Skip to main content
SpringerLink
Log in

Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions

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

Abstract

In this paper, we establish the generalized Cauchy theorems on the para-sphere and the generalized Cauchy integral formulae on the strong para-sphere in Clifford analysis. As applications, the generalized Cauchy theorems and the generalized Cauchy integral formulae on the closed smooth surface and the cylindroid with crooked tips are respectively obtained. And these directly result in the Painlevé theorem and the generalization of the Sochocki–Plemelj formula for the difference of boundary values in Clifford analysis. Then, by using these results the Riemann jump boundary value problems and Dirichlet boundary value problems for regular functions in Clifford analysis are discussed. Some singular integral equations are also solved and the inversion formula for Cauchy principal value is obtained by the results based on these boundary value problems solved.

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

References

  1. Ahlfors, L.V.: Complex Analysis. McGraw-Hill Book Company, New York (1978)

    Google Scholar 

  2. Begehr, H., Zhang, Z., Du, J.: On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Acta Math. Sci. 23B(1), 95–103 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Bernstein, S.: On the left linear Riemann problem in Clifford analysis. Bull. Belg. Math. Soc. 3, 557–576 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Bernstein, S.: On the index of Clifford algebra valued singular integral operators and the left linear Riemann problem. Complex Var. Theory App. 35, 33–64 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blaya, R.A., Reyes, J.B., Peña, D.: Clifford Cauchy type integrals on Ahlfors-David regular surfaces in \(\mathbb{R}^{m+1}\). Adv. Appl. Clifford Algebra 13(2), 133–156 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blaya, R.A., Reyes, J.B., Peña, D.: Jump problem and removable singularies for monogenic functions. J. Geom. Anal. 17(1), 1–13 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blaya, R.A., Reyes, J.B.: 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 

  8. Blaya, R.A., Reyes, J.B.: The quaternionic Riemann problem with a natural geometric condition on the boundary. Complex Var. Theory Appl. 42(2), 135–149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman (Advanced Publishing Program), Boston (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Inc., Englewood Cliffs, New Jersey (1976)

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

    Book  Google Scholar 

  14. Courant, R., John, F.: Introduction to Calculus and Analysis II/2. Springer-Verlag, Berlin, Heidelberg (2000)

  15. Delanghe, R.: On regular-analytic functions with values in a Clifford algebra. Math. Ann. 185, 91–111 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delanghe, R.: On the singularities of functions with values in a Clifford algebra. Math. Ann. 196, 293–319 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator. Kluwer, Dordrecht (1992)

    Book  MATH  Google Scholar 

  18. Du, J., Xu, N., Zhang, Z.: Boundary behavior of Cauchy-type integrals in Clifford analysis. Acta Math. Sci. 29B(1), 210–224 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Du, J., Xu, N.: On boundary behavior of the Cauchy type integrals with values in a universal Clifford algebra. Adv. Appl. Clifford Algebra 21, 49–87 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Du, J., Zhang, Z.: A Cauchy’s integral formula for functions with values in a universal Clifford algebra and its applications. Complex Var. Theory App. 47(10), 915–928 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gakhov, F.D.: Boundary Value Problems. Nauka, Moscow (1977)

    MATH  Google Scholar 

  22. Gilbert, R.P., Buchnan, J.L.: First Order Elliptic Systems: A Function Theoretic Approch. Academic, New York (1983)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Gu, L., Du, J., Zhang, Z.: Riemann boundary value problems for triharmonic functions in Clifford analysis. Adv. Appl. Clifford Algebras 23, 77–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser Verlag, Basel (1990)

    Book  MATH  Google Scholar 

  26. Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester, England (1997)

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

    MathSciNet  MATH  Google Scholar 

  28. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, London (1934)

    MATH  Google Scholar 

  29. Hile, G.N.: Hypercomplex function theory applied to partial differential equations. Ph.D. thesis, Indiana Univ., Bloomington (1972)

  30. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, London (1985)

    Book  MATH  Google Scholar 

  31. Iftimie, V.: Fonctions hypercomplexes. Bull. Math. Soc. Sci. Math. R. S. Roum. 9(57), 279–332 (1965)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  33. Lu, J.K.: Complex Variable Methods in Plane Elasticity. World Scientific, Singapore (1995)

    Book  MATH  Google Scholar 

  34. Luo, W., Du, J.: The Gauss–Green theorem in Clifford analysis and its applications. Acta Math. Sci. 35B(1), 235–254 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Markushevich, A.I.: Theory of Functions of a Complex Variable. Chelsea Publishing Co., New York (1977)

  36. Muskhelishvili, N.I.: Some Basic Problems of Mathmetical Theory of Elasticity. Noordhoff, Groningen (1963)

    MATH  Google Scholar 

  37. Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (1968)

    MATH  Google Scholar 

  38. Obolashvili, E.: Higher Order Partial Differential Equations in Clifford Analysis. Progress in Mathematical Physics, vol 28. Birkhäuser, Boston (2003)

  39. Privalov, I.I.: Introduction to the Theory of Functions of a Complex Variable. Nauka, Moscow (1984)

    MATH  Google Scholar 

  40. Shapiro, M.: On analogues of the Riemann boundary value problem for a class of hyperholomorphic functions. In: Wen, G.C., Zhao, Z. (eds.) Integral Equations and Boundary Value Problems, pp. 184–188. World Scientific (1991)

  41. Shapiro, M., Vasilievski, N.: Quaternionic \(\psi \)-hyperholomorphic functions, singular integral operators with quaternionic Cauchy kernel and analogues of the Riemann boundary value problem I, \(\psi \)-hyperholomorphic function theory. Complex Var. Theory App. 27, 17–46 (1995)

  42. Shapiro, M.N., Vasilievski, N.: Quaternionic \(\psi \)-hyperholomorphic functions, singular integral operators with quaternionic Cauchy kernel and analogues of the Riemann boundary value problem II, Algebras of singular integral operators and Riemann type boundary value problem. Complex Var. Theory App. 27, 67–96 (1995)

  43. Si, Z., Du, J.: The Hilbert boundary value problem for generalized analytic functions in Clifford analysis. Acta Math. Sci. 33B(2), 393–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Stern, I.: Boundary value problems for generalized Cauchy Riemann systems in the space. In: Krichmann, R., Tutschke, W. (eds.) Boundary Value and Initial Value Problem in Complex Analysis, pp. 159–183. Pitman Res. Notes Math. (1991)

  45. Xu, Z.: Helmholtz equation and boundary value problems. In: Begehr, H., Jeffrey, A. (eds.) Partial Differential Equations with Complex Analysis, pp. 204–214. Pitman Res. Notes Math. Series, vol. 262, Longman Scientific & Technical (1992)

  46. Xu, Z.: On Riemann boundary value problems for regular functions in Clifford algebra. Chin. Sci. Bull. 32(23), 476–477 (1987)

    Google Scholar 

  47. Xu, Z.: On linear and non-linear Riemann–Hilbert problems for regular functions with values in Clifford algebras. Chin. Ann. Math. 11B(3), 349–358 (1990)

    MathSciNet  MATH  Google Scholar 

  48. Xu, Z., Zhou, C.: On boundary value problems of Riemann–Hilbert type for monogenic functions in a half space of \(R^{m}\) \((m\ge 2)\). Complex Var. Theory Appl. 22, 181–193 (1993)

    Article  MATH  Google Scholar 

  49. Yeh, R.Z.: Hyperholomorphic functions and second order partial differential equations. Trans. Am. Math. Soc. 325, 287–318 (1991)

    MathSciNet  MATH  Google Scholar 

  50. Yeh, R.Z.: Analysis and applications of holomorphic functions in higher dimensions. Trans. Am. Math. Soc. 345, 151–177 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhang, Z.: Möbius transformation and Poisson integral representation for monogenic functions. Acta Math. Sin. 56(4), 487–504 (2013)

    MATH  Google Scholar 

  52. Zhang, Z., Du, J.: On certain Riemann boundary value problems and singular integral equations in Clifford analysis. Chin. J. Contemp. Math. 22(3), 237–244 (2001)

    MathSciNet  Google Scholar 

  53. Zorich, V.A.: Mathematical Analysis II. Springer, Berlin Springer-Verlag, Berlin, Heidelberg (2003)

Download references

Acknowledgements

The first version of this work was done while the authors worked at the Wuhan University in fall term 2010. The revision of this work is done while the first author as a postdoctoral student and the second author as a research scholar were visiting the University of Macau in summer term 2015. The authors are very grateful to Professor Tao Qian for his helpful support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Wuhan University, Wuhan, 430072, China

    Weiyu Luo & Jinyuan Du

  2. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, China

    Weiyu Luo

Authors
  1. Weiyu Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinyuan Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinyuan Du.

Additional information

Communicated by Frank Sommen.

This work was supported by NNSF of China (#11171260) and RFDP of Higher Education of China (#20100141110054).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, W., Du, J. Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions. Adv. Appl. Clifford Algebras 27, 2531–2583 (2017). https://doi.org/10.1007/s00006-017-0790-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-017-0790-2

Mathematics Subject Classification

Keywords

Search

Navigation