Skip to main content
SpringerLink
Log in

Well-Posedness of Jump Problem on Non-rectifiable Curves

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We investigate the well-posedness of a jump problem on non-rectifiable curves with Hölder boundary data. The jump problem is a special case of Riemann boundary value problem. However, the classic results are obtained in the case of piecewise smooth boundary. There are some results for non-rectifiable boundary (see our previous papers in bibliography), but still there is a large field of questions waiting for answers.

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

    MATH  Google Scholar 

  2. Jian-Ke, L.: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (1993)

    MATH  Google Scholar 

  3. Muskhelishvili, N.I.: Singular Integral Equations. Nauka Publishers, Moscow (1962)

    MATH  Google Scholar 

  4. Kats, B.A.: Riemann problem on a closed Jordan curve. Izvestija vuzov. Matem. 4, 68–80 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Falconer, K.J.: Fractal Geometry, 3rd edn. Wiley, Hoboken (2014)

    MATH  Google Scholar 

  6. Kats, B.A.: The Riemann boundary value problem on non-rectifiable curves and related questions. Complex Var. Elliptic Equ. 59(8), 1053–1069 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Katz, D.B.: New metric characteristics of non-rectifiable curves with applications. Sib. Math. J. 57(2), 364–372 (2016)

    Article  MathSciNet  Google Scholar 

  8. Katz, D.B.: Local and weighted Marcinkiewicz exponents with applications. J. Math. Anal. Appl. 440(1), 74–85 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kats, B.A.: Riemann problem on a closed Jordan curve. Semi-continuous statement. Izvestija vuzov. Matem. 5, 14–22 (1985)

    Google Scholar 

  10. Kats, B.A., Katz, D.B.: Marcinkiewicz exponents and integrals over non-rectifiable paths. Math. Methods Appl. Sci. 39(12), 3402–3410 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Katz, D.B., Kats, B.A.: Interactions of germs with applications. Math. Methods Appl. Sci. (2017). https://doi.org/10.1002/mma.4362

  12. Stein, E.M.: Singular Integrals and Differential Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  13. Vekua, I.N.: Generalized Analytical Functions, vol. 521. Nauka, Moscow (1988)

    Google Scholar 

  14. Painleve, P.: Sur les Lignes Singulieres des Fonctions Analytiques. Gauthier-Villars, Paris (1887)

    MATH  Google Scholar 

  15. Dolzhenko, E.P.: On “erasing” of singularities of analytic functions. Uspekhi Matem. Nauk 18(4), 135–142 (1963)

    Google Scholar 

  16. Schippers, E., Staubach, W.: Well-posedness of a Riemann–Hilbert problem on d-regular quasidisks. Ann. Acad. Sci. Fenn. Math. 42, 141–147 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kremlevskaya str. 18, Kazan, Russia, 420138

    Boris A. Kats

  2. Mathematical Modeling of Nonequilibrium Processes in Oil and Gas Production Laboratory, Kazan (Volga Region) Federal University, Kremlevskaya str. 18, Kazan, Russia, 420138

    David B. Katz

Authors
  1. David B. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris A. Kats
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris A. Kats.

Additional information

Communicated by Laurent Baratchart.

The research is partially supported by Russian Foundation for Basic Researches and Government of Republic Tatarstan, Grant No. 18-41-160003.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katz, D.B., Kats, B.A. Well-Posedness of Jump Problem on Non-rectifiable Curves. Complex Anal. Oper. Theory 13, 1873–1882 (2019). https://doi.org/10.1007/s11785-018-0868-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-018-0868-4

Keywords

Search

Navigation