Skip to main content
SpringerLink
Log in

The Dirichlet and Riemann–Hilbert Problems in Smirnov Classes with Variable Exponent in Doubly Connected Domains

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

The Riemann–Hilbert problem Re [a(t)Φ+(t)] = b(t) is studied in Smirnov classes with variable exponent in a domain D. Both simply and doubly connected domains with piecewise smooth boundary Γ are considered under the assumption that Γ consists of simple smooth arcs A k A k+1, the tangents of which at the points A k form angles πν k , 0 ≤ ν k ≤ 2, k = 1, . . . , n, A n+1 = A 1 . Various conditions (necessary, necessary and sufficient) are found for the problem to be solvable and Noetherian. In some cases, the index is calculated. In the case of solvability, the solutions in a simply connected domain are constructed in an explicit form. Special attention is given to the Dirichlet problem, i.e. the case a(t) ≡ 1. Bibliography: 21 titles.

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. L. Diening and M. Ružička, “Calderón–Zygmund operators on generalized Lebesgue spaces L p(·) and problems related to fluid dynamics,” J. Reine Angew. Math. 563, 197–220 (2003).

    MATH  MathSciNet  Google Scholar 

  2. M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000).

    Google Scholar 

  3. A. Yu. Karlovich, “Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces” J. Integral Equations Appl. 15, No. 3, 263–320 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  4. V. Kokilashvili and S. Samko, “Singular integral equations in the Lebesgue spaces with variable exponent,” Proc. A. Razmadze Math. Inst. 131, 61–78 (2003).

    MATH  MathSciNet  Google Scholar 

  5. V. V. Zhikov and S. E. Pastukhova, “On the improved integrability of the gradient of solutions of elliptic equations with a variable nonlinearity exponent” [in Russian], Mat. Sb. 199, No. 12, 19–52 (2008); English transl.: Sb. Math. 199, No. 11-12, 1751–1782 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. Kokilashvili, V. Paatashvili, and S. Samko, “Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in Lp( · )(Γ),” Bound. Value Probl. No. 1, 43–71 (2005).

  7. V. Kokilashvili and V. Paatashvili, “The Riemann–Hilbert problem in weighted classes of Cauchy type integrals with density from L p( · )(Γ),” Complex Anal. Oper. Theory 2, No. 4, 569–591 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Kokilashvili and V. Paatashvili, “The Riemann–Hilbert problem in a domain with piecewise smooth boundaries in weight classes of Cauchy type integrals with a density from variable exponent Lebesgue spaces,” Georgian Math. J. 16, No. 4, 737–755 (2009).

    MATH  MathSciNet  Google Scholar 

  9. V. Kokilashvili and V. Paatashvili, “On Hardy classes of analytic functions with a variable exponent,” Proc. A. Razmadze Math. Inst. 142, 134–137 (2006).

    MATH  MathSciNet  Google Scholar 

  10. V. Kokilashvili and V. Paatashvili, “On variable Hardy and Smirnov classes of analytic functions,” Georgian Int. J. Sci. Technol. Med. 1, No. 2, 181–195 (2008).

    MATH  MathSciNet  Google Scholar 

  11. G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, “The Dirichlet problem for variable exponent Smirnov class harmonic functions in doubly connected domains,” Mem. Differ. Equ. Math. Phys. 52, 131–156 (2011).

    MATH  MathSciNet  Google Scholar 

  12. V. Kokilashvili and V. Paatashvili, “The Dirichlet problem for harmonic functions from variable exponent Smirnov classes in domains with piecewise smooth boundary” [in Russian], Probl. Mat. Anal. 52, 100–117 (2010); English transl.: J. Math. Sci., New York 172, No. 3, 401–421 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. Kokilashvili and V. Paatashvili, Boundary Value Problems for Analytic and Harmonic Functions in Nonstandard Banach Function Spaces, Nova Science Publ., New York (2012).

    MATH  Google Scholar 

  14. V. Maz’ya and A. Soloviev, “L p -theory of boundary integral equations on a contour with outward peak,” Integral Equ. Operator Theory 32, No. 1, 75–100 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  15. V. Maz’ya and A. Soloviev, “A direct method for boundary integral equations on a contour with a peak,” Georgian Math. J. 10, No. 3, 573–593 (2003).

    MATH  MathSciNet  Google Scholar 

  16. V. G. Maz’ya and A. Soloviev, Boundary Integral Equations on Contours with Peaks, Birkhäuser, Basel (2010).

    Book  MATH  Google Scholar 

  17. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1956).

    Google Scholar 

  18. G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, “Boundary value problems for analytic and harmonic functions in domains with nonsmooth boundaries. Applications to conformal mappings,” Mem. Differ. Equ. Math. Phys. 14, 1–195, (1998),

    MATH  MathSciNet  Google Scholar 

  19. V. Kokilashvili, V. Paatashvili, and S. Samko, “Boundedness in Lebesgue spaces with variable exponent of the Cauchy singular operator on Carleson curves,” In: Modern Operator Theory and Applications, pp. 167–186, Birkhäuser, Basel (2006).

    Google Scholar 

  20. V. Paatashvili, “On some properties of analytic functions from Smirnov class with a variable exponent,” Mem. Differ. Equ. Math. Phys. 56, 73–88 (2012).

    MathSciNet  Google Scholar 

  21. F. V. Atkinson, “The normal solubility of linear equations in normed spaces” [in Russian], Mat. Sb. 28, 3–14 (1951).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 2, University Str., Tbilisi, 0143, Georgia

    V. Kokilashvili & V. Paatashvili

Authors
  1. V. Kokilashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Paatashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Kokilashvili.

Additional information

Translated from Problemy Matematicheskogo Analiza 75, April 2014, pp. 71–80.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kokilashvili, V., Paatashvili, V. The Dirichlet and Riemann–Hilbert Problems in Smirnov Classes with Variable Exponent in Doubly Connected Domains. J Math Sci 198, 735–746 (2014). https://doi.org/10.1007/s10958-014-1822-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-014-1822-2

Keywords

Search

Navigation