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Compound Riemann Hilbert Boundary Value Problems in Complex and Quaternionic Analysis

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Abstract

The aim of this paper is the study of a class of compound boundary value problems for the homogeneous Dirac equation in two and three dimensions where one of the two boundary conditions (linear conjugation) is loaded. It is shown how the lack of commutativity inherent in the quaternionic product, paradoxically relaxes the conditions to guarantee the solvability of considered problems. Some examples illustrating the results are presented.

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References

  1. Abreu Blaya R., Bory Reyes J.: On a reduction procedure to solve a loaded Riemann Hilbert boundary value problem. Cienc. Mat. (Havana) 16(1), 35–41 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Abreu Blaya R., Bory Reyes J.: Boundary value problems for quaternionic monogenic functions on non-smooth surfaces. Adv. Appl. Clifford Algebras 9(1), 1–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abreu Blaya R., Bory Reyes J.: On the Riemann Hilbert type problems in Clifford analysis. Adv. Appl. Clifford Algebras 11(1), 15–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abreu Blaya R., Peña Peña D., Bory Reyes J.: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 17(1), 1–14 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernstein, S.: Riemann–Hilbert problems in Clifford analysis. Clifford analysis and its applications (Prague, 2000), vol. 18, NATO Sci. Ser. II Math. Phys. Chem., vol. 25. Kluwer Acad. Publ., Dordrecht (2001)

  6. Bernstein, S.: The quaternionic Riemann problem. Function spaces (Edwardsville, IL, 1998), vol. 6983, Contemp. Math., vol. 232. Amer. Math. Soc., Providence (1999)

  7. Bory Reyes J., Abreu Blaya R.: The quaternionic Riemann problem with a natural geometric condition on the boundary. Complex Variables Theory Appl. 42(2), 135–149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bory Reyes, J., Abreu Blaya, R.: On the Cauchy type integral and the Riemann problem. Clifford algebras and their applications in mathematical physics, vol. 2 (Ixtapa, 1999), pp. 81–94, Progr. Phys., vol. 19. Birkhauser, Boston (2000)

  9. Gajov, F.D.: Boundary Value Problems, 3rd edn. Nauka, Moskow, English transl. of 2nd edn. Pergamon Press, Oxford; Addison-Wesley, Reading (1966)

  10. Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäusser, Boston (1990)

  11. Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)

  12. He, F., Ku, M., Kähler, U., Sommen, F., Bernstein, S.: Riemann–Hilbert problems for monogenic functions in axially symmetric domains. Bound. Value Probl. 2016, 22 (2016)

  13. Kravchenko, V.V.: Applied Quaternionic Analysis. Heldemann Verlag, Berlin (2003)

  14. Lu, J.K.: Boundary Value Problems for Analytic Functions. World Scientific Publishing, Singapore (1993)

  15. Oboloshvili, E.: Effective solution of some boundary value problems in two and three dimensional cases. In: Proceeding of Functional Analytic Methods in Complex Analysis and Applications to Partial Diff Equations. World Scientific, Singapore, pp. 149–172 (1988)

  16. Oboloshvili, E.: Boundary and initial problems in clifford analysis. In: Sprössig, W., et al. (eds.) Proceedings of the Symposium on Analytical and Numerical Methods in Quaternionic and Clifford Analysis, June 5–7, 1996, Seiffen, pp. 145–152. TU Bergakademie Freiberg, Freiberg (1996)

  17. Ryan J.: Clifford Algebras in Analysis and Related Topics. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  18. Mushelisvili, N.I.: Singular Integral Equations, Nauka, Moskow (1968) [English transl. of 1st edn., Noodhoff, Groningen (1953)]

  19. Selim M.S.: A nonlinear loaded Hilbert problem. Azerbaidzhan. Gos. Univ. Uchen. Zap. 1, 100–104 (1978)

    MathSciNet  Google Scholar 

  20. Selim M.S.: A nonlinear loaded boundary value problem of Hilbert’s type. Azerbaidzhan. Gos. Univ. Uchen. Zap. 6, 84–89 (1978)

    MathSciNet  Google Scholar 

  21. Shapiro M.V., Vasilevski N.L.: Quaternionic \({\psi}\)-hyperholomorphic functions, singular integral operators and boundary value problems. I. \({\psi}\)-hyperholomorphic function theory. Complex Variables Theory Appl. 27(1), 17–46 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto Politécnico Nacional SEPI-ESIME-ZAC, Mexico, D.F., 07738, Mexico

    Juan Bory Reyes

  2. Facultad de Informática y Matemática, Universidad de Holguín, Holguín, 80100, Cuba

    Carlos Daniel Tamayo Castro & Ricardo Abreu Blaya

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  1. Juan Bory Reyes
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  2. Carlos Daniel Tamayo Castro
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  3. Ricardo Abreu Blaya
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Correspondence to Juan Bory Reyes.

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Bory Reyes, J., Tamayo Castro, C.D. & Blaya, R.A. Compound Riemann Hilbert Boundary Value Problems in Complex and Quaternionic Analysis. Adv. Appl. Clifford Algebras 27, 977–991 (2017). https://doi.org/10.1007/s00006-016-0710-x

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  • DOI: https://doi.org/10.1007/s00006-016-0710-x

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