Duke Mathematical Journal

The method of layer potentials in electromagnetic scattering theory on nonsmooth domains

Marius Mitrea

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J. Volume 77, Number 1 (1995), 111-133.

Dates
First available: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077286148

Mathematical Reviews number (MathSciNet)
MR1317629

Zentralblatt MATH identifier
0833.35138

Digital Object Identifier
doi:10.1215/S0012-7094-95-07705-9

Subjects
Primary: 78A45: Diffraction, scattering [See also 34E20 for WKB methods]
Secondary: 35Q60: PDEs in connection with optics and electromagnetic theory

Citation

Mitrea, Marius. The method of layer potentials in electromagnetic scattering theory on nonsmooth domains. Duke Mathematical Journal 77 (1995), no. 1, 111--133. doi:10.1215/S0012-7094-95-07705-9. http://projecteuclid.org/euclid.dmj/1077286148.


Export citation

References

  • [1] R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339–379.
  • [2] R. Brown and Z. Shen, The initial-Dirichlet problem for a fourth-order parabolic equation in Lipschitz cylinders, Indiana Math. J. 39 (1990), no. 4, 1313–1353.
  • [3] A. Calderón, The multipole expansion of radiation fields, J. Rational Mech. Anal. 3 (1954), 523–537.
  • [4] A. Calderón, Boundary value problems for the Laplace equation in Lipschitzian domains, Recent Progress in Fourier Analysis (El Escorial, 1983), North-Holland Math Stud., vol. 111, North-Holland, Amsterdam, 1985, pp. 33–48.
  • [5] R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $L\sp2$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387.
  • [6] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure and Applied Mathematics, Wiley, New York, 1983.
  • [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Appl. Math. Sci., vol. 93, Springer-Verlag, Berlin, 1992.
  • [8] B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465.
  • [9] B. Dahlberg, C. Kenig, and G. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135.
  • [10] B. Dahlberg, C. Kenig, and G. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 795–818.
  • [11]1 R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 1, Springer-Verlag, Berlin, 1990.
  • [11]2 R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988.
  • [11]3 R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 3, Springer-Verlag, Berlin, 1990.
  • [11]4 R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 4, Springer-Verlag, Berlin, 1990.
  • [12] L. Escauriaza, E. Fabes, and G. Verchota, On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1069–1076.
  • [13] E. Fabes, M. Jodeit, and N. Rivière, Potential techniques for boundary value problems on $C^1$ domains, Acta Math. 141 (1978), no. 3-4, 165–186.
  • [14] E. Fabes, C. Kenig, and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 769–793.
  • [15] E. Fabes, M. Sand, and J. Seo, The spectral radius of the classical layer potentials on convex domains, Partial Differential Equations with Minimal Smoothness and Applications (Chicago, IL, 1990), IMA Vol. Math. Appl., vol. 42, Springer-Verlag, New York, 1992, pp. 129–137.
  • [16] P. Hähner, An exterior boundary-value problem for the Maxwell equations with boundary data in a Sobolev space, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), no. 3-4, 213–224.
  • [17] D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. 4 (1981), no. 2, 203–207.
  • [18] D. Jerison and C. Kenig, Boundary value problems on Lipschitz domains, Studies in Partial Differential Equations, MAA Studies in Mathematics, vol. 23, Math. Assoc. America, Washington D.C., 1982, pp. 1–68.
  • [19] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976.
  • [20] C. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing Lectures in Harmonic Analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton University Press, Princeton, 1986, pp. 131–183.
  • [21] O. D. Kellogg, Foundation of Potential Theory, Dover Publications, Inc., New York, 1954.
  • [22] R. Leis, Initial Boundary Value Problems in Mathematical Physics, Wiley, New York, 1986.
  • [23] M. Mitrea, Boundary value problems and Hardy spaces associated to the Helmholtz equation on Lipschitz domains, preprint.
  • [24] M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Math., vol. 1575, Springer-Verlag, Berlin, 1994.
  • [25] M. Mitrea, R. Torres, and G. Welland, Layer potential techniques in electromagnetism, preprint.
  • [26] C. Müller, Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern, Math. Ann. 123 (1951), 345–378.
  • [27] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves, Revised and enlarged translation from the German. Die Grundlehren der mathematischen Wissenschaften, Band 155, Springer-Verlag, New York, 1969.
  • [28] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris, 1967.
  • [29] P. Ola, L. Päivärinta, and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J. 70 (1993), no. 3, 617–653.
  • [30] R. Torres and G. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces, preprint.
  • [31] G. Verchota, Layer Potentials and Boundary Value Problems for Laplace's equation on Lipschitz Domains, thesis, University of Minnesota, 1982.
  • [32] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611.
  • [33] H. Weyl, Kapazität von Strahlungsfeldern, Math. Z. 55 (1952), 187–198.