The Maxwell Equations

  • David Colton
  • Rainer Kress
Part of the Applied Mathematical Sciences book series (AMS, volume 93)


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.


  1. 12.
    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).MathSciNetCrossRefGoogle Scholar
  2. 13.
    Angell, T.S., and Kirsch, A.: The conductive boundary condition for Maxwell’s equations. SIAM J. Appl. Math. 52, 1597–1610 (1992).MathSciNetCrossRefGoogle Scholar
  3. 34.
    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).MathSciNetCrossRefGoogle Scholar
  4. 71.
    Calderón, A.P.: The multipole expansions of radiation fields. J. Rat. Mech. Anal. 3, 523–537 (1954).MathSciNetzbMATHGoogle Scholar
  5. 93.
    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).MathSciNetCrossRefGoogle Scholar
  6. 97.
    Colton, D., and Kress, R.: Dense sets and far field patterns in electromagnetic wave propagation. SIAM J. Math. Anal. 16, 1049–1060 (1985).MathSciNetCrossRefGoogle Scholar
  7. 104.
    Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefGoogle Scholar
  8. 144.
    Ganesh, M., and Hawkins, S. C.: A spectrally accurate algorithm for electromagnetic scattering in three dimensions. Numer. Algorithms 43, 25–60 (2006).MathSciNetCrossRefGoogle Scholar
  9. 145.
    Ganesh, M., and Hawkins, S. C.: An efficient surface integral equation method for the time-harmonic Maxwell equations. ANZIAM J. 48, C17–C33 (2007).MathSciNetCrossRefGoogle Scholar
  10. 146.
    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).MathSciNetCrossRefGoogle Scholar
  11. 171.
    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).MathSciNetCrossRefGoogle Scholar
  12. 172.
    Hähner, P.: Eindeutigkeits- und Regularitätssätze für Randwertprobleme bei der skalaren und vektoriellen Helmholtzgleichung. Dissertation, Göttingen 1990.Google Scholar
  13. 223.
    Jones, D.S.: Methods in Electromagnetic Wave Propagation. Clarendon Press, Oxford 1979.Google Scholar
  14. 224.
    Jones, D.S.: Acoustic and Electromagnetic Waves. Clarendon Press, Oxford 1986.Google Scholar
  15. 234.
    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).MathSciNetCrossRefGoogle Scholar
  16. 256.
    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).MathSciNetCrossRefGoogle Scholar
  17. 260.
    Kress, R.: On the boundary operator in electromagnetic scattering. Proc. Royal Soc. Edinburgh 103A, 91–98 (1986).MathSciNetCrossRefGoogle Scholar
  18. 298.
    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).MathSciNetCrossRefGoogle Scholar
  19. 299.
    Le Louër, F.: A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles. ANZIAM 59, E1–E49 (2018).CrossRefGoogle Scholar
  20. 311.
    Martensen, E.: Potentialtheorie. Teubner-Verlag, Stuttgart 1968.zbMATHGoogle Scholar
  21. 313.
    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).CrossRefGoogle Scholar
  22. 320.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford, 2003.CrossRefGoogle Scholar
  23. 328.
    Müller, C.: Zur mathematischen Theorie elektromagnetischer Schwingungen. Abh. deutsch. Akad. Wiss. Berlin 3, 5–56 (1945/46).Google Scholar
  24. 330.
    Müller, C.: Randwertprobleme der Theorie elektromagnetischer Schwingungen. Math. Z. 56, 261–270 (1952).MathSciNetCrossRefGoogle Scholar
  25. 332.
    Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin 1969.CrossRefGoogle Scholar
  26. 337.
    Nédélec, J.C.; Acoustic and Electromagnetic Equations. Springer, Berlin 2001.CrossRefGoogle Scholar
  27. 349.
    Pieper, M.: Spektralrandintegralmethoden zur Maxwell-Gleichung. Dissertation, Göttingen 2007.Google Scholar
  28. 351.
    Pieper, M.: Vector hyperinterpolation on the sphere. J. Approx. Theory 156, 173–186 (2009).MathSciNetCrossRefGoogle Scholar
  29. 375.
    Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.zbMATHGoogle Scholar
  30. 393.
    Silver, S.: Microwave Antenna Theory and Design. M.I.T. Radiation Laboratory Series Vol. 12, McGraw-Hill, New York 1949.Google Scholar
  31. 399.
    Stratton, J.A., and Chu, L.J.: Diffraction theory of electromagnetic waves. Phys. Rev. 56, 99–107 (1939).CrossRefGoogle Scholar
  32. 411.
    van Bladel, J.: Electromagnetic Fields. Hemisphere Publishing Company, Washington 1985.Google Scholar
  33. 426.
    Weyl, H.: Kapazität von Strahlungsfeldern. Math. Z. 55, 187–198 (1952).Google Scholar
  34. 429.
    Wilcox, C.H.: An expansion theorem for electromagnetic fields. Comm. Pure Appl. Math. 9, 115–134 (1956).MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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