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Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell's and Helmholtz's equations

Authors: Emilio Marmolejo-Olea, Irina Mitrea, Marius Mitrea and Qiang Shi
Journal: Trans. Amer. Math. Soc. 364 (2012), 4369-4424
MSC (2010): Primary 30G35, 35C15, 35F15, 35J56, 42B20, 42B30, 42B37; Secondary 30E20, 31B10, 35F45, 35J25, 45B05, 65N80
Published electronically: March 29, 2012
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Abstract: The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in $ \mathbb{R}^3$ is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.

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  • [AS72] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
  • [Ag97] M.S.Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79, Partial differential equations, IX, pp.1-144 and pp.275-281, Springer, Berlin, 1997. MR 1481215 (99a:35056)
  • [AK92] T.Angell and A.Kirsch, The conductive boundary condition for Maxwell's equations, SIAM J. Appl. Math., 52 (1992), 1597-1610. MR 1191352 (93k:78009)
  • [BD54] R.B.Barrar and C.L.Dolph, On a three dimensional transmission problem of electromagnetic theory, J. Rational Mech. Anal., 3 (1954), 725-743. MR 0064639 (16:313f)
  • [BDS82] F.Brackx, R.Delanghe and F.Sommen, Clifford Analysis, Research Notes in Mathematics, Vol.76, Pitman, Boston, MA, 1982. MR 697564 (85j:30103)
  • [Br89] R.Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math., 111 (1989), 339-379. MR 987761 (90d:35118)
  • [Ca54] A.Calderón, The multipole expansion of radiation fields, J. Rat. Mech. Anal., 3 (1954), 523-537. MR 0063540 (16:136a)
  • [Ca77] A.P.Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA., 74 (1977), no. 4, 1324-1327. MR 0466568 (57:6445)
  • [Ca80] A.P.Calderón, Commutators, singular integrals on Lipschitz curves and applications, pp. 85-96 in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980. MR 562599 (82f:42016)
  • [CK83] D.Colton and R.Kress, Integral Equation Methods in Scattering Theory, Wiley Interscience Publications, New York, 1983. MR 700400 (85d:35001)
  • [CK92] D.Colton and R.Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Applied Math. Series, No.93, 1992. MR 1183732 (93j:35124)
  • [CMM82] R.Coifman, A.McIntosh, and Y.Meyer, L'intégrale de Cauchy definit un opérateur borné sur $ L^2$ pour les courbes Lipschitziennes, Annals of Math., 116 (1982), 361-388. MR 672839 (84m:42027)
  • [Da77] J. E.Dahlberg, On estimates of harmonic measure, Arch. Rat. Mech. Anal., 65 (1977), 275-288. MR 0466593 (57:6470)
  • [DK87] B.Dahlberg and C.Kenig, Hardy spaces and the $ L^p$-Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math., 125 (1987), 437-465. MR 890159 (88d:35044)
  • [DK90] B.E.J.Dahlberg and C.E.Kenig, $ L^p$ estimates for the three-dimensional systems of elastostatics on Lipschitz domains, Analysis and Partial Differential Equations, pp.621-634, Lecture Notes in Pure and Appl. Math., Vol.122, Dekker, New York, 1990. MR 1044810 (91h:35053)
  • [DKV88] B.E.J.Dahlberg, C.E.Kenig and G.C.Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57 (1988), no. 3, 795-818. MR 975122 (90d:35259)
  • [DL90] R.Dautray and J.-L.Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol.4, Springer-Verlag, 1990. MR 1081946 (91h:00004b)
  • [DMM06] R.Duduchava, D.Mitrea and D.Mitrea, Differential operators and boundary value problems on surfaces, Mathematische Nachrichten, 9-10 (2006), 996-1023.
  • [EFV92] 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), 1069-1076. MR 1092919 (92j:35020)
  • [EM04] L.Escauriaza and M.Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains, J. Funct. Anal., 216 (2004), no. 1, 141-171. MR 2091359 (2005f:35065)
  • [ES93] L.Escauriaza and J.K.Seo, Regularity properties of solutions to transmission problems, Trans. Amer. Math. Soc., 338 (1993), 405-430. MR 1149120 (93j:35039)
  • [FJR78] E.Fabes, M.Jodeit and N.Rivière, Potential techniques for boundary value problems on $ C^1$ domains, Acta Math., 141 (1978), 165-186. MR 501367 (80b:31006)
  • [FKV88] E.B.Fabes, C.E.Kenig and G.C.Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), no. 3, 769-793. MR 975121 (90d:35258)
  • [GK96] T.Gerlach and R.Kress, Uniqueness in inverse obstacle scattering with conductive boundary condition, Inverse Problems, 12 (1996), 619-625. MR 1413422 (97i:35184a)
  • [GeMi08] F.Gesztesy and M.Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, pp.105-173 in the Proceedings of Symposia in Pure Mathematics, Vol.79, Amer. Math. Soc., 2008. MR 2500491 (2010k:35087)
  • [GM91] J.Gilbert and M.A.Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 26, Cambridge Univ. Press, Cambridge 1991. MR 1130821 (93c:42027)
  • [H-T09] S.Hofmann, E.Marmolejo-Olea, M.Mitrea, S.Perez-Esteva, and M.Taylor, Hardy spaces, singular integrals and the geometry of Euclidean domains of locally finite perimeter, Journal Geometric and Functional Analysis, 19, (2009), no. 3, 842-882. MR 2563770 (2011a:42021)
  • [HMT10] S.Hofmann, M.Mitrea, and M.Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, International Mathematics Research Notices, Oxford University Press, 2010 (14), 2567-2865. MR 2669659
  • [JM95] B.Jawerth and M.Mitrea, Higher dimensional scattering theory on $ C^1$ and Lipschitz domains, Amer. J. of Math., 117 (1995), 929-963. MR 1342836 (96h:35143)
  • [Ke94] C.E.Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Mathematics, No. 83, AMS, Providence, RI, 1994. MR 1282720 (96a:35040)
  • [KM88] R.E.Kleinman and P.A.Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math., 48 (1988), 307-325. MR 933037 (89f:35053)
  • [KP98] A.Kirsch and L.Päivärinta, On recovering obstacles inside inhomogeneities, Math. Methods Appl. Sci., 21 (1998), 619-651. MR 1615992 (99b:35214)
  • [KMR01] V.A.Kozlov, V.G.Maz'ya and J.Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Amer. Math. Soc., Providence, RI, 2001. MR 1788991 (2001i:35069)
  • [La93] S.Lang, Real and Function Analysis, third edition, Springer-Verlag, 1993. MR 1216137 (94b:00005)
  • [LRU66] O.A.Ladyzenskaja, V.J.Rivkind and N.N.Ural'ceva, The classical solvability of diffraction problems, Proc. Steklov Inst. Math., 92 (1966), 132-166. MR 0211050 (35:1932)
  • [MM03] E.Marmolejo-Olea and M.Mitrea, Harmonic analysis for general first order differential operators in Lipschitz domains, pp.91-114 in ``Clifford Algebras: Application to Mathematics, Physics, and Engineering'', Birkhäuser Progress in Mathematical Physics Series, 2003. MR 2025974 (2004k:58033)
  • [MO93] P.A.Martin and P.Ola, Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle, Proc. of the Royal Soc. of Edinburgh, 123 (1993), 185-208. MR 1204856 (94c:78008)
  • [McMM97] A.McIntosh, D.Mitrea and M. Mitrea, Rellich type identities for one-sided monogenic functions in Lipschitz domains and applications, pp.135-143 in Proceedings of the Symposium ``Analytical and Numerical Methods in Quaternions and Clifford Analysis'', Seiffen, W.Sprössig and K. Gürlebeck eds., Technical University of Freiberg, 1997.
  • [McM99] A.McIntosh and M. Mitrea, Clifford algebras and Maxwell's equations in Lipschitz domains, Math. Meth. Appl. Sci., 22 (1999), 1599-1620. MR 1727215 (2001c:30046)
  • [MM98] D.Mitrea and M.Mitrea, Uniqueness for inverse conductivity and transmission problems in the class of Lipschitz domains, Comm. Partial Differential Equations, 23 (1998), 1419-1448. MR 1642603 (99f:35225)
  • [MMP97] D.Mitrea, M.Mitrea and J.Pipher, Vector potential theory on non-smooth domains in $ {\mathbb{R}}^3$ and applications to electromagnetic scattering, J. Fourier Anal. and Appl., 3 (1997), no. 2, 131-192. MR 1438894 (99e:31009)
  • [MMS06] D.Mitrea, M.Mitrea and Q. Shi, Variable coefficient transmission problems and singular integral operators on non-smooth manifolds, J. Integral Equations Appl., 18 (2006), no. 3, 361-397. MR 2269727 (2007k:35095)
  • [MMT01] D.Mitrea, M.Mitrea and M.Taylor, Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds, Mem. Amer. Math. Soc., Vol.150, No.713, 2001. MR 1809655 (2002g:58026)
  • [Mi02] M.Mitrea, Boundary value problems for Dirac operators and Maxwell's equations in nonsmooth domains, Mathematical Methods in the Applied Sciences, 25 (2002), no.16-18, 1355-1369. MR 1949501 (2003k:35198)
  • [Mi01] M.Mitrea, Generalized Dirac operators on non-smooth manifolds and Maxwell's equations, Journal of Fourier Analysis and Applications, 7 (2001), no.3, 207-256. MR 1835281 (2002k:58046)
  • [Mi95] M.Mitrea, The method of layer potentials in electro-magnetic scattering theory on non-smooth domains, Duke Math. J., 77 (1995), no. 1, 111-133. MR 1317629 (96b:78035)
  • [Mi94] M.Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics, No.1575, Springer-Verlag, Berlin, Heidelberg, New York, 1994. MR 1295843 (96e:31005)
  • [MW10] M.Mitrea and M.Wright, Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains, to appear in Astérisque, Societé Mathématique de France, 2011.
  • [Mu51] C.Müller, Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern, Math. Ann., 123 (1951), 345-378. MR 0045030 (13:514c)
  • [Mu69] C.Müller, Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, Berlin, Heidelberg, New York, 1969. MR 0253638 (40:6852)
  • [NS99] S.Nicaise and A.-M.Sändig, Transmission problems for the Laplace and elasticity operators: Regularity and boundary integral formulation, Math. Models Methods Appl. Sci., 9 (1999), 855-898. MR 1702865 (2000i:35022)
  • [Re89a] S.Rempel, Corner singularity for transmission problems in three dimensions, Integral Equations Operator Theory, 12 (1989), 835-854. MR 1018215 (91f:35082)
  • [Re89b] S.Rempel, Elliptic pseudodifferential operators on manifolds with corners and edges, pp.202-211, in ``Function Spaces, Differential Operators and Nonlinear Analysis,'' Pitman Res. Notes Math. Ser., Vol.211, Longman Sci. Tech., Harlow, 1989. MR 1041119 (91b:58253)
  • [Rei93] M.Reissel, On a transmission boundary value problem for the time-harmonic Maxwell equations without displacement currents, SIAM J. Math. Anal., 24 (1993), 1440-1457. MR 1241153 (94i:35186)
  • [Sa52] W.K.Saunders, On solutions of Maxwell equations in an exterior region, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 342-348. MR 0053763 (14:823a)
  • [Seo97] J.K.Seo, Regularity for solutions of transmission problems across internal non-smooth boundary, pp.189-199 in Proceedings of Miniconference of Partial Differential Equations and Applications, (Seoul, 1995), Lecture Notes Ser., 38, Seoul Nat. Univ., Seoul, 1997. MR 1449919 (98d:35043)
  • [St70] E.M.Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No.30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [Sh91] Z.Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, Amer. J. Math., 113 (1991), 293-373. MR 1099449 (92a:35133)
  • [Ve84] G. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. MR 769382 (86e:35038)
  • [We52a] H.Weyl, Kapazität von Strahlungsfeldern, Math. Zeit., 55 (1952), 187-198. MR 0049782 (14:225c)
  • [We52b] H.Weyl, Die natürlichen Randwertaufgaben im Aussenraum für Strahlungsfeldern beliebiger Dimensionen und beliebiger Ranges, Math. Zeit., 56 (1952), 105-119. MR 0054524 (14:933g)
  • [Wi87] P.Wilde, Transmission problems for the vector Helmholtz equation, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 61-76. MR 890043 (88g:35064)

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Additional Information

Emilio Marmolejo-Olea
Affiliation: Instituto de Matemáticas Unidad Cuernavaca, Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca, Morelos, 62251, México

Irina Mitrea
Affiliation: Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, Pennsylvania 19122

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Qiang Shi
Affiliation: Department of Mathematics, Computer Science and Economics, Emporia State University, Emporia, Kansas 66801

Keywords: Transmission boundary value problems, Lipschitz domains, Dirac operator, Maxwell system, Helmholtz operator, Hardy spaces, Cauchy operator, boundary layer potentials, Clifford algebras, Clifford analysis
Received by editor(s): March 13, 2010
Received by editor(s) in revised form: April 15, 2011
Published electronically: March 29, 2012
Article copyright: © Copyright 2012 American Mathematical Society

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