Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell’s and Helmholtz’s equations
HTML articles powered by AMS MathViewer
- by Emilio Marmolejo-Olea, Irina Mitrea, Marius Mitrea and Qiang Shi PDF
- Trans. Amer. Math. Soc. 364 (2012), 4369-4424 Request permission
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.References
- M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
- M. S. Agranovich, Elliptic boundary problems, Partial differential equations, IX, Encyclopaedia Math. Sci., vol. 79, Springer, Berlin, 1997, pp. 1–144, 275–281. Translated from the Russian by the author. MR 1481215, DOI 10.1007/978-3-662-06721-5_{1}
- T. S. Angell and A. Kirsch, The conductive boundary condition for Maxwell’s equations, SIAM J. Appl. Math. 52 (1992), no. 6, 1597–1610. MR 1191352, DOI 10.1137/0152092
- 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 64639, DOI 10.1512/iumj.1954.3.53035
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
- Russell M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339–379. MR 987761, DOI 10.2307/2374513
- A. P. Calderón, The multipole expansion of radiation fields, J. Rational Mech. Anal. 3 (1954), 523–537. MR 63540, DOI 10.1512/iumj.1954.3.53026
- A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 466568, DOI 10.1073/pnas.74.4.1324
- A.-P. Calderón, Commutators, singular integrals on Lipschitz curves and applications, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 85–96. MR 562599
- David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
- David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1992. MR 1183732, DOI 10.1007/978-3-662-02835-3
- R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387 (French). MR 672839, DOI 10.2307/2007065
- Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
- Björn E. J. Dahlberg and Carlos E. 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. MR 890159, DOI 10.2307/1971407
- 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, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 621–634. MR 1044810
- 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, DOI 10.1215/S0012-7094-88-05735-3
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 4, Springer-Verlag, Berlin, 1990. Integral equations and numerical methods; With the collaboration of Michel Artola, Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard and Bruno Scheurer; Translated from the French by John C. Amson. MR 1081946
- R. Duduchava, D. Mitrea and D. Mitrea, Differential operators and boundary value problems on surfaces, Mathematische Nachrichten, 9-10 (2006), 996–1023.
- L. Escauriaza, E. B. 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. MR 1092919, DOI 10.1090/S0002-9939-1992-1092919-1
- Luis Escauriaza and Marius Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains, J. Funct. Anal. 216 (2004), no. 1, 141–171. MR 2091359, DOI 10.1016/j.jfa.2003.12.005
- Luis Escauriaza and Jin Keun Seo, Regularity properties of solutions to transmission problems, Trans. Amer. Math. Soc. 338 (1993), no. 1, 405–430. MR 1149120, DOI 10.1090/S0002-9947-1993-1149120-8
- E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), no. 3-4, 165–186. MR 501367, DOI 10.1007/BF02545747
- 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, DOI 10.1215/S0012-7094-88-05734-1
- Thomas Gerlach and Rainer Kress, Uniqueness in inverse obstacle scattering with conductive boundary condition, Inverse Problems 12 (1996), no. 5, 619–625. MR 1413422, DOI 10.1088/0266-5611/12/5/006
- Fritz Gesztesy and Marius Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 105–173. MR 2500491, DOI 10.1090/pspum/079/2500491
- John E. Gilbert and Margaret A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, vol. 26, Cambridge University Press, Cambridge, 1991. MR 1130821, DOI 10.1017/CBO9780511611582
- Steve Hofmann, Emilio Marmolejo-Olea, Marius Mitrea, Salvador Pérez-Esteva, and Michael Taylor, Hardy spaces, singular integrals and the geometry of Euclidean domains of locally finite perimeter, Geom. Funct. Anal. 19 (2009), no. 3, 842–882. MR 2563770, DOI 10.1007/s00039-009-0015-5
- Steve Hofmann, Marius Mitrea, and Michael Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Not. IMRN 14 (2010), 2567–2865. MR 2669659, DOI 10.1093/imrn/rnp214
- Björn Jawerth and Marius Mitrea, Higher-dimensional electromagnetic scattering theory on $C^1$ and Lipschitz domains, Amer. J. Math. 117 (1995), no. 4, 929–963. MR 1342836, DOI 10.2307/2374954
- Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720, DOI 10.1090/cbms/083
- R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48 (1988), no. 2, 307–325. MR 933037, DOI 10.1137/0148016
- Andreas Kirsch and Lassi Päivärinta, On recovering obstacles inside inhomogeneities, Math. Methods Appl. Sci. 21 (1998), no. 7, 619–651. MR 1615992, DOI 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.3.CO;2-G
- V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991, DOI 10.1090/surv/085
- Serge Lang, Real and functional analysis, 3rd ed., Graduate Texts in Mathematics, vol. 142, Springer-Verlag, New York, 1993. MR 1216137, DOI 10.1007/978-1-4612-0897-6
- O. A. Ladyženskaja, V. Ja. Rivkind, and N. N. Ural′ceva, Solvability of diffraction problems in the classical sense, Trudy Mat. Inst. Steklov. 92 (1966), 116–146 (Russian). MR 0211050
- Emilio Marmolejo-Olea and Marius Mitrea, Harmonic analysis for general first order differential operators in Lipschitz domains, Clifford algebras (Cookeville, TN, 2002) Prog. Math. Phys., vol. 34, Birkhäuser Boston, Boston, MA, 2004, pp. 91–114. MR 2025974
- P. A. Martin and Petri Ola, Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 1, 185–208. MR 1204856, DOI 10.1017/S0308210500021296
- 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.
- Alan McIntosh and Marius Mitrea, Clifford algebras and Maxwell’s equations in Lipschitz domains, Math. Methods Appl. Sci. 22 (1999), no. 18, 1599–1620. MR 1727215, DOI 10.1002/(SICI)1099-1476(199912)22:18<1599::AID-MMA95>3.3.CO;2-D
- Dorina Mitrea and Marius Mitrea, Uniqueness for inverse conductivity and transmission problems in the class of Lipschitz domains, Comm. Partial Differential Equations 23 (1998), no. 7-8, 1419–1448. MR 1642603, DOI 10.1080/03605309808821388
- Dorina Mitrea, Marius Mitrea, and Jill Pipher, Vector potential theory on nonsmooth domains in $\textbf {R}^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl. 3 (1997), no. 2, 131–192. MR 1438894, DOI 10.1007/s00041-001-4053-0
- Dorina Mitrea, Marius Mitrea, and Qiang 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, DOI 10.1216/jiea/1181075395
- Dorina Mitrea, Marius Mitrea, and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc. 150 (2001), no. 713, x+120. MR 1809655, DOI 10.1090/memo/0713
- Marius Mitrea, Boundary value problems for Dirac operators and Maxwell’s equations in non-smooth domains, Math. Methods Appl. Sci. 25 (2002), no. 16-18, 1355–1369. Clifford analysis in applications. MR 1949501, DOI 10.1002/mma.375
- Marius Mitrea, Generalized Dirac operators on nonsmooth manifolds and Maxwell’s equations, J. Fourier Anal. Appl. 7 (2001), no. 3, 207–256. MR 1835281, DOI 10.1007/BF02511812
- Marius Mitrea, The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J. 77 (1995), no. 1, 111–133. MR 1317629, DOI 10.1215/S0012-7094-95-07705-9
- Marius Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Mathematics, vol. 1575, Springer-Verlag, Berlin, 1994. MR 1295843, DOI 10.1007/BFb0073556
- 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.
- Claus Müller, Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern, Math. Ann. 123 (1951), 345–378 (German). MR 45030, DOI 10.1007/BF02054960
- Claus Müller, Foundations of the mathematical theory of electromagnetic waves, Die Grundlehren der mathematischen Wissenschaften, Band 155, Springer-Verlag, New York-Heidelberg, 1969. Revised and enlarged translation from the German. MR 0253638
- Serge Nicaise and Anna-Margarete Sändig, Transmission problems for the Laplace and elasticity operators: regularity and boundary integral formulation, Math. Models Methods Appl. Sci. 9 (1999), no. 6, 855–898. MR 1702865, DOI 10.1142/S0218202599000403
- Stephan Rempel, Corner singularity for transmission problems in three dimensions, Integral Equations Operator Theory 12 (1989), no. 6, 835–854. MR 1018215, DOI 10.1007/BF01196880
- S. Rempel, Elliptic pseudodifferential operators on manifolds with corners and edges, Function spaces, differential operators and nonlinear analysis (Sodankylä, 1988) Pitman Res. Notes Math. Ser., vol. 211, Longman Sci. Tech., Harlow, 1989, pp. 202–211. MR 1041119
- Martin Reissel, On a transmission boundary value problem for the time-harmonic Maxwell equations without displacement currents, SIAM J. Math. Anal. 24 (1993), no. 6, 1440–1457. MR 1241153, DOI 10.1137/0524083
- William K. Saunders, On solutions of Maxwell’s equations in an exterior region, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 342–348. MR 53763, DOI 10.1073/pnas.38.4.342
- Jin Keun Seo, Regularity for solutions of transmission problems across internal non-smooth boundary, Proceedings of Miniconference of Partial Differential Equations and Applications (Seoul, 1995) Lecture Notes Ser., vol. 38, Seoul Nat. Univ., Seoul, 1997, pp. 189–199. MR 1449919
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Zhong Wei 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), no. 2, 293–373. MR 1099449, DOI 10.2307/2374910
- Gregory 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. MR 769382, DOI 10.1016/0022-1236(84)90066-1
- Hermann Weyl, Kapazität von Strahlungsfeldern, Math. Z. 55 (1952), 187–198 (German). MR 49782, DOI 10.1007/BF01268654
- Hermann Weyl, Die natürlichen Randwertaufgaben im Aussenraum für Strahlungsfelder beliebiger Dimension und beliebigen Ranges, Math. Z. 56 (1952), 105–119 (German). MR 54524, DOI 10.1007/BF01175027
- Peter Wilde, Transmission problems for the vector Helmholtz equation, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 61–76. MR 890043, DOI 10.1017/S0308210500021910
Similar Articles
- Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30G35, 35C15, 35F15, 35J56, 42B20, 42B30, 42B37, 30E20, 31B10, 35F45, 35J25, 45B05, 65N80
- Retrieve articles in all journals with MSC (2010): 30G35, 35C15, 35F15, 35J56, 42B20, 42B30, 42B37, 30E20, 31B10, 35F45, 35J25, 45B05, 65N80
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
- Email: emilio@matcuer.unam.mx
- Irina Mitrea
- Affiliation: Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, Pennsylvania 19122
- MR Author ID: 634131
- Email: imitrea@temple.edu
- Marius Mitrea
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 341602
- ORCID: 0000-0002-5195-5953
- Email: mitream@missouri.edu
- Qiang Shi
- Affiliation: Department of Mathematics, Computer Science and Economics, Emporia State University, Emporia, Kansas 66801
- Email: qshi@emporia.edu
- Received by editor(s): March 13, 2010
- Received by editor(s) in revised form: April 15, 2011
- Published electronically: March 29, 2012
- © Copyright 2012 American Mathematical Society
- 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
- DOI: https://doi.org/10.1090/S0002-9947-2012-05606-6
- MathSciNet review: 2912458