Riemann Boundary Value Problems with Reflection on the Real Axis and Related Singular Integral Equations

Abstract

We prove a theorem on the equivalence of a scalar Riemann boundary value problem with a shift-reflection on the real axis and a matrix Riemann boundary value problem without the shift. A relationship between the boundary value problem in question and a boundary value problem with orientation-preserving shift-rotation on the unit circle is established. The operator identities constructed by the present authors in their preceding papers and permitting one to eliminate the shift-reflection from the integral equation corresponding to the boundary value problem are the main research tool.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1

    Karelin, A.A., On a relation between singular integral operators with a Carleman linear-fractional shift and matrix characteristic operators without shift, Bol. Soc. Mat. Mex., 2001, vol. 7, no. 3, pp. 235–246.

    MathSciNet  MATH  Google Scholar 

  2. 2

    Karelin, A.A., Singular integral operators with coefficients of a special structure related to operator equalities, Complex Anal. Oper. Theory, 2008, vol. 2, no. 4, pp. 549–567.

    MathSciNet  Article  Google Scholar 

  3. 3

    Spitkovsky, I.M. and Tashbaev, A.M., Factorization of piecewise constant matrix functions with 3 points of discontinuity in the classes \(L_{p,\rho } \) and some of its applications, Dokl. Math., 1990, vol. 40, no. 1, pp. 80–85.

    MathSciNet  MATH  Google Scholar 

  4. 4

    Karelin, A.A., On the operator equality and some of its applications, Proc. A. Razmadze Math. Inst., 2002, vol. 128, pp. 105–116.

    MathSciNet  MATH  Google Scholar 

  5. 5

    Karelin, A.A., Applications of operator equalities to singular integral operators and to Riemann boundary value problems, Math. Nachr., 2007, vol. 280, no. 9–10, pp. 1108–1117.

    MathSciNet  Article  Google Scholar 

  6. 6

    Karelin, A.A., Pérez Lechuga, G., and Tarasenko, A.A., Riemann problem and singular integral equations with coefficients generated by piecewise constant functions, Differ. Equations, 2008, vol. 44, no. 9, pp. 1225–1235.

    MathSciNet  Article  Google Scholar 

  7. 7

    Litvinchuk, G.S., Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift, Dordrecht–Boston–London: Springer, 2000.

    Google Scholar 

  8. 8

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

    Google Scholar 

  9. 9

    Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1968.

    Google Scholar 

  10. 10

    Gohberg, I. and Krupnik, N., One-Dimensional Linear Singular Integral Equations. Operator Theory: Advances and Applications. Vol. 53 , Basel–Boston–Berlin: Birkhäuser, 1992.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to A. A. Karelin or A. A. Tarasenko.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Karelin, A.A., Tarasenko, A.A. Riemann Boundary Value Problems with Reflection on the Real Axis and Related Singular Integral Equations. Diff Equat 57, 100–110 (2021). https://doi.org/10.1134/S0012266121010080

Download citation