Abstract
Up until now, we have considered only the direct and inverse obstacle scattering problem for time-harmonic acoustic waves. In the following two chapters, we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves. As in our analysis on acoustic scattering, we begin with an outline of the solution of the direct problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Angell, T.S., Colton, D., and Kress, R.: Far field patterns and inverse scattering problems for imperfectly conducting obstacles. Math. Proc. Camb. Phil. Soc. 106, 553–569 (1989).
Angell, T.S., and Kirsch, A.: The conductive boundary condition for Maxwell’s equations. SIAM J. Appl. Math. 52, 1597–1610 (1992).
Blöhbaum, J.: Optimisation methods for an inverse problem with time-harmonic electromagnetic waves: an inverse problem in electromagnetic scattering. Inverse Problems 5, 463–482 (1989).
Calderón, A.P.: The multipole expansions of radiation fields. J. Rat. Mech. Anal. 3, 523–537 (1954).
Colton, D., and Kirsch, A.: The use of polarization effects in electromagnetic inverse scattering problems. Math. Meth. in the Appl. Sci. 15, 1–10 (1992).
Colton, D., and Kress, R.: Dense sets and far field patterns in electromagnetic wave propagation. SIAM J. Math. Anal. 16, 1049–1060 (1985).
Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.
Ganesh, M., and Hawkins, S. C.: A spectrally accurate algorithm for electromagnetic scattering in three dimensions. Numer. Algorithms 43, 25–60 (2006).
Ganesh, M., and Hawkins, S. C.: An efficient surface integral equation method for the time-harmonic Maxwell equations. ANZIAM J. 48, C17–C33 (2007).
Ganesh, M., and Hawkins, S. C.: A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces. J. Comput. Phys. 227, 4543–4562 (2008).
Hähner, P.: An exterior boundary-value problem for the Maxwell equations with boundary data in a Sobolev space. Proc. Roy. Soc. Edinburgh 109A, 213–224 (1988).
Hähner, P.: Eindeutigkeits- und Regularitätssätze für Randwertprobleme bei der skalaren und vektoriellen Helmholtzgleichung. Dissertation, Göttingen 1990.
Jones, D.S.: Methods in Electromagnetic Wave Propagation. Clarendon Press, Oxford 1979.
Jones, D.S.: Acoustic and Electromagnetic Waves. Clarendon Press, Oxford 1986.
Kirsch, A.: Surface gradients and continuity properties for some integral operators in classical scattering theory. Math. Meth. in the Appl. Sci. 11, 789–804 (1989).
Knauff, W., and Kress, R.: On the exterior boundary value problem for the time-harmonic Maxwell equations. J. Math. Anal. Appl. 72, 215–235 (1979).
Kress, R.: On the boundary operator in electromagnetic scattering. Proc. Royal Soc. Edinburgh 103A, 91–98 (1986).
Le Louër, F.: Spectrally accurate numerical solution of hypersingular boundary integral equations for three-dimensional electromagnetic wave scattering problems. J. Comput. Phys. 275, 662–666 (2014).
Le Louër, F.: A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles. ANZIAM 59, E1–E49 (2018).
Martensen, E.: Potentialtheorie. Teubner-Verlag, Stuttgart 1968.
Mautz, J.R., and Harrington, R.F.: A combined-source solution for radiating and scattering from a perfectly conducting body. IEEE Trans. Ant. and Prop. AP-27, 445–454 (1979).
Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford, 2003.
Müller, C.: Zur mathematischen Theorie elektromagnetischer Schwingungen. Abh. deutsch. Akad. Wiss. Berlin 3, 5–56 (1945/46).
Müller, C.: Randwertprobleme der Theorie elektromagnetischer Schwingungen. Math. Z. 56, 261–270 (1952).
Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin 1969.
Nédélec, J.C.; Acoustic and Electromagnetic Equations. Springer, Berlin 2001.
Pieper, M.: Spektralrandintegralmethoden zur Maxwell-Gleichung. Dissertation, Göttingen 2007.
Pieper, M.: Vector hyperinterpolation on the sphere. J. Approx. Theory 156, 173–186 (2009).
Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.
Silver, S.: Microwave Antenna Theory and Design. M.I.T. Radiation Laboratory Series Vol. 12, McGraw-Hill, New York 1949.
Stratton, J.A., and Chu, L.J.: Diffraction theory of electromagnetic waves. Phys. Rev. 56, 99–107 (1939).
van Bladel, J.: Electromagnetic Fields. Hemisphere Publishing Company, Washington 1985.
Weyl, H.: Kapazität von Strahlungsfeldern. Math. Z. 55, 187–198 (1952).
Wilcox, C.H.: An expansion theorem for electromagnetic fields. Comm. Pure Appl. Math. 9, 115–134 (1956).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Colton, D., Kress, R. (2019). The Maxwell Equations. In: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-030-30351-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-30351-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30350-1
Online ISBN: 978-3-030-30351-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)