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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bergh, J. and Lofström, J. (1976).Interpolation Spaces. An Introduction. Springer-Verlag, Berlin.
Brown, R. (1989). The method of layer potentials for the heat equation in Lipschitz cylinders.Amer. J. Math. 111, 339–379.
Brown, R. (1990). The initial-Neumann problem for the heat equation in Lipschitz cylinders.Trans. Amer. Math. Soc. 320, 1–52.
Brown, R.The Neumann problem on Lipschitz domains in Hardy spaces of order less than one, (to appear).
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.
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.
Birman, M. Sh. and Solomyak, M. Z. (1987).L 2-Theory of the Maxwell operator in arbitrary domains.Russian Math. Surveys 42, 75–96.
Calderón, A. (1954). The multipole expansion of radiation fields.J. Rational Mech. Anal. 3, 523–537.
Calderón, A. (1985). Boundary value problems in Lipschitz domains.Recent Progress in Fourier Analysis. Elsevier, Amsterdam, 33–48.
Calderón, A. (1980). Commutators, singular integrals on Lipschitz curves and applications.Proc. Int. Congress Math., Helsinki 19781, 85–96.
Carleman, T. (1916).Über das Neumann-Poincarésche Problem für ein Gebiet mit Ecken. Thesis, Uppsala.
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.
Coifman, R. and Weiss, G. (1977). Extensions of Hardy spaces and their use in analysis.Bull. Amer. Math. Soc. (N.S.) 83, 569–645.
Colton, D. and Kress, R. (1983).Integral Equation Methods in Scattering Theory. Wiley, New York.
Colton, D. and Kress, R. (1992).Inverse Acoustic and Electromagnetic Scattering Theory. Appl. Math. Sci. 93. Springer-Verlag, New York.
Costabel, M. (1990). A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains.Math. Meth. Appl. Sci. 12, 365–368.
Dahlberg, B. E. J. (1977). On estimates of harmonic measure.Arch. Rational Mech. Anal. 65, 272–288.
Dahlberg, B. E. J. (1979). On the Poisson integral for Lipschitz andC 1 domains.Studia Math. 66, 13–24.
Dahlberg, B. E. J. (1979).L q-Estimates for Green potentials in Lipschitz domains.Math. Scand. 44, 149–170.
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.
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.
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.
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.
Dautray, R. and Lions, J.-L. (1990).Mathematical Analysis and Numerical Methods for Science and Technology, Vols. 1–4. Springer-Verlag, Berlin and Heidelberg.
Elschner, J. (1992). The double layer potential operator over polyhedral domains I: Solvability in weighted Sobolev spaces.Appl. Anal. 45, 117–134.
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.
Fabes, E. and Kenig, C. (1981). On the Hardy spaceH 1 of aC 1 domain.Ark. Mat. 19, 1–22.
Fabes, E., Jodeit, M., and Rivière, N. (1978). Potential techniques for boundary value problems onC 1 domains.Acta Math. 141, 165–186.
Fabes, E., Kenig, C., and Verchota, G. (1988). The Dirichlet problem for the Stokes system on Lipschitz domains.Duke Math. J. 5, 769–793.
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.
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.
Folland, G. B. (1976).Introduction to Partial Differential Equations. Princeton University Press, Princeton, NJ.
Girault, V. and Raviart, P.-A. (1986).Finite Element Methods for Navier Stokes Equations. Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo.
Grisvard, P. (1985).Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, London, and Melbourne.
Isakov, V. (1990). On uniqueness in the inverse transmission scattering problem.Comm. Partial Differential Equations 15, 1565–1587.
Isakov, V. (1993). Uniqueness and stability in multi-dimensional inverse problems.Inverse Problems 9, 579–621.
Iwaniec, T., Mitrea, M., and Scott, C. (1996). Boundary estimates for harmonic forms.Proc. Amer. Math. Soc. 124, No. 5, 1467–1471.
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.
Jawerth, B. and Mitrea, M. (1995). Higher dimensional scattering theory onC 1 and Lipschitz domains.Amer. J. Math. 117, No. 4, 929–963.
Jerison, D. and Kenig, C. (1981). The Neumann problem on Lipschitz domains.Bull. Amer. Math. Soc. 4, 203–207.
Jerison, D. and Kenig, C. (1989). The functional calculus for the Laplacian on Lipschitz domains.J. Eq. Deriv. Part., IV.1–IV.10.
Jerison, D. and Kenig, C. (1995). The inhomogeneous Dirichlet problem in Lipschitz domains.J. Funct. Anal. 130, No. 1, 161–219.
Jones, D. S. (1988). The eigenvalues of a cavity resonator.Quart. J. Mech. Appl. Math. 41, 469–477.
Kato, T. (1976).Perturbation Theory for Linear Operators. Springer-Verlag, New York.
Kenig, C. (1986). Elliptic boundary value problems on Lipschitz domains.Beijing Lectures in Harmonic Analysis. Ann. of Math. Stud. 112, 131–183.
Kenig, C. (1994).Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. AMS, CBMS, Providence, RI.
Kenig, C. and Pipher, J. (1993). The Neumann problem for elliptic equations with non-smooth coefficients.Inventiones Math. 113, 447–509.
Kellogg, O. D. (1954).Foundation of Potential Theory. Dover, New York.
Kirsch, A. and Kress, R. (1993). Uniqueness in inverse obstacle scattering.Inverse Problems 9, 285–299.
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.
Král, J. (1980).Integral Operators in Potential Theory.Lecture Notes in Math. 823. Springer-Verlag, Berlin, Heidelberg, and New York.
Lax, P. and Phillips, R. (1967).Scattering Theory. Academic Press, New York.
Leis, R. (1986).Initial Boundary Value Problems in Mathematical Physics. Wiley, Chichester.
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.
Hofmann, S. and Mitrea, M. In preparation.
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.
Lions, J.-L. and Magenes, E. (1972).Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York, Heidelberg, and Berlin.
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.
McIntosh, A. and Mitrea, M. (1995). A Clifford algebra approach to elliptic and parabolic boundary value problems in nonsmooth domains, (preprint).
Fabes, E., Mendez, O. and Mitrea, M. (1996), in preparation.
Mitrea, D. (1995). Thesis. University of Minnesota.
Mitrea, M. (1994).Clifford Wavelets, Singular Integrals, and Hardy Spaces.Lecture Notes in Math. 1575. Springer-Verlag, New York.
Mitrea, M. (1994). Electromagnetic scattering on nonsmooth domains.Math. Res. Lett. 1, 639–646.
Mitrea, M. (1995). The method of layer potentials in electro-magnetic scattering theory on non-smooth domains.Duke Math. J. 77, 111–133.
Mitrea, M. (1995). Initial boundary value problems for the parabolic Maxwell system in Lipschitz cylinders.Indiana Math. J. 44, No. 3, 797–813.
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.
Müller, C. (1951). Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern.Math. Ann. 123, 345–378.
Müller, C. (1969).Foundations of the Mathematical Theory of Electromagnetic Waves. Springer-Verlag, New York.
Müller, C. and Niemeyer, H. (1961). Greensche Tensoren und asymptotische Gesetze der elektromagnetischen Hohlraumschwingungen.Arch. Rational Mech. Anal. 7, 305–348.
Nečas, J. (1967).Les méthodes directes en théorie des équations élliptique. Academia, Prague.
Payne, L. and Weinberger, H. (1954). New bounds in harmonic and biharmonic problems.J. Math. Phys. 33, 291–307.
Picard, R. (1984). An elementary proof for a compact embedding result in generalized electromagnetic theory.Math. Z. 187, 151–164.
Pipher, J. and Verchota, G. (1992). The Dirichlet problem inL p for biharmonic functions on Lipschitz domains.Amer. J. Math. 114, 923–972.
Pipher, J. and Verchota, G. (1995). Dilation invariant estimates and the boundary Garding inequality.Ann. of Math. (to appear).
Radon, J. Über die Randwertaufgaben beim logarithmischen potential.Sitzungsber. Akad. Wiss. 128, 1123–1167.
Ramm, A. (1986).Scattering by Obstacles. D. Reidel, Dordrecht.
Reed, M. and Simon, B. (1979).Methods of Modern Mathematical Physics. Vol. III Scattering Theory. Academic Press, New York.
Rellich, F. (1940). Darstellung der eigenwerte von Δu + λu durch ein Radintegral.Math. Z. 46, 635–646.
Safarov, Yu. G. (1984). On the asymptotic behavior of the spectrum of the Maxwell operator.J. Soviet Math. 27, 2655–2661.
Saranen, J. (1982). On an inequality of Friedrichs.Math. Scand. 51, 310–322.
Saunders, W. K. (1952). On solutions of Maxwell’s equations in an exterior region.Proc. Nat. Acad. Sci. U.S.A. 38, 342–348.
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.
Shen, Z. (1995). A note on the Dirichlet problem for the Stokes system in Lipschitz domains.Proc. Amer. Math. Soc. 123, 801–811.
Spencer, R. S. (1994).Series Solutions and Spectral Properties of Boundary Integral Equations. Thesis. University of Minnesota.
Stein, E.M. (1970).Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ.
Stephan, E. P. (1986). Boundary integral equations for magnetic screens inR 3.Proc. Roy. Soc. Edinburgh Sect. A 102, 188–210.
Taylor, M. E. (1996).Partial Differential Equations. Springer-Verlag, New York.
Temam, R. (1977).Theory and Numerical Analysis of the Navier-Stokes Equations. North-Holland, Amsterdam.
Uhlmann, G. (1992). Inverse boundary value problems and applications.Astérisque 207. Soc. Math. France, Montrouge.
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.
Varopoulos, N. (1977). A remark on functions of bounded mean oscillations and bounded harmonic functions.Pacific J. Math. 74, 257–259.
Verchota, G. (1982).Layer Potentials and Boundary Value Problems for Laplace’s Equation on Lipschitz Domains. Thesis. University of Minnesota.
Verchota, G. (1984). Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains.J. Funct. Anal. 59, 572–611.
Weber, C. (1980). A local compactness theorem for Maxwell’s equations.Math. Meth. Appl. Sci. 2, 12–25.
Weck, N. (1974). Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries.J. Math. Anal. Appl. 46, 410–437.
Weyl, H. (1911). Über die asymptotische verteilung der Eigenwerte.Gott. Nach. 110–117.
Weyl, H. (1912). Über das Spectrum der Hohlraumstrahlung.J. Reine Angew. Math. 141, 163–181.
Weyl, H. (1952). Kapazität von Strahlungsfeldern.Math. Z. 55, 187–198.
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02649132