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Journal of Vibration Testing and System Dynamics 7(3) (2023) 285--306 | DOI:10.5890/JVTSD.2023.09.004

K. H. Leem$^1$, G. Pelekanos$^1$, V. Sevroglou$^2$

$^1$ Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA,

$^2$ Department of Statistics and Insurance Science, University of Piraeus, GR 15834, Piraeus, Greece

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Abstract

In this paper the direct scattering problem by an impenetrable obstacle embedded in a given homogeneous background chiral medium is studied. Incident electromagnetic waves are propagated in a homogeneous chiral environment. We assume that the chirality measures of the background and exterior medium are both distinct positive constants. Our scatterer has a smooth boundary that is divided into two open disjoint parts for which an impedance boundary condition on the one part of the boundary, and a perfectly conducting boundary condition on the other part, are satisfied. Uniqueness results for the above scattering problem, using the Bohren decomposition into Beltrami fields, are established. Consequently, we introduce a \emph{chiral Calderon} operator, for which its basic properties are proved, and its connection with the existence of our problems solution is presented. The well-posedness of our problem is completed by proving the continuous dependence of the solution on the boundary data. Finally, some discussion and conclusions are given.

References

1. [1]	Lakhtakia, A., Varadan, V.K., and Varadan, V.V. (1989), Time-harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics, Springer-Berlin.

2. [2]	Lakhtakia, A. (1994), Beltrami Fields in Chiral Media, World Scientific, Singapore.

3. [3]	Roach, G.F., Stratis, I.G., and Yannacopoulos, A.N. (2012), Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton Series in Applied Mathematics, Princeton University Press, New Jersey.

4. [4]	Athanasiadis, C.E., Costakis, G., and Stratis, I.G. (2000), Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment, IMA Journal of Applied Mathematics, 64(3), 245-258.

5. [5]	Lakhtakia, A., Varadan, V.K., and Varadan, V.V. (1991), Surface integral equations for scattering by PEC scatterers in isotropic chiral media, International Journal of Engineering Science, 29(2), 179-185.

6. [6]	Lindell, I.V., Sihvola, A.H., Tretyakov, S.A., and Viitanen, A.J. (1994), Electromagnetic Waves in Chiral and Bi-isotropic Media, Artech House, Boston.

7. [7]	Athanasiadis, C.E., Martin, P.A., and Stratis, I.G. (1999), Electromagnetic scattering by a homogeneous chiral obstacle: boundary integral equations and low-chirality approximations, SIAM Journal on Applied Mathematics, 59(5), 1745-1762.

8. [8]	Athanasiadis, C.E., Costakis, G., and Stratis, I.G. (2000), On some properties of Beltrami fields in chiral media, Reports on Mathematical Physics, 45(2), 257-270.

9. [9]	Ammari, H. and Nedelec, J.C. (1998), Time-harmonic electromagnetic fields in thin chiral curved layers, SIAM Journal on Mathematical Analysis, 29(2), 395-423.

10. [10]	Ammari, H., Hamdache, K., and Nedelec, J.C. (1999), Chirality in the Maxwell equations by the dipole approximation method, SIAM Journal on Applied Mathematics, 59(6), 2045-2059.

11. [11]	Athanasiadis, C.E. and Kardasi, E. (2005), Beltrami Herglotz functions for electromagnetic scattering, Applicable Analysis, 84(2), 145-163.

12. [12]	Tai, C.T. (1994), Dyadic Green Functions in Electromagnetic Theory, 2nd edition, IEEE Press, New York.

13. [13]	Colton, D. and Kress, R. (1983), Integral Equation Methods in Scattering Theory, Wiley.

14. [14]	Stephan, E.P. (1987), Boundary integral equations for screen problems in $\mathbb{R}^3$, Integral Eqns Operator Theory, 10, 236-257.

15. [15]	Cakoni, F. and and Colton, D. (2005), Qualitative Methods in Inverse Electromagnetic Scattering Theory, Springer-Verlag, New York.

16. [16]	Athanasiadis, C.E. (2005), On the far field patterns for electromagnetic scattering by a chiral obstacle in chiral environment, Journal of Mathematical Analysis and Applications, 309(2), 517-533.

17. [17]	Cakoni, F., Colton, D., and Darringrand, E. (2003), A Uniqueness theorem for an inverse electromagnetic scattering problem in homogeneous anisotropic media, Proceedings of the Edinburgh Mathematical Society, 46(2), 293-314.

18. [18]	Cakoni, F. and Darringrand, E. (2005), The inverse electromagnetic scattering problem for a mixed boundary value problem for screens, Journal of Computational and Applied Mathematics, 174(2), 251-269.

19. [19]	Cakoni, F. and Haddar, H, (2007), Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data, The Journal of Integral Equations and Applications, 19, 359--389.

20. [20]	Collino, F., Fares, M. and Haddar, H. (2003), Numerical and analytical studies of the linear sampling method in electromagnetic scattering problems, Inverse Problems, 19, 1279-1299.

21. [21]	Colton, D. and Kirsch, A. (1997), A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12, 383-393.

22. [22]	Colton, D., Pianna, M., and Potthast, R. (1997), A simple method using Morozov's discrepancy principle for solving inverse scattering problem, Inverse Problems, 13, 1477-1493.

23. [23]	Kirsch, A. and Grinberg, N. (2008), The Factorization Method for Inverse Problems, Oxford University Press, Oxford.

24. [24]	Athanasiadis, C.E., Sevroglou, V.I., and Skourogiannis, K.I. (2012), The direct electromagnetic scattering problem by a mixed impedance screen in chiral media, Applicable Analysis, 91(11), 1-11.

25. [25]	Athanasiadis, C.E., Sevroglou, V., and Skourogiannis, K.I. (2015), The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media, Inverse Problems in Image, 9(4), 951-970.

26. [26]	Athanasiadis, C.E., Natrosvili, D., Sevroglou, V., and Stratis, I.G. (2011), A boundary integral equations approach for direct mixed impedance problems in elasticity, The Journal of Integral Equations and Applications, 23(2), 183-222.

27. [27]	McLean, W. (2003), Strongly Elliptic Systems and Boundary Integral equations, Cambridge University Press, Cambridge.

28. [28]	Monk, P. (2003), Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford.

29. [29]	Colton, D. and Kress, R. (1992), Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag. % %

30. [30]	Ammari, H. and Nedelec, J.C. (1998), Time-harmonic electromagnetic fields in thin chiral curved layers, SIAM Journal on Mathematical Analysis, 29(2), 395-423. % %

31. [31]	Ammari, H., Hamdache, K., and Nedelec, J.C. (1999), Chirality in the Maxwell equations by the dipole approximation method, SIAM Journal on Applied Mathematics, 59(6), 2045-2059. % % %

32. [32]	Athanasiadis, C.E., Martin, P.A., and Stratis, I.G. (1999), Electromagnetic scattering by a homogeneous chiral obstacle: boundary integral equations and low-chirality approximations, SIAM Journal on Applied Mathematics, 59(5), 1745-1762. % % % % %

33. [33]	Athanasiadis, C.E., Costakis, G., and Stratis, I.G. (2000), %Electromagnetic scattering by a homogeneous chiral %obstacle in a chiral environment, IMA Journal of Applied Mathematics, 64(3), 245-258. % % %

34. [34]	Athanasiadis, C.E., Costakis, G., and Stratis, I.G. (2000), % On some properties of Beltrami fields in chiral media, % Reports on Mathematical Physics, 45(2), 257-270. % % % %

35. [35]	Athanasiadis, C.E. (2005), On the far field patterns for electromagnetic scattering by a chiral obstacle in chiral environment, %Journal of Mathematical Analysis and Applications, 309(2), 517-533. % % %

36. [36]	Athanasiadis, C.E. and Kardasi, E. (2005), %Beltrami Herglotz functions for electromagnetic scattering, %Applicable Analysis, 84(2), 145-163. % %

37. [37]	Athanasiadis, C.E., Natrosvili, D., Sevroglou, V., and Stratis, I.G. (2011), %A boundary integral equations approach for direct mixed impedance %problems in elasticity, The Journal of Integral Equations and Applications, 23(2), 183-222. % %

38. [38]	Athanasiadis, C.E., Sevroglou, V.I., and Skourogiannis, K.I. (2012), The direct electromagnetic scattering problem by a mixed impedance screen %in chiral media, Applicable Analysis, 91(11), 1-11. % % %

39. [39]	Athanasiadis, C.E., Sevroglou, V., and Skourogiannis, K.I. (2015), The inverse electromagnetic scattering problem by a mixed impedance screen %in chiral media, Inverse Problems in Image, 9(4), 951-970. % %

40. [40]	Cakoni, F., Colton, D., and Darringrand, E. (2003), A Uniqueness theorem for an inverse electromagnetic scattering problem in homogeneous anisotropic media, %Proceedings of the Edinburgh Mathematical Society, 46(2), 293-314. % % %

41. [41]	Cakoni, F. and Colton, D. (2003), The inverse electromagnetic scattering problem for screens, Inverse Problems, 19, 627-642. % %

42. [42]	Cakoni, F. and and Colton, D. (2005), Qualitative Methods in Inverse Electromagnetic Scattering Theory, Springer-Verlag, New York. % %

43. [43]	Cakoni, F. and Darringrand, E. (2005), The inverse electromagnetic scattering problem for a mixed boundary value problem for screens, %Journal of Computational and Applied Mathematics, 174(2), 251-269. % %

44. [44]	Cakoni, F. and Haddar, H, (2007), Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data, The Journal of Integral Equations and Applications, 19, 359--389. % %

45. [45]	Collino, F., Fares, M. and Haddar, H. (2003), Numerical and analytical studies of the linear sampling method in electromagnetic scattering problems, Inverse Problems, 19, 1279-1299. % %

46. [46]	Colton, D. and Kress, R. (1983), Integral Equation Methods in Scattering Theory, Wiley. % % %

47. [47]	Colton, D. and Kress, R. (1992), %Inverse Acoustic and Electromagnetic Scattering Theory, %Springer-Verlag. % %

48. [48]	Colton, D. and Kirsch, A. (1997), %A simple method for solving inverse scattering problems in the resonance region, %Inverse Problems, 12, 383-393. % %

49. [49]	Colton, D. Pianna, M., and Potthast, R. (1997), %A simple method using Morozov's discrepancy principle for solving inverse scattering problem, %Inverse Problems, 13, 1477-1493. % %

50. [50]	Colton, D., Haddar, H., and Pianna, M. (2003), %The linear sampling method in inverse electromagnetic scattering theory, %Inverse problems, 19, 105--137. % %

51. [51]	Kirsch, A. and Grinberg, N. (2008), %The Factorization Method for Inverse Problems, Oxford University Press, Oxford. % %

52. [52]	Lakhtakia, A., Varadan, V.K., and Varadan, V.V. (1989), %Time-harmonic Electromagnetic Fields in Chiral Media, %Lecture Notes in Physics, Springer-Berlin. % %

53. [53]	Lakhtakia, A., Varadan, V.K., and Varadan, V.V. (1991), %Surface integral equations for scattering by PEC scatterers in isotropic chiral %media, International Journal of Engineering Science,, 29(2), 179-185. % %

54. [54]	Lakhtakia, A. (1994), Beltrami Fields in Chiral Media, World Scientific, Singapore. % % %

55. [55]	Lindell, I.V., Sihvola, A.H., Tretyakov, S.A., and Viitanen, A.J. (1994), %Electromagnetic Waves in Chiral and Bi-isotropic Media, %Artech House, Boston. % %

56. [56]	McLean, W. (2003), Strongly Elliptic Systems and Boundary Integral equations, Cambridge University Press, Cambridge. % %

57. [57]	Monk, P. (2003), Finite Element Methods for Maxwell's Equations, %Clarendon Press, Oxford. % %

58. [58]	Roach, G.F., Stratis, I.G., and Yannacopoulos, A.N. (2012), %Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media %Electromagnetics, Princeton Series in Applied Mathematics, Princeton University Press, New Jersey. % %

59. [59]	Stephan, E.P. (1987), %Boundary integral equations for screen problems in $\mathbb{R}^3$, %Integral Eqns Operator Theory, 10, 236-257. % %

60. [60]	Tai, C.T. (1994), Dyadic Green Functions in Electromagnetic Theory, 2nd edition, IEEE Press, New York.