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Original Articles

Hilbert transform for the three-dimensional Vekua equation

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Pages 1797-1824 | Received 10 Mar 2018, Accepted 27 Nov 2018, Published online: 20 Dec 2018

References

  • Axelsson A, Kou KI, Qian T. Hilbert transforms and the Cauchy integral in Euclidean space. Studia Math. 2009;193:161–187. doi: 10.4064/sm193-2-4
  • Qian T. Hilbert transforms on the sphere and Lipschitz surfaces. In: Sabadini I, Shapiro M, Sommen F, editors. Hypercomplex analysis. trends in mathematics. Basel: Birkhäuser; 2008. p. 259–275. DOI:10.1007/978-3-7643-9893-4_16
  • Qian T, Yang Y. Hilbert transforms on the sphere with the Clifford algebra setting. J Fourier Anal Appl. 2009;15:753–774. DOI:10.1007/s00041-009-9062-4
  • Calderón AP. On an inverse boundary value problem. In: Seminar on numerical analysis and its applications to continuous physics; Río de Janeiro, 1980. Rio de Janeiro: Soc. Brasil. Mat.; 1980. p. 65–73.
  • Kenig C, Sjöstrand S, Uhlmann G. The Calderón problem with partial data. Ann Math. 2007;165(2):567–591. DOI: 10.5802/jedp.9 doi: 10.4007/annals.2007.165.567
  • Uhlmann G. Electrical impedance tomography and Calderón problem. Inverse Probl. 2009;25(12):123011. doi: 10.1088/0266-5611/25/12/123011
  • Holder D. Electrical impedance tomography. Bristol: Institute of Physics; 2005.
  • Brackx F, De Knock B, De Schepper H, et al. On the interplay between the Hilbert transform and conjugate harmonic functions. Math Methods Appl Sci. 2006;29:1435–1450. doi: 10.1002/mma.735
  • Brackx F, De Schepper H, Eelbode D. A new Hilbert transform on the unit sphere in Rm. Complex Var Elliptic Equ. 2006;51(5–6):453–462. doi: 10.1080/17476930500481574
  • Brackx F, De Schepper H. The Hilbert transform on the unit sphere in Rm. In: Sabadini I, Shapiro M, Sommen F, editors. Hypercomplex Analysis. Trends in Mathematics. Basel: Birkhäuser; 2008:11–36.
  • Delgado BB, Porter RM. General solution of the inhomogeneous div-curl system and consequences. Adv Appl Clifford Algebras. 2017;27:3015–3037. DOI:10.1007/s00006-017-0805-z
  • Gürlebeck K, Habetha K, Sprößig W. Holomorphic functions in the plane and n-dimensional space. Basel: Birkhäuser; 2008.
  • Adams RA, Fournier JJF. Sobolev spaces. New York: Academic Press; 1978.
  • Brezis H. Functional analysis, sobolev spaces and partial differential equations. New York: Springer; 2011. DOI:10.1007/978-0-387-70914-7
  • Grisvard P. Elliptical problems in nonsmooth domains. Boston (MA): Pitman (Advanced Publishing Program); 1985.
  • Chen G, Zhou J. Boundary element methods. New York: Academic Press; 1992.
  • McLean V. Strongly elliptic systems and boundary integral operators. 1st ed. Cambridge: Cambridge University Press; 2000.
  • Gürlebeck K, Sprößig W. Quaternionic analysis and elliptic boundary value problems. Berlin: Birkhäuser; 1990.
  • Gürlebeck K, Habetha K, Sprößig W. Application of holomorphic functions in two and higher dimensions. Basel: Birkhäuser; 2016.
  • Mikhlin SG, Prössdorf S. Singular integral operators. Berlin: Springer-Verlag; 1986.
  • Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory. Berlin: Springer-Verlag; 1992.
  • Knudsen K. On the inverse conductivity problem [Ph.D. thesis]. Aalborg University; 2002.
  • Dahlberg B, Kenig C. Hardy spaces and the Neumann problem in Lp for Laplace's equation in Lipschitz domains. Ann Math. 1987;125:437–465. doi: 10.2307/1971407
  • Kenig CE. Harmonic analysis techniques for second order elliptic boundary value problems. Rhode Island: American Mathematical Society; 1994. (CBMS regional conference series in mathematics; 83).
  • Verchota G. Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains. J Funct Anal. 1984;59:572–611. doi: 10.1016/0022-1236(84)90066-1
  • Kravchenko VV, Shapiro MV. Integral representations for spatial models of mathematical physics. Harlow: Addison Wesley Longman; 1996.
  • Fabes E, Jodeit M, Lewis J. Double layer potentials for domains with corners and edges. Indiana Univ Math J. 1977;26:95–114. doi: 10.1512/iumj.1977.26.26007
  • Kubrusly CS. Spectral theory of operators on Hilbert spaces. New York: Birkhäuser; 2012.
  • Behrndt J, ter Elst AFM. Dirichlet-to-Neumann maps on bounded Lipschitz domains. J Differ Equ. 2015;259:5903–5926. DOI:10.1016/j.jde.2015.07.012
  • Dautray R, Lions J-L. Mathematical analysis and numerical methods for science and technology. Vol. 3. New York: Springer-Verlag; 1985.
  • Girault V, Raviart PA. Finite element methods for the Navier-Stokes equations. Berlin: Springer-Verlag; 1986.
  • Amrouche C, Bernardi C, Dauge M, et al. Vector potentials in three-dimensional nonsmooth domains. Math Meth Appl Sci. 1998;21:823–864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
  • Kíek M, Neittaanmäki P. On the validity of Friedrichs' inequalities. Math Scand. 1984;54(1):17–26.
  • Saranen J. On an inequality of Friedrichs. Math Scand. 1982;51(2):310–322. doi: 10.7146/math.scand.a-11983
  • Weber Ch, Werner P. A local compactness theorem for Maxwell's equations. Math Meth Appl Sci. 1980;2:12–25. doi: 10.1002/mma.1670020103
  • Berger MS. IX Nonlinearity and functional analysis: lectures on nonlinear problems in mathematical physics. New York: Academic Press; 1977.
  • Bers L. Theory of pseudo-analytic functions. New York: New York University; 1953.
  • Vekua IN. Generalized analytic functions. Moscow: Nauka (in Russian); 1959. English translation. Oxford: Pergamon Press; 1962.
  • Berglez P. On generalized derivatives and formal powers for pseudoanalytic functions. Le Mat. 2007;62:29–36.
  • Malonek H. Generalizing the (F,G)-derivative in the sense of Bers. In: Dietrich V, et al., editors. Clifford algebras and their application in mathematical physics. Vol. 94. Dordrecht: Springer; 1998. p. 247–257.
  • Kravchenko VV. Applied pseudoanalytic function theory. Basel: Birkhäuser; 2009. (Frontiers in mathematics).
  • Isakov V. Inverse problems for partial differential equations. Berlin: Springer-Verlag; 1998.
  • Mikhailov VP. Partial differential equations. Moscow: Mir Publishers; 1978.
  • Evans LC. Partial differential equations. Providence (RI): American Mathematical Society; 1998.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. 2nd ed. Berlin: Springer-Verlag; 1983.
  • Salo M. Calderón problem. Inside out II, MSRI Publications 60, 2012. Available from http://users.jyu.fi/salomi/lecturenotes/calderon_lectures.pdf
  • Sylvester J, Uhlmann G. A global uniqueness theorem for an inverse boundary value problem. Ann Math. 1987;125:153–169. doi: 10.2307/1971291
  • Kravchenko VV, Tremblay S. Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators. Math Methods Appl Sci. 2011;34:1999–2010. DOI:10.1002/mma.1500
  • Astala K, Päivärinta L. A boundary integral equation for Calderón's inverse conductivity problem. In: Proceedings of the 7th International Conference on Harmonic Analysis. Collectanea Mathematica; 2006. p. 127–139.
  • Astala K, Päivärinta L. Calderón's inverse conductivity problem in the plane. Ann Math. 2006;163:265–299. doi: 10.4007/annals.2006.163.265
  • Sylvester J, Uhlmann G. Inverse boundary value problems at the boundary-continuous dependence. Comm Pure Appl Math. 1988;41:197–219. doi: 10.1002/cpa.3160410205
  • Alessandrini G. Stable determination of conductivity by boundary measurements. Appl Anal. 1988;27(1–3):153–172. DOI:10.1080/03605302.2015.1007379 doi: 10.1080/00036818808839730
  • Stein E, Weiss G. On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math. 1960;103:25–62. doi: 10.1007/BF02546524

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