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doi: 10.3934/cpaa.2022080
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Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion

1. 

Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Tunisia

2. 

GAMA Laboratory LR21ES10, Faculty of Sciences of Bizerte, 7021 Zarzouna, Bizerte, University of Carthage, Tunisia

*Corresponding author

Received  June 2021 Revised  January 2022 Early access April 2022

In this paper, we derive rigorously asymptotic formulas for perturbations in the eigenfrequencies of a Maxwell system due to small changes in the interface of a smooth inclusion. Taking advantage of small perturbations, we use a rigorous asymptotic analysis to develop an asymptotic formula for the case where the eigenvalue of the reference problem is simple or multiple. We show that our asymptotic formulas can be expressed in terms of the electric permittivity and the profile function $ h $ modelling the shape perturbation. We assume that our results are ambitious tools to solve the inverse problem of identifying interface changes (deformations) of inclusions, given eigenvalues measurements.

Citation: 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
References:
[1]

H. AmmariE. BerettaE. FranciniH. Kang and M. Lim, Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case, J. Math. Pure Appl., 94 (2010), 322-339.  doi: 10.1016/j.matpur.2010.02.001.

[2]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. 

[3]

H. AmmariH. KangM. Lim and H. Zribi, Layer potential techniques in spectral analysis. Part I: complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions, Trans. Amer. Math. Soc., 362 (2010), 2901-2922. 

[4]

H. AmmariM. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of imperfections of small diameter II.The full Maxwell equations, J. Math. Pures Appl., 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.

[5]

H. Ammari and D. Volkov, Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter, Asymp. Anal., 30 (2002), 331-350. 

[6]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., North-Holland, Amsterdam, 197A.

[7]

M. Sh. Birman and M. Z. Solomyak, $L_2$ Theory of the Maxwell operator in an arbitrary domain, Russian Math. Surveys, 42 (1987), 75-96. 

[8]

H. BoujlidaH. Haddar and M. Khenissi, The asymptotic of transmission eigenvalues for a domain with a thin coating, SIAM J. Appl. Math., 78 (2018), 2348-2369.  doi: 10.1137/17M1154126.

[9]

S. C. BrennerF. Li and L. Y. Sung, Nonconforming Maxwell eigensolvers, J. Sci. Comput, 40 (2009), 51-85.  doi: 10.1007/s10915-008-9266-9.

[10]

F. CakoniN. Chaulet and H. Haddar, Asymptotic analysis of the transmission eigenvalue problem for a Dirichlet obstacle coated by a thin layer of non-absorbing media, IMA J. Appl. Math., 80 (2014), 1063-1098.  doi: 10.1093/imamat/hxu045.

[11]

D. Colton and Y. J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inver. Prob., 29 (2013), 6 pp. doi: 10.1088/0266-5611/29/10/104008.

[12]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci., (1990), 365–36A. doi: 10.1002/mma.1670120406.

[13]

M. Costabel and M. Dauge, Maxwell and Lamé eigenvalues on polyhedra, Math. Meth. Appl. Sci., 22 (1999), 243-258. 

[14]

M. DarbasJ. Heleine and S. Lohrengel, Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency, Inver. Prob. Sci. Eng., 28 (2019), 459-496.  doi: 10.1080/17415977.2019.1588896.

[15]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.

[16]

C. Daveau, A. Khelifi and I. Balloumi, Asymptotic behaviors for eigenvalues and eigenfunctions associated to Stokes operator in the presence of small boundary perturbations, Math. Phys. Anal. Geom., 20 (2017), 25 pp. doi: 10.1007/s11040-017-9243-3.

[17]

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

[18]

J. S. Hesthaven and T. Warburton, High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem, Phil. Trans. R. Soc. Lond. A, 362 (2004), 493-524.  doi: 10.1098/rsta.2003.1332.

[19]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

[20]

D. V. Korikov, Asymptotics of Maxwell system eigenvalues in a domain with small cavities, Algebr. Anal., 31 (2019), 18-71.  doi: 10.1090/spmj/1582.

[21]

V. Kozlov, On the Hadamard formula for nonsmooth domains, J. Differ. Equ., 230 (2006), 532-555.  doi: 10.1016/j.jde.2006.0A.004.

[22]

R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner and Wiley, Stuttgart, 1986. doi: 10.1007/978-3-663-10649-4.

[23]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[24] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[25]

J. H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the stokes system in two space dimensions, Adv. Differ. Equ., 6 (2001), 987-1023. 

[26]

J. E. Osborn, Spectral approximation for compact operators, Math. Comp., 29 (1975), 712-725. 

[27]

J. Sanchez Hubert and E. Sanchez Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Asymptotic Methods, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-73782-4.

[28]

C. Weber, A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci, 2 (1980), 12-25.  doi: 10.1002/mma.1670020103.

[29]

C. Wieners and J. Xin, Boundary element approximation for Maxwell's eigenvalue problem, Math. Meth. Appl. Sci., 36 (2013), 2524-2539.  doi: 10.1002/mma.2772.

show all references

References:
[1]

H. AmmariE. BerettaE. FranciniH. Kang and M. Lim, Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case, J. Math. Pure Appl., 94 (2010), 322-339.  doi: 10.1016/j.matpur.2010.02.001.

[2]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. 

[3]

H. AmmariH. KangM. Lim and H. Zribi, Layer potential techniques in spectral analysis. Part I: complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions, Trans. Amer. Math. Soc., 362 (2010), 2901-2922. 

[4]

H. AmmariM. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of imperfections of small diameter II.The full Maxwell equations, J. Math. Pures Appl., 80 (2001), 769-814.  doi: 10.1016/S0021-7824(01)01217-X.

[5]

H. Ammari and D. Volkov, Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter, Asymp. Anal., 30 (2002), 331-350. 

[6]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., North-Holland, Amsterdam, 197A.

[7]

M. Sh. Birman and M. Z. Solomyak, $L_2$ Theory of the Maxwell operator in an arbitrary domain, Russian Math. Surveys, 42 (1987), 75-96. 

[8]

H. BoujlidaH. Haddar and M. Khenissi, The asymptotic of transmission eigenvalues for a domain with a thin coating, SIAM J. Appl. Math., 78 (2018), 2348-2369.  doi: 10.1137/17M1154126.

[9]

S. C. BrennerF. Li and L. Y. Sung, Nonconforming Maxwell eigensolvers, J. Sci. Comput, 40 (2009), 51-85.  doi: 10.1007/s10915-008-9266-9.

[10]

F. CakoniN. Chaulet and H. Haddar, Asymptotic analysis of the transmission eigenvalue problem for a Dirichlet obstacle coated by a thin layer of non-absorbing media, IMA J. Appl. Math., 80 (2014), 1063-1098.  doi: 10.1093/imamat/hxu045.

[11]

D. Colton and Y. J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues, Inver. Prob., 29 (2013), 6 pp. doi: 10.1088/0266-5611/29/10/104008.

[12]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci., (1990), 365–36A. doi: 10.1002/mma.1670120406.

[13]

M. Costabel and M. Dauge, Maxwell and Lamé eigenvalues on polyhedra, Math. Meth. Appl. Sci., 22 (1999), 243-258. 

[14]

M. DarbasJ. Heleine and S. Lohrengel, Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency, Inver. Prob. Sci. Eng., 28 (2019), 459-496.  doi: 10.1080/17415977.2019.1588896.

[15]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.

[16]

C. Daveau, A. Khelifi and I. Balloumi, Asymptotic behaviors for eigenvalues and eigenfunctions associated to Stokes operator in the presence of small boundary perturbations, Math. Phys. Anal. Geom., 20 (2017), 25 pp. doi: 10.1007/s11040-017-9243-3.

[17]

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

[18]

J. S. Hesthaven and T. Warburton, High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem, Phil. Trans. R. Soc. Lond. A, 362 (2004), 493-524.  doi: 10.1098/rsta.2003.1332.

[19]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

[20]

D. V. Korikov, Asymptotics of Maxwell system eigenvalues in a domain with small cavities, Algebr. Anal., 31 (2019), 18-71.  doi: 10.1090/spmj/1582.

[21]

V. Kozlov, On the Hadamard formula for nonsmooth domains, J. Differ. Equ., 230 (2006), 532-555.  doi: 10.1016/j.jde.2006.0A.004.

[22]

R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner and Wiley, Stuttgart, 1986. doi: 10.1007/978-3-663-10649-4.

[23]

N. G. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206. 

[24] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[25]

J. H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the stokes system in two space dimensions, Adv. Differ. Equ., 6 (2001), 987-1023. 

[26]

J. E. Osborn, Spectral approximation for compact operators, Math. Comp., 29 (1975), 712-725. 

[27]

J. Sanchez Hubert and E. Sanchez Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Asymptotic Methods, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-73782-4.

[28]

C. Weber, A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci, 2 (1980), 12-25.  doi: 10.1002/mma.1670020103.

[29]

C. Wieners and J. Xin, Boundary element approximation for Maxwell's eigenvalue problem, Math. Meth. Appl. Sci., 36 (2013), 2524-2539.  doi: 10.1002/mma.2772.

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