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  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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

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