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Inverse source problems without (pseudo) convexity assumptions
Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast
1. | Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA |
2. | Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA |
In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [
References:
[1] |
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. |
[2] |
F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974461.ch1. |
[3] |
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. |
[4] |
F. Cakoni, H. Harris and S. Moskow,
The imaging of small perturbations in an anisotropic media, Computers and Mathematics with Applications, 74 (2017), 2769-2783.
doi: 10.1016/j.camwa.2017.06.050. |
[5] |
F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp.
doi: 10.1088/0266-5611/29/10/104014. |
[6] |
F. Cakoni, S. Moskow and S. Rome,
The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions, Inverse Probl. Imaging, 9 (2015), 725-748.
doi: 10.3934/ipi.2015.9.725. |
[7] |
L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b138375. |
[8] |
A. Kirsch,
A note on Sylvester's proof of discreteness of interior transmission eigenvalues, C. R. Math. Acad. Sci. Paris, 354 (2016), 377-382.
doi: 10.1016/j.crma.2016.01.015. |
[9] |
A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp.
doi: 10.1088/0266-5611/29/10/104011. |
[10] |
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. |
[11] |
S. Moskow, Nonlinear eigenvalue approximation for compact operators, J. Math. Phys., 56 (2015), 113512, 11pp.
doi: 10.1063/1.4936304. |
[12] |
J. E. Osborn,
Spectral approximation for compact operators, Math. Comput., 29 (1975), 712-725.
doi: 10.1090/S0025-5718-1975-0383117-3. |
[13] |
V. Petkov and G. Vodev,
Asymptotics of the number of the interior transmission eigenvalues, J. Spectr. Theory, 7 (2017), 1-31.
doi: 10.4171/JST/154. |
[14] |
L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28pp.
doi: 10.1088/0266-5611/29/10/104001. |
[15] |
L. Robbiano,
Counting function for interior transmission eigenvalues, Math. Control Relat. Fields, 6 (2016), 167-183.
doi: 10.3934/mcrf.2016.6.167. |
[16] |
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] |
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. |
[2] |
F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, vol. 88 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974461.ch1. |
[3] |
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. |
[4] |
F. Cakoni, H. Harris and S. Moskow,
The imaging of small perturbations in an anisotropic media, Computers and Mathematics with Applications, 74 (2017), 2769-2783.
doi: 10.1016/j.camwa.2017.06.050. |
[5] |
F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp.
doi: 10.1088/0266-5611/29/10/104014. |
[6] |
F. Cakoni, S. Moskow and S. Rome,
The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions, Inverse Probl. Imaging, 9 (2015), 725-748.
doi: 10.3934/ipi.2015.9.725. |
[7] |
L. Hörmander, The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer-Verlag, Berlin, 2005.
doi: 10.1007/b138375. |
[8] |
A. Kirsch,
A note on Sylvester's proof of discreteness of interior transmission eigenvalues, C. R. Math. Acad. Sci. Paris, 354 (2016), 377-382.
doi: 10.1016/j.crma.2016.01.015. |
[9] |
A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp.
doi: 10.1088/0266-5611/29/10/104011. |
[10] |
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. |
[11] |
S. Moskow, Nonlinear eigenvalue approximation for compact operators, J. Math. Phys., 56 (2015), 113512, 11pp.
doi: 10.1063/1.4936304. |
[12] |
J. E. Osborn,
Spectral approximation for compact operators, Math. Comput., 29 (1975), 712-725.
doi: 10.1090/S0025-5718-1975-0383117-3. |
[13] |
V. Petkov and G. Vodev,
Asymptotics of the number of the interior transmission eigenvalues, J. Spectr. Theory, 7 (2017), 1-31.
doi: 10.4171/JST/154. |
[14] |
L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, 28pp.
doi: 10.1088/0266-5611/29/10/104001. |
[15] |
L. Robbiano,
Counting function for interior transmission eigenvalues, Math. Control Relat. Fields, 6 (2016), 167-183.
doi: 10.3934/mcrf.2016.6.167. |
[16] |
J. Sylvester,
Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44 (2012), 341-354.
doi: 10.1137/110836420. |


Domain | |
Background Transmission Eigenvalue | 7.12761 |
Background Coefficient | 6.29 |
Perturbed Coefficient | 24 |
Parameter | 50.72217 |
Domain | |
Background Transmission Eigenvalue | 7.12761 |
Background Coefficient | 6.29 |
Perturbed Coefficient | 24 |
Parameter | 50.72217 |
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