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Surface integral formulation of the interior transmission problem. (English) Zbl 1366.78019

Summary: We consider a surface integral formulation of the so-called interior transmission problem that appears in the study of inverse scattering problems from dielectric inclusions. In the case where the magnetic permeability contrast is zero, the main originality of our approach consists in still using classical potentials for the Helmholtz equation but in weaker trace space solutions. One major outcome of this study is to establish Fredholm properties of the problem for relaxed assumptions on the material coefficients. For instance, we allow the contrast to change sign inside the medium. We also show how one can retrieve discreteness results for transmission eigenvalues in some particular situations.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

References:

[1] M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables , Dover Publications, New York, 1964. · Zbl 0171.38503
[2] 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. Math. 11 - 12 (2011), 647-651. · Zbl 1244.35099 · doi:10.1016/j.crma.2011.05.008
[3] F. Cakoni and D. Colton, Qualitative methods in inverse scattering theory , in Interaction of mechanics and mathematics, An introduction , Springer-Verlag, Berlin, 2006. · Zbl 1099.78008
[4] 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 Equat. Appl. 21 (2009), 203-227. · Zbl 1173.35722 · doi:10.1216/JIE-2009-21-2-203
[5] —, The interior transmission problem for regions with cavities , SIAM J. Math. Anal. 42 (2010), 145-162. · Zbl 1209.35135 · doi:10.1137/090754637
[6] 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. · Zbl 1115.78008 · doi:10.1088/0266-5611/23/2/004
[7] F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles , Inverse Prob. Imag. 6 (2012), 373-398. · Zbl 1253.35214 · doi:10.3934/ipi.2012.6.373
[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. · Zbl 1210.35282 · 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. · Zbl 1168.35448 · doi:10.1080/00036810802713966
[10] F. Collino, A. Cossonnière, M’B. Fares and H. Haddar, A numerical method to compute transmission eigenvalues based on surface integral equations , 2012,
[11] D. Colton and R. Kress, Inverse acoustic and eletromagnetic scattering theory , 2nd edition, Springer, New York, 1998. · Zbl 0893.35138
[12] D. Colton, L.Päivärinta and J. Sylvester, The interior transmission problem , Inverse Prob. Imag. 1 (2007), 13-28. · Zbl 1130.35132 · doi:10.3934/ipi.2007.1.13
[13] A Cossonnière, Valeurs propres de transmission et leur utilisation dans l’identification d’inclusions à partir de mesures électromagnétiques , Ph.D. thesis, University of Toulouse, 2011.
[14] A. Cossonnière and H. Haddar, The electromagnetic interior transmission problem for regions with cavities , SIAM J. Math. Anal. 43 (2011), 1698-1715. · Zbl 1229.78014 · doi:10.1137/100813890
[15] M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators , SIAM J. Math. Anal. 43 (2011), 2630-2639. · Zbl 1233.35148 · doi:10.1137/110827867
[16] G. Hsiao and W. Wendland, Boundary integral equations , Springer, New York, 2008. · Zbl 1157.65066
[17] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator , Inverse Prob. 14 (1998), 1489-1512. · Zbl 0919.35147 · doi:10.1088/0266-5611/14/6/009
[18] —, A note on Sylvester’s proof of discreteness of interior transmission eigenvalues , 2011.
[19] A. Kirsch and N. Grinberg, The factorization method for inverse problems , Oxford Lect. Ser. Math. Appl. 36 , Oxford University Press, Oxford, 2008. · Zbl 1222.35001
[20] E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media , SIAM J. Math. Anal. 44 (2012), 1165-1174. · Zbl 1245.35126 · doi:10.1137/11084738X
[21] —, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards , J. Phys. Math. Theor. 45 (2012), 125202 (10 pp.). · Zbl 1245.81290 · doi:10.1088/1751-8113/45/12/125202
[22] W. McLean, Strongly elliptic systems and boundary integral equations , Cambridge University Press, Cambridge, 2000. · Zbl 0948.35001
[23] J.C. Nédélec, Acoustic and electromagnetic equations , Appl. Math. Sci. 144 , Springer, New York, 2001.
[24] L. Päivärinta and J. Sylvester, Transmission eigenvalues , SIAM J. Math. Anal. 40 (2008), 738-753. · Zbl 1159.81411 · doi:10.1137/070697525
[25] J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators , SIAM J. Math. Anal. 44 (2012), 373-398. \noindentstyle · Zbl 1238.81172 · doi:10.1137/110836420
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