Skip to main content
SpringerLink
Log in

Vector potential theory on nonsmooth domains in R3 and applications to electromagnetic scattering

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space.

The main goal is to develop the corresponding theory for Lp-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bergh, J. and Lofström, J. (1976).Interpolation Spaces. An Introduction. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  2. Brown, R. (1989). The method of layer potentials for the heat equation in Lipschitz cylinders.Amer. J. Math. 111, 339–379.

    Article  MATH  MathSciNet  Google Scholar 

  3. Brown, R. (1990). The initial-Neumann problem for the heat equation in Lipschitz cylinders.Trans. Amer. Math. Soc. 320, 1–52.

    Article  MATH  MathSciNet  Google Scholar 

  4. Brown, R.The Neumann problem on Lipschitz domains in Hardy spaces of order less than one, (to appear).

  5. Brown, R. and Shen, Z. (1990). The initial-Dirichlet problem for a fourth-order parabolic equation in Lipschitz cylinders.Indiana Univ. Math. J. 39, 1313–1353.

    Article  MATH  MathSciNet  Google Scholar 

  6. Birman, M. Sh. and Solomyak, M. Z. (1987). The Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary.Vestnik Leningrads. Univ. Mat. Mekh. Astronom. 20, 23–28.

    MathSciNet  Google Scholar 

  7. Birman, M. Sh. and Solomyak, M. Z. (1987).L 2-Theory of the Maxwell operator in arbitrary domains.Russian Math. Surveys 42, 75–96.

    Article  MATH  MathSciNet  Google Scholar 

  8. Calderón, A. (1954). The multipole expansion of radiation fields.J. Rational Mech. Anal. 3, 523–537.

    MathSciNet  Google Scholar 

  9. Calderón, A. (1985). Boundary value problems in Lipschitz domains.Recent Progress in Fourier Analysis. Elsevier, Amsterdam, 33–48.

    Google Scholar 

  10. Calderón, A. (1980). Commutators, singular integrals on Lipschitz curves and applications.Proc. Int. Congress Math., Helsinki 19781, 85–96.

    Google Scholar 

  11. Carleman, T. (1916).Über das Neumann-Poincarésche Problem für ein Gebiet mit Ecken. Thesis, Uppsala.

  12. Coifman, R., McIntosh, A., and Meyer, Y. (1982). L’intégrale de Cauchy définit un opérateur borné surL 2 pour les courbes Lipschitziennes.Ann. of Math. (2)116, 361–387.

    Article  MathSciNet  Google Scholar 

  13. Coifman, R. and Weiss, G. (1977). Extensions of Hardy spaces and their use in analysis.Bull. Amer. Math. Soc. (N.S.) 83, 569–645.

    MATH  MathSciNet  Google Scholar 

  14. Colton, D. and Kress, R. (1983).Integral Equation Methods in Scattering Theory. Wiley, New York.

    MATH  Google Scholar 

  15. Colton, D. and Kress, R. (1992).Inverse Acoustic and Electromagnetic Scattering Theory. Appl. Math. Sci. 93. Springer-Verlag, New York.

    Google Scholar 

  16. Costabel, M. (1990). A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains.Math. Meth. Appl. Sci. 12, 365–368.

    Article  MATH  MathSciNet  Google Scholar 

  17. Dahlberg, B. E. J. (1977). On estimates of harmonic measure.Arch. Rational Mech. Anal. 65, 272–288.

    MathSciNet  Google Scholar 

  18. Dahlberg, B. E. J. (1979). On the Poisson integral for Lipschitz andC 1 domains.Studia Math. 66, 13–24.

    MATH  MathSciNet  Google Scholar 

  19. Dahlberg, B. E. J. (1979).L q-Estimates for Green potentials in Lipschitz domains.Math. Scand. 44, 149–170.

    MATH  MathSciNet  Google Scholar 

  20. Dahlberg, B. and Kenig, C. (1987). Hardy spaces and theL p-Neumann problem for Laplace’s equation in a Lipschitz domain.Ann. of Math. 125, 437–465.

    Article  MathSciNet  Google Scholar 

  21. Dahlberg, B. and Kenig, C. (1989).L p estimates for the three-dimension system of elastostatics on Lipschitz domains.Lecture Notes in Pure and Appl. Math. (C. Sadosky ed.)122, 621–634.

    MathSciNet  Google Scholar 

  22. Dahlberg, B., Kenig, C., and Verchota, G. (1986). The Dirichlet problem for the biharmonic equation in a Lipschitz domain.Ann. Inst. Fourier (Grenoble) 36, 109–135.

    MATH  MathSciNet  Google Scholar 

  23. Dahlberg, B., Kenig, C., and Verchota, G. (1988). Boundary value problems for the systems of elastostatics in Lipschitz domains.Duke Math. J. 57, 795–818.

    Article  MATH  MathSciNet  Google Scholar 

  24. Dautray, R. and Lions, J.-L. (1990).Mathematical Analysis and Numerical Methods for Science and Technology, Vols. 1–4. Springer-Verlag, Berlin and Heidelberg.

    Google Scholar 

  25. Elschner, J. (1992). The double layer potential operator over polyhedral domains I: Solvability in weighted Sobolev spaces.Appl. Anal. 45, 117–134.

    MATH  MathSciNet  Google Scholar 

  26. Fabes, E. B. (1988). Layer potential methods for boundary value problems on Lipschitz domains.Potential Theory, Surveys and Problems (J. Král et al., eds.).Lecture Notes in Math. 1344. Springer-Verlag, New York, 55–80.

    Chapter  Google Scholar 

  27. Fabes, E. and Kenig, C. (1981). On the Hardy spaceH 1 of aC 1 domain.Ark. Mat. 19, 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  28. Fabes, E., Jodeit, M., and Rivière, N. (1978). Potential techniques for boundary value problems onC 1 domains.Acta Math. 141, 165–186.

    Article  MATH  MathSciNet  Google Scholar 

  29. Fabes, E., Kenig, C., and Verchota, G. (1988). The Dirichlet problem for the Stokes system on Lipschitz domains.Duke Math. J. 5, 769–793.

    Article  MathSciNet  Google Scholar 

  30. Fabes, E., Sand, M., and Seo, J. (1992). The spectral radius of the classical layer potentials on convex domains.Partial Differential Equations with Minimal Smoothness and Applications. Springer, New York, 129–137.

    Google Scholar 

  31. Fefferman, R. A., Kenig, C. E., and Pipher, J. (1991). The theory of weights and the Dirichlet problem for elliptic equations.Ann. Math. (2)134, 65–124.

    Article  MATH  MathSciNet  Google Scholar 

  32. Folland, G. B. (1976).Introduction to Partial Differential Equations. Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  33. Girault, V. and Raviart, P.-A. (1986).Finite Element Methods for Navier Stokes Equations. Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo.

    MATH  Google Scholar 

  34. Grisvard, P. (1985).Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, London, and Melbourne.

    MATH  Google Scholar 

  35. Isakov, V. (1990). On uniqueness in the inverse transmission scattering problem.Comm. Partial Differential Equations 15, 1565–1587.

    MATH  MathSciNet  Google Scholar 

  36. Isakov, V. (1993). Uniqueness and stability in multi-dimensional inverse problems.Inverse Problems 9, 579–621.

    Article  MATH  MathSciNet  Google Scholar 

  37. Iwaniec, T., Mitrea, M., and Scott, C. (1996). Boundary estimates for harmonic forms.Proc. Amer. Math. Soc. 124, No. 5, 1467–1471.

    Article  MATH  MathSciNet  Google Scholar 

  38. Jawerth, B. and Mitrea, M. (1995). On the spectra of the higher dimensional Maxwell operators.Harmonic Analysis and Operator Theory (G. Mendoza et al., eds.).Contemp. Math. Amer. Math. Soc., Providence, RI., 309–315.

    Google Scholar 

  39. Jawerth, B. and Mitrea, M. (1995). Higher dimensional scattering theory onC 1 and Lipschitz domains.Amer. J. Math. 117, No. 4, 929–963.

    Article  MATH  MathSciNet  Google Scholar 

  40. Jerison, D. and Kenig, C. (1981). The Neumann problem on Lipschitz domains.Bull. Amer. Math. Soc. 4, 203–207.

    Article  MATH  MathSciNet  Google Scholar 

  41. Jerison, D. and Kenig, C. (1989). The functional calculus for the Laplacian on Lipschitz domains.J. Eq. Deriv. Part., IV.1–IV.10.

    Google Scholar 

  42. Jerison, D. and Kenig, C. (1995). The inhomogeneous Dirichlet problem in Lipschitz domains.J. Funct. Anal. 130, No. 1, 161–219.

    Article  MATH  MathSciNet  Google Scholar 

  43. Jones, D. S. (1988). The eigenvalues of a cavity resonator.Quart. J. Mech. Appl. Math. 41, 469–477.

    Article  MATH  MathSciNet  Google Scholar 

  44. Kato, T. (1976).Perturbation Theory for Linear Operators. Springer-Verlag, New York.

    MATH  Google Scholar 

  45. Kenig, C. (1986). Elliptic boundary value problems on Lipschitz domains.Beijing Lectures in Harmonic Analysis. Ann. of Math. Stud. 112, 131–183.

    MathSciNet  Google Scholar 

  46. Kenig, C. (1994).Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. AMS, CBMS, Providence, RI.

    MATH  Google Scholar 

  47. Kenig, C. and Pipher, J. (1993). The Neumann problem for elliptic equations with non-smooth coefficients.Inventiones Math. 113, 447–509.

    Article  MATH  MathSciNet  Google Scholar 

  48. Kellogg, O. D. (1954).Foundation of Potential Theory. Dover, New York.

    Google Scholar 

  49. Kirsch, A. and Kress, R. (1993). Uniqueness in inverse obstacle scattering.Inverse Problems 9, 285–299.

    Article  MATH  MathSciNet  Google Scholar 

  50. Kondrat’ev, V. A. (1967). Boundary value problems for elliptic equations in regions with conical or angular points.Trans. Moscow Math. Soc. 16, 227–313.

    MATH  Google Scholar 

  51. Král, J. (1980).Integral Operators in Potential Theory.Lecture Notes in Math. 823. Springer-Verlag, Berlin, Heidelberg, and New York.

    MATH  Google Scholar 

  52. Lax, P. and Phillips, R. (1967).Scattering Theory. Academic Press, New York.

    MATH  Google Scholar 

  53. Leis, R. (1986).Initial Boundary Value Problems in Mathematical Physics. Wiley, Chichester.

    MATH  Google Scholar 

  54. Hofmann, S. and Lewis, J. L. (1995).Solvability and Representation by Caloric Layer Potentials in Time-Varying Domains. Univ. of Kentucky Research Report 95–01.

  55. Hofmann, S. and Mitrea, M. In preparation.

  56. Lewis, J. L. and Murray, M. A. M. (1995).The Method of Layer Potentials for the Heat Equation in Time-Varying Domains. Mem. Amer. Math. Soc. No. 545.

  57. Lions, J.-L. and Magenes, E. (1972).Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York, Heidelberg, and Berlin.

    Google Scholar 

  58. Maz’ya, V. G. (1991).Boundary Integral Equations.Encyclopedia Math. Sci. Anal. IV. (V. G. Maz’ya and S. M. Nikol’skii, eds.),27. Springer-Verlag, 130–233.

    Google Scholar 

  59. McIntosh, A. and Mitrea, M. (1995). A Clifford algebra approach to elliptic and parabolic boundary value problems in nonsmooth domains, (preprint).

  60. Fabes, E., Mendez, O. and Mitrea, M. (1996), in preparation.

  61. Mitrea, D. (1995). Thesis. University of Minnesota.

  62. Mitrea, M. (1994).Clifford Wavelets, Singular Integrals, and Hardy Spaces.Lecture Notes in Math. 1575. Springer-Verlag, New York.

    MATH  Google Scholar 

  63. Mitrea, M. (1994). Electromagnetic scattering on nonsmooth domains.Math. Res. Lett. 1, 639–646.

    MATH  MathSciNet  Google Scholar 

  64. Mitrea, M. (1995). The method of layer potentials in electro-magnetic scattering theory on non-smooth domains.Duke Math. J. 77, 111–133.

    Article  MATH  MathSciNet  Google Scholar 

  65. Mitrea, M. (1995). Initial boundary value problems for the parabolic Maxwell system in Lipschitz cylinders.Indiana Math. J. 44, No. 3, 797–813.

    Article  MATH  MathSciNet  Google Scholar 

  66. Mitrea, M., Torres, R., and Welland, G. (1955). Regularity and approximation results for the Maxwell problem onC 1 and Lipschitz domains.Proc. Conf. on Clifford Algebras in Analysis (J. Ryan, ed.),Studies in Advanced Mathematics. C.R.C. Press Inc., Boca Raton, FL, 297–308.

    Google Scholar 

  67. Müller, C. (1951). Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern.Math. Ann. 123, 345–378.

    Article  MathSciNet  Google Scholar 

  68. Müller, C. (1969).Foundations of the Mathematical Theory of Electromagnetic Waves. Springer-Verlag, New York.

    MATH  Google Scholar 

  69. Müller, C. and Niemeyer, H. (1961). Greensche Tensoren und asymptotische Gesetze der elektromagnetischen Hohlraumschwingungen.Arch. Rational Mech. Anal. 7, 305–348.

    Article  MATH  MathSciNet  Google Scholar 

  70. Nečas, J. (1967).Les méthodes directes en théorie des équations élliptique. Academia, Prague.

    Google Scholar 

  71. Payne, L. and Weinberger, H. (1954). New bounds in harmonic and biharmonic problems.J. Math. Phys. 33, 291–307.

    MathSciNet  Google Scholar 

  72. Picard, R. (1984). An elementary proof for a compact embedding result in generalized electromagnetic theory.Math. Z. 187, 151–164.

    Article  MATH  MathSciNet  Google Scholar 

  73. Pipher, J. and Verchota, G. (1992). The Dirichlet problem inL p for biharmonic functions on Lipschitz domains.Amer. J. Math. 114, 923–972.

    Article  MATH  MathSciNet  Google Scholar 

  74. Pipher, J. and Verchota, G. (1995). Dilation invariant estimates and the boundary Garding inequality.Ann. of Math. (to appear).

  75. Radon, J. Über die Randwertaufgaben beim logarithmischen potential.Sitzungsber. Akad. Wiss. 128, 1123–1167.

  76. Ramm, A. (1986).Scattering by Obstacles. D. Reidel, Dordrecht.

    MATH  Google Scholar 

  77. Reed, M. and Simon, B. (1979).Methods of Modern Mathematical Physics. Vol. III Scattering Theory. Academic Press, New York.

    Google Scholar 

  78. Rellich, F. (1940). Darstellung der eigenwerte von Δu + λu durch ein Radintegral.Math. Z. 46, 635–646.

    Article  MATH  MathSciNet  Google Scholar 

  79. Safarov, Yu. G. (1984). On the asymptotic behavior of the spectrum of the Maxwell operator.J. Soviet Math. 27, 2655–2661.

    Article  MATH  MathSciNet  Google Scholar 

  80. Saranen, J. (1982). On an inequality of Friedrichs.Math. Scand. 51, 310–322.

    MATH  MathSciNet  Google Scholar 

  81. Saunders, W. K. (1952). On solutions of Maxwell’s equations in an exterior region.Proc. Nat. Acad. Sci. U.S.A. 38, 342–348.

    Article  MathSciNet  Google Scholar 

  82. Shen, Z. (1991). Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders.Amer. J. Math. 113, 293–373.

    Article  MATH  MathSciNet  Google Scholar 

  83. Shen, Z. (1995). A note on the Dirichlet problem for the Stokes system in Lipschitz domains.Proc. Amer. Math. Soc. 123, 801–811.

    Article  MATH  MathSciNet  Google Scholar 

  84. Spencer, R. S. (1994).Series Solutions and Spectral Properties of Boundary Integral Equations. Thesis. University of Minnesota.

  85. Stein, E.M. (1970).Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ.

    MATH  Google Scholar 

  86. Stephan, E. P. (1986). Boundary integral equations for magnetic screens inR 3.Proc. Roy. Soc. Edinburgh Sect. A 102, 188–210.

    MathSciNet  Google Scholar 

  87. Taylor, M. E. (1996).Partial Differential Equations. Springer-Verlag, New York.

    Google Scholar 

  88. Temam, R. (1977).Theory and Numerical Analysis of the Navier-Stokes Equations. North-Holland, Amsterdam.

    Google Scholar 

  89. Uhlmann, G. (1992). Inverse boundary value problems and applications.Astérisque 207. Soc. Math. France, Montrouge.

    Google Scholar 

  90. Uhlmann, G. (1994). Inverse boundary value problems for first order perturbations of the Laplions.Lectures in Appl. Math. 30. Amer. Math. Soc., Providence, RI, 245–258.

    Google Scholar 

  91. Varopoulos, N. (1977). A remark on functions of bounded mean oscillations and bounded harmonic functions.Pacific J. Math. 74, 257–259.

    MathSciNet  Google Scholar 

  92. Verchota, G. (1982).Layer Potentials and Boundary Value Problems for Laplace’s Equation on Lipschitz Domains. Thesis. University of Minnesota.

  93. Verchota, G. (1984). Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains.J. Funct. Anal. 59, 572–611.

    Article  MATH  MathSciNet  Google Scholar 

  94. Weber, C. (1980). A local compactness theorem for Maxwell’s equations.Math. Meth. Appl. Sci. 2, 12–25.

    Article  MATH  Google Scholar 

  95. Weck, N. (1974). Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries.J. Math. Anal. Appl. 46, 410–437.

    Article  MATH  MathSciNet  Google Scholar 

  96. Weyl, H. (1911). Über die asymptotische verteilung der Eigenwerte.Gott. Nach. 110–117.

  97. Weyl, H. (1912). Über das Spectrum der Hohlraumstrahlung.J. Reine Angew. Math. 141, 163–181.

    MATH  Google Scholar 

  98. Weyl, H. (1952). Kapazität von Strahlungsfeldern.Math. Z. 55, 187–198.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri at Columbia, 65211, Columbia, Missouri

    Dorina Mitrea

  2. Department of Mathematics, University of Missouri at Columbia, 65211, Columbia, Missouri

    Marius Mitrea

  3. Department of Mathematics, Brown University, 02912, Providence, Rhode Island

    Jill Pipher

Authors
  1. Dorina Mitrea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marius Mitrea
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jill Pipher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mitrea, D., Mitrea, M. & Pipher, J. Vector potential theory on nonsmooth domains in R3 and applications to electromagnetic scattering. The Journal of Fourier Analysis and Applications 3, 131–192 (1997). https://doi.org/10.1007/BF02649132

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02649132

Math Subject Classifications

Keywords and Phrases

Search

Navigation