Skip to main content
SpringerLink
Log in

Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in \({\mathbb {R}}^{n+1}\)

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems \(\big (\)for short RHBVPs\(\big )\) for axially monogenic functions on Euclidean space \({\mathbb {R}}^{n+1},n\in {\mathbb {N}}\). With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to \(\big ({\mathcal {D}}-\alpha \big )\phi =0,\alpha \in {\mathbb {R}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Abreu, L.D., Feichtinger, H.G.: Function spaces of poly-analytic functions, Harmonic and Complex Analysis and its Applications, Trends in Mathematics, pp. 1–38 (2014)

  2. Abreu Blaya, R., Bory Reyes, J., Peña-Peña, D.: Jump problem and removable singularities for monogenic functions. J. Geom. Anal. Theory Appl. 17(1), 1–13 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balk, M.B.: On Poly-analytic Functions. Akademie Verlag, Berlin (1991)

    Google Scholar 

  4. Begehr, H., Schmersau, D.: The Schwarz problem for poly-analytic functions. ZAA 24(2), 341–351 (2005)

    MATH  Google Scholar 

  5. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Res Notes Math, vol. 76. Pitman, London (1982)

    MATH  Google Scholar 

  6. Bu, Y.D., Du, J.Y.: The RH boundary value problem for the \(k\)-monogenic functions. J. Math. Anal. Appl. 347, 633–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cerejeiras, P., Kähker, U., Ku, M.: On the Riemann boundary value problem for null solutions to iterated generalized Cauchy–Riemann operator in Clifford analysis. Results Math. 63(3–4), 1375–1394 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Colombo, F., Sabadinia, I., Sommen, F.: The Fueter mapping theorem in integral form and the \({\cal{F} }\)-functional calculus. Math. Methods Appl. Sci. 33, 2050–2066 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Conway, J.B.: Functions of One Complex Variable. Springer, New York (1978)

    Book  Google Scholar 

  11. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  12. Delanghe, R., Sommen, F., Soucék, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic, Dordrecht (1992)

    Book  MATH  Google Scholar 

  13. Fokas, A.S.: A Unified Approach to Boundary Value Problems. University of Cambridge, Cambridge (2008)

    Book  MATH  Google Scholar 

  14. Fueter, R.: Die Funktionentheorie der Differentialgleichungen \(\Delta u\)=0 und \(\Delta \Delta u\)=0 mit vier reellen Variablen. Comment. Math. Helv. 7, 307–330 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gakhov, F.D.: Boundary Value Problems. Pergamon, Oxford (1966)

    Book  MATH  Google Scholar 

  16. Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. University of Cambridge, Cambridge (1991)

    Book  MATH  Google Scholar 

  17. Gong, Y.F., Du, J.Y.: A kind of Riemann and Hilbert boundary value problem for left monogenic functions in \({\mathbb{R} }^{m}(m\ge 2)\). Complex Var. Theory Appl. 49(5), 303–318 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Gürlebeck, K., Zhang, Z.X.: Some Riemann boundary value problems in Clifford analysis. Math. Methods Appl. Sci. 33, 287–302 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. He, F.L., Ku, M., Kähker, U., Sommen, F., Bernstein, S.: Riemann–Hilbert problems for monogenic functions in axially symmetric domains. Bound. Value Probl. 22, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  20. He, F.L., Ku, M., Kähker, 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), 1990–2000 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hilbert, D.: Über eine Anwendung der Integralgleichungen auf ein Problem Funktionentheorie, Verhandl. der III Int. Math. Kongr., pp. 223–240 (1904)

  22. Ku, M., Kähker, U., Wang, D.S.: Riemann boundary value problems on the sphere in Clifford analysis. Adv. Appl. Clifford Algebra 22(2), 365–390 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ku, M., Kähker, U.: Riemann boundary value problems on half space in Clifford analysis. Math. Methods Appl. Sci. 35(18), 2141–2156 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Ku, M., Fu, Y.X., Kähker, 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ku, M., Wang, Y., He, F.L., Kähker, U.: Riemann–Hilbert problems for monogenic functions on upper half ball of \({\mathbb{R} }^4\). Adv. Appl. Clifford Algebra 27(3), 2493–2508 (2017)

    Article  MATH  Google Scholar 

  26. Ku, M., He, F.L., Wang, Y.: Riemann Hilbert problems for Hardy space of meta-analytic functions on the unit disc. Complex Anal. Oper. Theory 12, 457–474 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ku, M., He, F.L., He, X.L.: Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equation on upper half ball. Complex Var. Ellipt. Equ. 65(11), 1902–1918 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  29. Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Leyden (1977)

    Book  Google Scholar 

  30. Riemann, B.: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Adalbert Rente, Göttingen (1851)

    Google Scholar 

  31. Rodin, Y.L.: Generalized Analytic Functions on Riemann Surfaces. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  32. Sommen, F.: On a generalization of Fueter’s theorem. ZAA 19, 899–902 (2000)

    MathSciNet  MATH  Google Scholar 

  33. Vekua, I.N.: Generalized Analytic Functions. Pergamon, London (1962)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, Z.Y., Zhou, Z.P.: On boundary value problems of Riemann–Hilbert type for monogenic functions in a half space of \({\mathbb{R} }^{m}\left(m\ge 2\right)\). Complex Var. Ellipt. Equ. 22(3–4), 181–193 (1993)

    Google Scholar 

Download references

Acknowledgements

We would like to thank Prof. Uwe Kaehler for sharing his idea without reservation when we started to study RHBVPs with variable coefficients for monogenic functions in high-dimensional Euclidean space several years ago.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Qian Huang & Fuli He

  2. Department of Computing Science, University of Radboud, 6525 EC, Nijmegen, The Netherlands

    Min Ku

Authors
  1. Qian Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fuli He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Min Ku
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fuli He.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Q., He, F. & Ku, M. Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in \({\mathbb {R}}^{n+1}\). Adv. Appl. Clifford Algebras 33, 23 (2023). https://doi.org/10.1007/s00006-023-01264-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-023-01264-0

Keywords

Mathematics Subject Classification

Search

Navigation