• Previous Article
    Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions
  • IPI Home
  • This Issue
  • Next Article
    Recovering a bounded elastic body by electromagnetic far-field measurements
doi: 10.3934/ipi.2022020
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A spectral target signature for thin surfaces with higher order jump conditions

1. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

2. 

Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, USA

3. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

In Memory of Professor Victor Isakov

Received  January 2022 Revised  March 2022 Early access April 2022

In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in $ {\mathbb R}^m $, $ m = 2, 3 $ from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open $ m-1 $ dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach

Citation: Fioralba Cakoni, Heejin Lee, Peter Monk, Yangwen Zhang. A spectral target signature for thin surfaces with higher order jump conditions. Inverse Problems and Imaging, doi: 10.3934/ipi.2022020
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.

[2]

H. AmmariJ. GarnierH. KangW.-K. Park and K. Sølna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68-91.  doi: 10.1137/100800130.

[3]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.

[4]

L. Audibert, F. Cakoni and H. Haddar, New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data, Inverse Problems, 33 (2017), 125011, 28 pp. doi: 10.1088/1361-6420/aa982f.

[5]

L. Audibert, L. Chesnel, H. Haddar and K. Napal, Qualitative indicator functions for imaging crack networks using acoustic waves, SIAM J. Sci. Comput., 43 (2021), B271–B297. doi: 10.1137/20M134650X.

[6]

C. E. BaumE. J. RothwellK. Chen and D. P. Nyquist, The singularity expansion method and its application to target identification, Proceedings of the IEEE, 79 (1991), 1481-1492. 

[7]

E. Beretta, E. Francini, E. Kim and J.-Y. Lee, Algorithm for the determination of a linear crack in an elastic body from boundary measurements, Inverse Problems, 26 (2010), 085015, 13 pp. doi: 10.1088/0266-5611/26/8/085015.

[8]

M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem., 35 (2011), 223-235.  doi: 10.1016/j.enganabound.2010.08.007.

[9]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.

[10]

F. CakoniS. Cogar and P. Monk, A spectral approach to nondestructive testing via electromagnetic waves, IEEE Transactions on Antennas and Propagation, 69 (2021), 8689-8697.  doi: 10.1109/TAP.2021.3090810.

[11]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295.  doi: 10.1088/0266-5611/19/2/303.

[12]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, Springer, 188. New York, 2014, https://mathscinet-ams-org.proxy.libraries.rutgers.edu/mathscinet-getitem?mr=3137429.

[13]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.

[14]

F. CakoniD. Colton and H. Haddar, Transmission eigenvalues, Notices Amer. Math. Soc., 68 (2021), 1499-1510.  doi: 10.1090/noti2350.

[15]

F. Cakoni, D. Colton, S. Meng and P. Monk, Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math., 76 (2016), 1737–1763, https://mathscinet-ams-org.proxy.libraries.rutgers.edu/mathscinet-getitem?mr=3542029. doi: 10.1137/16M1058704.

[16]

F. CakoniD. 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.

[17]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14 pp. doi: 10.1088/0266-5611/26/7/074004.

[18]

F. CakoniI. de TeresaH. Haddar and P. Monk, Nondestructive testing of the delaminated interface between two materials, SIAM J. Appl. Math., 76 (2016), 2306-2332.  doi: 10.1137/16M1064167.

[19]

F. Cakoni, I. de Teresa and P. Monk, Nondestructive testing of delaminated interfaces between two materials using electromagnetic interrogation, Inverse Problems, 34 (2018), 065005, 36 pp. doi: 10.1088/1361-6420/aabb1c.

[20]

F. Cakoni, P. Monk and Y. Zhang, Target signatures for thin surfaces, Inverse Problems, 38 (2021), 025011, 28 pp. doi: 10.1088/1361-6420/ac4154.

[21]

F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, Comm. Pure Appl. Math..

[22]

S. Cogar, A modified transmission eigenvalue problem for scattering by a partially coated crack, Inverse Problems, 34 (2018), 115003, 29 pp. doi: 10.1088/1361-6420/aadb20.

[23]

S. Cogar, Analysis of a trace class Stekloff eigenvalue problem arising in inverse scattering, SIAM J. Appl. Math., 80 (2020), 881-905.  doi: 10.1137/19M1295155.

[24]

S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems, 33 (2017), 125002, 31 pp. doi: 10.1088/1361-6420/aa9418.

[25]

S. Cogar, D. Colton and P. Monk, Using eigenvalues to detect anomalies in the exterior of a cavity, Inverse Problems, 34 (2018), 085006, 27 pp. doi: 10.1088/1361-6420/aac8ef.

[26]

S. Cogar and P. B. Monk, Modified electromagnetic transmission eigenvalues in inverse scattering theory, SIAM J. Math. Anal., 52 (2020), 6412-6441.  doi: 10.1137/20M134006X.

[27]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Fourth edition, Applied Mathematical Sciences, 93. Springer, Cham, 2019]. doi: 10.1007/978-3-030-30351-8.

[28]

I. de Teresa Truebs, Asymptotic Methods in Inverse Scattering for Inhomogeneous Media, Thesis (Ph.D.)–University of Delaware, 2017,196 pp.

[29]

B. DelourmeH. Haddar and P. Joly, Approximate models for wave propagation across thin periodic interfaces, J. Math. Pures Appl., 98 (2012), 28-71.  doi: 10.1016/j.matpur.2012.01.003.

[30]

B. DelourmeH. Haddar and P. Joly, On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems, Math. Models Methods Appl. Sci., 23 (2013), 2433-2464.  doi: 10.1142/S021820251350036X.

[31]

H. HaddarP. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: The scalar case, Math. Models Methods Appl. Sci., 15 (2005), 1273-1300.  doi: 10.1142/S021820250500073X.

[32]

H. HaddarP. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: The case of Maxwell's equations, Math. Models Methods Appl. Sci., 18 (2008), 1787-1827.  doi: 10.1142/S0218202508003194.

[33]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Problems, 30 (2014), 035016, 21 pp. doi: 10.1088/0266-5611/30/3/035016.

[34]

V. Isakov, Inverse Problems for Partial Differential Equations, Third edition, Applied Mathematical Sciences, 127. Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5.

[35]

A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105.  doi: 10.1088/0266-5611/16/1/308.

[36]

H. Lee, Inverse Scattering Problems for Obstacles with Higher Order Boundary Conditions, Rutgers, The State University of New Jersey, New Brunswick, 2022.

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[38]

O. ÖzdemirH. Haddar and A. Yaka, Reconstruction of the electromagnetic field in layered media using the concept of approximate transmission conditions, IEEE Trans. Antennas and Propagation, 59 (2011), 2964-2972.  doi: 10.1109/TAP.2011.2158967.

[39]

F. Pourahmadian, B. B. Guzina and H. Haddar, Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures, Inverse Problems, 33 (2017), 055007, 32 pp. doi: 10.1088/1361-6420/33/5/055007.

[40]

M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, Res. Math. Sci., 8 (2021), Paper No. 58, 19 pp. doi: 10.1007/s40687-021-00294-z.

[41]

J. Schöberl, Netgen/NGSolve, Technical University of Vienna, 2022, Netgen version 6.2.2105, https://ngsolve.org/downloads.

[42]

N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042.  doi: 10.1137/070711542.

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. doi: 10.1090/chel/369.

[2]

H. AmmariJ. GarnierH. KangW.-K. Park and K. Sølna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68-91.  doi: 10.1137/100800130.

[3]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.

[4]

L. Audibert, F. Cakoni and H. Haddar, New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data, Inverse Problems, 33 (2017), 125011, 28 pp. doi: 10.1088/1361-6420/aa982f.

[5]

L. Audibert, L. Chesnel, H. Haddar and K. Napal, Qualitative indicator functions for imaging crack networks using acoustic waves, SIAM J. Sci. Comput., 43 (2021), B271–B297. doi: 10.1137/20M134650X.

[6]

C. E. BaumE. J. RothwellK. Chen and D. P. Nyquist, The singularity expansion method and its application to target identification, Proceedings of the IEEE, 79 (1991), 1481-1492. 

[7]

E. Beretta, E. Francini, E. Kim and J.-Y. Lee, Algorithm for the determination of a linear crack in an elastic body from boundary measurements, Inverse Problems, 26 (2010), 085015, 13 pp. doi: 10.1088/0266-5611/26/8/085015.

[8]

M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems, Eng. Anal. Bound. Elem., 35 (2011), 223-235.  doi: 10.1016/j.enganabound.2010.08.007.

[9]

Y. Boukari and H. Haddar, The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging, 7 (2013), 1123-1138.  doi: 10.3934/ipi.2013.7.1123.

[10]

F. CakoniS. Cogar and P. Monk, A spectral approach to nondestructive testing via electromagnetic waves, IEEE Transactions on Antennas and Propagation, 69 (2021), 8689-8697.  doi: 10.1109/TAP.2021.3090810.

[11]

F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Problems, 19 (2003), 279-295.  doi: 10.1088/0266-5611/19/2/303.

[12]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, Springer, 188. New York, 2014, https://mathscinet-ams-org.proxy.libraries.rutgers.edu/mathscinet-getitem?mr=3137429.

[13]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.

[14]

F. CakoniD. Colton and H. Haddar, Transmission eigenvalues, Notices Amer. Math. Soc., 68 (2021), 1499-1510.  doi: 10.1090/noti2350.

[15]

F. Cakoni, D. Colton, S. Meng and P. Monk, Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math., 76 (2016), 1737–1763, https://mathscinet-ams-org.proxy.libraries.rutgers.edu/mathscinet-getitem?mr=3542029. doi: 10.1137/16M1058704.

[16]

F. CakoniD. 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.

[17]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26 (2010), 074004, 14 pp. doi: 10.1088/0266-5611/26/7/074004.

[18]

F. CakoniI. de TeresaH. Haddar and P. Monk, Nondestructive testing of the delaminated interface between two materials, SIAM J. Appl. Math., 76 (2016), 2306-2332.  doi: 10.1137/16M1064167.

[19]

F. Cakoni, I. de Teresa and P. Monk, Nondestructive testing of delaminated interfaces between two materials using electromagnetic interrogation, Inverse Problems, 34 (2018), 065005, 36 pp. doi: 10.1088/1361-6420/aabb1c.

[20]

F. Cakoni, P. Monk and Y. Zhang, Target signatures for thin surfaces, Inverse Problems, 38 (2021), 025011, 28 pp. doi: 10.1088/1361-6420/ac4154.

[21]

F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, Comm. Pure Appl. Math..

[22]

S. Cogar, A modified transmission eigenvalue problem for scattering by a partially coated crack, Inverse Problems, 34 (2018), 115003, 29 pp. doi: 10.1088/1361-6420/aadb20.

[23]

S. Cogar, Analysis of a trace class Stekloff eigenvalue problem arising in inverse scattering, SIAM J. Appl. Math., 80 (2020), 881-905.  doi: 10.1137/19M1295155.

[24]

S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems, 33 (2017), 125002, 31 pp. doi: 10.1088/1361-6420/aa9418.

[25]

S. Cogar, D. Colton and P. Monk, Using eigenvalues to detect anomalies in the exterior of a cavity, Inverse Problems, 34 (2018), 085006, 27 pp. doi: 10.1088/1361-6420/aac8ef.

[26]

S. Cogar and P. B. Monk, Modified electromagnetic transmission eigenvalues in inverse scattering theory, SIAM J. Math. Anal., 52 (2020), 6412-6441.  doi: 10.1137/20M134006X.

[27]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Fourth edition, Applied Mathematical Sciences, 93. Springer, Cham, 2019]. doi: 10.1007/978-3-030-30351-8.

[28]

I. de Teresa Truebs, Asymptotic Methods in Inverse Scattering for Inhomogeneous Media, Thesis (Ph.D.)–University of Delaware, 2017,196 pp.

[29]

B. DelourmeH. Haddar and P. Joly, Approximate models for wave propagation across thin periodic interfaces, J. Math. Pures Appl., 98 (2012), 28-71.  doi: 10.1016/j.matpur.2012.01.003.

[30]

B. DelourmeH. Haddar and P. Joly, On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems, Math. Models Methods Appl. Sci., 23 (2013), 2433-2464.  doi: 10.1142/S021820251350036X.

[31]

H. HaddarP. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: The scalar case, Math. Models Methods Appl. Sci., 15 (2005), 1273-1300.  doi: 10.1142/S021820250500073X.

[32]

H. HaddarP. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: The case of Maxwell's equations, Math. Models Methods Appl. Sci., 18 (2008), 1787-1827.  doi: 10.1142/S0218202508003194.

[33]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Problems, 30 (2014), 035016, 21 pp. doi: 10.1088/0266-5611/30/3/035016.

[34]

V. Isakov, Inverse Problems for Partial Differential Equations, Third edition, Applied Mathematical Sciences, 127. Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5.

[35]

A. Kirsch and S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Problems, 16 (2000), 89-105.  doi: 10.1088/0266-5611/16/1/308.

[36]

H. Lee, Inverse Scattering Problems for Obstacles with Higher Order Boundary Conditions, Rutgers, The State University of New Jersey, New Brunswick, 2022.

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[38]

O. ÖzdemirH. Haddar and A. Yaka, Reconstruction of the electromagnetic field in layered media using the concept of approximate transmission conditions, IEEE Trans. Antennas and Propagation, 59 (2011), 2964-2972.  doi: 10.1109/TAP.2011.2158967.

[39]

F. Pourahmadian, B. B. Guzina and H. Haddar, Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures, Inverse Problems, 33 (2017), 055007, 32 pp. doi: 10.1088/1361-6420/33/5/055007.

[40]

M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, Res. Math. Sci., 8 (2021), Paper No. 58, 19 pp. doi: 10.1007/s40687-021-00294-z.

[41]

J. Schöberl, Netgen/NGSolve, Technical University of Vienna, 2022, Netgen version 6.2.2105, https://ngsolve.org/downloads.

[42]

N. Zeev and F. Cakoni, The identification of thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math., 69 (2009), 1024-1042.  doi: 10.1137/070711542.

Figure 2.  In the left column we show the scatterer $ \Gamma $ (red curve) and the remainder of $ \partial D $ as a green curve. Asterisks show the position of the random source points $ z $ in $ D $. In the right column we show the average $ \ell_2 $ norm of the regularized solution of the modified far field equation against the eigenparameter $ \lambda $. The vertical lines mark the position of the true eigenvalues found by solving the interior eigenvalue problem. Top row: Dirichlet end condition. Bottom row: Neumann end condition
Figure 3.  The layout of this figure is the same as in Fig. 2 except that the scatter is now the quarter circle shown in the left column. The same parameter values are used. Top row: Dirichlet end condition. Bottom row: Neumann end condition
Figure 4.  Here we show the detection of eigenvalues for the half and quarter circle scatterers with Dirichlet end conditions and parameters given by $ \mu = 0.2 $, $ \beta = 1 $ and $ \alpha = -0.2 $. See Fig. 2 for a description of the symbols used. Top row: Half circle scatterer. Bottom: Quarter circle scatterer
Figure 5.  Here we show the detection of eigenvalues for the half and quarter circle scatterers with Dirichlet end conditions and parameters given by $ \mu = 0.2 = \beta = 2 $ and $ \alpha = -2 $. The domain $ D $ is now obtained by joining the end points of $ \Gamma $ by a straight line. See Fig. 2 for a description of the symbols used. Top row: Half circle scatterer. Bottom: Quarter circle scatterer
Figure 6.  An example of non-circular domains $ D $ containing $ \Gamma $. These domains are smoother than those in Fig. 5 and allow the approximation of more eigenvalues (the same parameters are used). Top row: Half circle scatterer. Bottom: Quarter circle scatterer
Figure 7.  Changes in the first five eigenvalues (in magnitude) computed by the finite element eigenvalue solver for the half-circle scatterer and disk $ D $ as functions of the parameters $ \alpha $, $ \beta $ and $ \mu $
Figure 8.  Predictions of the eigenvalues for the problem when $ \alpha = -2 $ and $ \mu = 2 $ and $ \beta = 0.4, 0.5 $ and 0.6. The shift in the eigenvalues predicted in Fig. 7 (middle graph) is evident in the large translation of the peaks for the three cases
[1]

Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems and Imaging, 2021, 15 (5) : 999-1014. doi: 10.3934/ipi.2021025

[2]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems and Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39

[3]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems and Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155

[4]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems and Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793

[5]

Annalena Albicker, Roland Griesmaier. Monotonicity in inverse scattering for Maxwell's equations. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022032

[6]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems and Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017

[7]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems and Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123

[8]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control and Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167

[9]

Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061

[10]

Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems and Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273

[11]

Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems and Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795

[12]

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

[13]

Abdessatar Khelifi, Siwar Saidani. Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022080

[14]

Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems and Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725

[15]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031

[16]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems and Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041

[17]

Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355

[18]

Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027

[19]

Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469

[20]

Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225

2021 Impact Factor: 1.483

Article outline

Figures and Tables

[Back to Top]