August  2021, 15(4): 723-744. doi: 10.3934/ipi.2021011

Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

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

Received  June 2020 Revised  October 2020 Published  August 2021 Early access  January 2021

Fund Project: This material is based upon work supported by the Army Research Office through the National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a

A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medium. Second, we leverage regularity results for Maxwell's equations in order to establish stability results for the eigenvalues with respect to the material coefficients, and we show that this generalized class of Stekloff eigenvalues converges to the standard class as the smoothing parameter approaches zero.

Citation: Samuel Cogar. Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification. Inverse Problems and Imaging, 2021, 15 (4) : 723-744. doi: 10.3934/ipi.2021011
References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, Vol. 65.

[2]

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, 28pp. doi: 10.1088/1361-6420/aa982f.

[3]

L. AudibertL. Chesnel and H. Haddar, Transmission eigenvalues with artificial background for explicit material index identification, C. R. Math. Acad. Sci. Paris, 356 (2018), 626-631.  doi: 10.1016/j.crma.2018.04.015.

[4]

L. Audibert, L. Chesnel and H. Haddar, Inside-outside duality with artificial backgrounds, Inverse Problems, 35 (2019), 104008, 26pp. doi: 10.1088/1361-6420/ab3244.

[5]

H. BiY. Zhang and Y. Yang, Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem, Comput. Math. Appl., 79 (2020), 1895-1913.  doi: 10.1016/j.camwa.2018.08.047.

[6]

A. BonitoJ. Guermond and F. Luddens, Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl., 408 (2013), 498-512.  doi: 10.1016/j.jmaa.2013.06.018.

[7]

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.

[8]

F. CakoniD. ColtonS. Meng and P. Monk, Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math., 76 (2016), 1737-1763.  doi: 10.1137/16M1058704.

[9]

J. CamañoC. Lackner and P. Monk, Electromagnetic Stekloff eigenvalues in inverse scattering, SIAM J. Math. Anal., 49 (2017), 4376-4401.  doi: 10.1137/16M1108893.

[10]

P. Ciarlet, On the approximation of electromagnetic fields by edge finite elements. Part 3: Sensitivity to coefficients, SIAM J. Math. Anal., 52 (2020), 3004-3038.  doi: 10.1137/19M1275383.

[11]

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

[12]

S. Cogar, New Eigenvalue Problems in Inverse Scattering, PhD thesis, University of Delaware, 2019, URL https://udspace.udel.edu/handle/19716/24732.

[13]

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.

[14]

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

[15]

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

[16]

S. Cogar, D. Colton and P. Monk, Eigenvalue problems in inverse electromagnetic scattering theory, in Maxwell's Equations: Analysis and Numerics (eds. U. Langer, D. Pauly and S. Repin), Radon Series on Computational and Applied Mathematics, De Gruyter, 2019, chapter 5, https://arXiv.org/abs/1805.06986.

[17]

S. Cogar and P. Monk, Modified electromagnetic transmission eigenvalues in inverse scattering theory, SIAM J. Math. Anal., 52 (2020), 6412–6441, https://arXiv.org/abs/2005.14277. doi: 10.1137/20M134006X.

[18]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[19]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[20]

F. Gylys-Colwell, An inverse problem for the Helmholtz equation, Inverse Problems, 12 (1996), 139-156.  doi: 10.1088/0266-5611/12/2/003.

[21]

M. Halla, Electromagnetic Stekloff eigenvalues: Approximation analysis, ESAIM: M2AN, to appear (2020), https://arXiv.org/abs/1909.00689.

[22]

M. Halla, Electromagnetic Stekloff eigenvalues: Existence and behavior in the selfadjoint case, preprint, URL https://arXiv.org/abs/1909.01983.

[23]

J. Jost, Riemannian Geometry and Geometric Analysis, Seventh edition, Universitext, Springer, Cham, 2017. doi: 10.1007/978-3-319-61860-9.

[24]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[25]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, vol. 190 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.

[26]

J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 094004, 20pp. doi: 10.1088/1361-6420/ab1be9.

[27]

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.

[28]

J.-C. Nédélec, Acoustic and Electromagnetic Equations, vol. 144 of Applied Mathematical Sciences, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[29]

J. R. Ringrose, Compact Non-Self-Adjoint Operators, Van Nostrand Reinhold Co., London, 1971.

[30] F. SayasT. Brown and M. Hassell, Variational Techniques for Elliptic Partial Differential Equations, CRC Press, 2019.  doi: 10.1201/9780429507069.

show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, Vol. 65.

[2]

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, 28pp. doi: 10.1088/1361-6420/aa982f.

[3]

L. AudibertL. Chesnel and H. Haddar, Transmission eigenvalues with artificial background for explicit material index identification, C. R. Math. Acad. Sci. Paris, 356 (2018), 626-631.  doi: 10.1016/j.crma.2018.04.015.

[4]

L. Audibert, L. Chesnel and H. Haddar, Inside-outside duality with artificial backgrounds, Inverse Problems, 35 (2019), 104008, 26pp. doi: 10.1088/1361-6420/ab3244.

[5]

H. BiY. Zhang and Y. Yang, Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem, Comput. Math. Appl., 79 (2020), 1895-1913.  doi: 10.1016/j.camwa.2018.08.047.

[6]

A. BonitoJ. Guermond and F. Luddens, Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl., 408 (2013), 498-512.  doi: 10.1016/j.jmaa.2013.06.018.

[7]

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.

[8]

F. CakoniD. ColtonS. Meng and P. Monk, Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math., 76 (2016), 1737-1763.  doi: 10.1137/16M1058704.

[9]

J. CamañoC. Lackner and P. Monk, Electromagnetic Stekloff eigenvalues in inverse scattering, SIAM J. Math. Anal., 49 (2017), 4376-4401.  doi: 10.1137/16M1108893.

[10]

P. Ciarlet, On the approximation of electromagnetic fields by edge finite elements. Part 3: Sensitivity to coefficients, SIAM J. Math. Anal., 52 (2020), 3004-3038.  doi: 10.1137/19M1275383.

[11]

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

[12]

S. Cogar, New Eigenvalue Problems in Inverse Scattering, PhD thesis, University of Delaware, 2019, URL https://udspace.udel.edu/handle/19716/24732.

[13]

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.

[14]

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

[15]

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

[16]

S. Cogar, D. Colton and P. Monk, Eigenvalue problems in inverse electromagnetic scattering theory, in Maxwell's Equations: Analysis and Numerics (eds. U. Langer, D. Pauly and S. Repin), Radon Series on Computational and Applied Mathematics, De Gruyter, 2019, chapter 5, https://arXiv.org/abs/1805.06986.

[17]

S. Cogar and P. Monk, Modified electromagnetic transmission eigenvalues in inverse scattering theory, SIAM J. Math. Anal., 52 (2020), 6412–6441, https://arXiv.org/abs/2005.14277. doi: 10.1137/20M134006X.

[18]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 4th edition, Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[19]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[20]

F. Gylys-Colwell, An inverse problem for the Helmholtz equation, Inverse Problems, 12 (1996), 139-156.  doi: 10.1088/0266-5611/12/2/003.

[21]

M. Halla, Electromagnetic Stekloff eigenvalues: Approximation analysis, ESAIM: M2AN, to appear (2020), https://arXiv.org/abs/1909.00689.

[22]

M. Halla, Electromagnetic Stekloff eigenvalues: Existence and behavior in the selfadjoint case, preprint, URL https://arXiv.org/abs/1909.01983.

[23]

J. Jost, Riemannian Geometry and Geometric Analysis, Seventh edition, Universitext, Springer, Cham, 2017. doi: 10.1007/978-3-319-61860-9.

[24]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[25]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, vol. 190 of Applied Mathematical Sciences, Springer, Cham, 2015. doi: 10.1007/978-3-319-11086-8.

[26]

J. Liu, Y. Liu and J. Sun, An inverse medium problem using Stekloff eigenvalues and a Bayesian approach, Inverse Problems, 35 (2019), 094004, 20pp. doi: 10.1088/1361-6420/ab1be9.

[27]

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.

[28]

J.-C. Nédélec, Acoustic and Electromagnetic Equations, vol. 144 of Applied Mathematical Sciences, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[29]

J. R. Ringrose, Compact Non-Self-Adjoint Operators, Van Nostrand Reinhold Co., London, 1971.

[30] F. SayasT. Brown and M. Hassell, Variational Techniques for Elliptic Partial Differential Equations, CRC Press, 2019.  doi: 10.1201/9780429507069.
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