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Advances in Applied Clifford Algebras

, Volume 27, Issue 3, pp 2493–2508 | Cite as

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of \({\mathbb {R}}^4\)

  • Min Ku
  • Ying Wang
  • Fuli He
  • Uwe Kähler
Article

Abstract

In this paper we are interested in finding solutions to Riemann–Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann–Hilbert problems for axially monogenic functions defined over the upper half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations.

Keywords

Riemann–Hilbert problems Generalized Cauchy–Riemann equation Quaternion analysis 

Mathematics Subject Classification

Primary: 30E25 35Q15 31A25 31B20 Secondary: 31B10 35J56 35J58 

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References

  1. 1.
    Abreu, L.D., Feichtinger, H.G.: Function spaces of poly-analytic functions. In: Harmonic and Complex Analysis and its Applications. Trends in Mathematics, pp. 1–38. Springer International Publishing (2014)Google Scholar
  2. 2.
    Abreu Blaya, R., Bory Reyes, J., Peña-Peña, D.: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. 17(1), 1–13 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akel, M., Begehr, H.: Schwarz problem for first order elliptic systems in unbounded sectors. Eurasian Math. J. 5(4), 6–24 (2014)MathSciNetGoogle Scholar
  4. 4.
    Balk, M.B.: On Poly-analytic Functions. Akademie Verlag, Berlin (1991)Google Scholar
  5. 5.
    Begehr, H.: Complex Analytic Methods for Partial Differential Equation: An Introductory Text. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Begehr, H., Vaitekhovich, T.: Harmonic boundary value problems in the half disc and half ring. Funct. Approx. 40(2), 251–282 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, London (1982)Google Scholar
  8. 8.
    Cerejeiras, P., Kähler, U., Ku, M.: On the Riemann boundary value problem for null solutions to iterated generalised Cauchy-Riemann operator in Clifford analysis. Results Math. 63(3–4), 1375–1394 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Inventiones Mathematicae 189(3), 515–580 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Colombo, F., Sabadini, I., Sommen, F.: The Fueter mapping theorem in integral form and the \(\cal{F}\)-functional calculus. Math. Methods Appl. Sci. 33, 2050–2066 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Common, A.K., Sommen, F.: Axial monogenic functions from holomorphic functions. J. Math. Anal. Appl. 179(2), 610–629 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  13. 13.
    Delanghe, R., Sommen, F., Soucěk, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic, Dordrecht (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fokas, A.S.: A Unified Approach to Boundary Value Problems. University of Cambridge, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta u=0\) und \(\Delta \Delta u=0\) mit vier reellen Variablen. Commentarii Mathematici Helvetici 7, 307–330 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    He, F., Ku, M., Dang, P., Kähler, U.: Riemann-Hilbert problems for poly-Hardy space on the unit ball. Complex Var. Elliptic Equ. 61(6), 772–790 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gakhov, F.D.: Boundary Value Problems. Pergamon, Oxford (1966)zbMATHGoogle Scholar
  18. 18.
    Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Basel (1990)Google Scholar
  19. 19.
    Gürlebeck, K., Zhongxiang, Z.: Some Riemann boundary value problems in Clifford analysis. Math. Methods Appl. Sci. 33, 287–302 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    He, F., Ku, M., Kähler, U., Sommen, F., Bernstein, S.: Riemann–Hilbert problems for monogenic functions in axially symmetric domains. Bound. Value Probl. 22, 1–11 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    He, F., Ku, M., Kähler, U., Sommen, F., Bernstein, S.: Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equations in axially symmetric domains. Comput. Math. Appl. 71(10), 1900–2000 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lu, J.: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (1993)zbMATHGoogle Scholar
  23. 23.
    Ku, M., Wang, D.: Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis. J. Math. Anal. Appl. 374(2), 442–457 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ku, M., Wang, D.: Solutions to polynomial Dirac equations on unbounded domains in Clifford analysis. Math. Methods Appl. Sci. 34, 418–427 (2011)Google Scholar
  25. 25.
    Ku, M., Du, J.: On integral representation of spherical \(k\)-regular functions in Clifford analysis. Adv. Appl. Clifford Algebras 19(1), 83–100 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ku, M., Kähler, U.: Riemann boundary value problems on half space in Clifford analysis. Math. Methods Appl. Sci. 35(18), 2141–2156 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ku, M.: Riemann boundary value problems on the sphere in Clifford analysis. Adv. Appl. Clifford Algebras 22(2), 365–390 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ku, M., Wang, D., Dong, L.: Solutions to polynomial generalised Bers-Vekua equations in Clifford analysis. Complex Anal. Oper. Theory 6, 407–424 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ku, M., Fu, Y., Kähler, U., Cerejeiras, P.: Riemann boundary value problems for iterated Dirac operator on the ball in Clifford analysis. Complex Anal. Oper. Theory 7(3), 673–693 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Leyden (1977)CrossRefzbMATHGoogle Scholar
  31. 31.
    Sommen, F.: On a generalization of Fueter’s theorem. Zeitschrift für Analysis und ihre Anwendungen 19, 899–902 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kähler, U., Ku, M., Qian, T.: Schwarz problems for poly-Hardy space on the unit ball. Results Math. (2016). doi: 10.1007/s00025-016-0575-2
  33. 33.
    Gong, Y., Du, J.: A kind of Riemann and Hilbert boundary value problem for left monogenic functions in \({\mathbb{R}}^{m}(m\ge 2)\). Complex Var. 49(5), 303–318 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Y., Du, J.: Mixed boundary value problems with a shift for a pair of meta-analytic and analytic functions. J. Math. Anal. Appl. 369(2), 510–524 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bu, Y., Du, J.: The RH boundary value problem for the \(k\)-monogenic functions. J. Math. Anal. Appl. 347, 633–644 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, Y.: Schwarz-type boundary value problems for the poly-analytic equation in the half unit disc. Complex Var. Elliptic Equ. 57(9), 983–993 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.CIDMA, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.School of Statistics and MathematicsZhongnan University of Economics and LawWuhanChina
  3. 3.School of Mathematics and StatisticsCentral South UniversityChangshaChina

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