromagnetic interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 43 (2011), 1698. doi: 10.1137/100813890. Google Scholar
[16]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Analysis, 42 (2010), 619. doi: 10.1137/100793748. Google Scholar
[17]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, Math Research Letters, 18 (2011), 279. Google Scholar
[18]	A. Kirsch, On the existence of transmission eigenvalues,, Inverse Problems and Imaging, 3 (2009), 155. Google Scholar
[19]	A. Kirsch and N. Grinberg, The factorization method for inverse problems,, in, 36 (2008). Google Scholar
[20]	A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities,, Math. Meth. Appl. Sci., 21 (1998), 2965. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P. Google Scholar
[21]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525. Google Scholar
[22]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420. Google Scholar

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