The Riemann problem on a ray for generalized analytic functions with a singular line

P. L. Shabalin ; R. R. Faizov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 1, pp. 58-69.

Résumé

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on a ray for a generalized Cauchy – Riemann equation with a singular coefficient. For the solution of this problem, we derived a formula for the general solution of the generalized Cauchy – Riemann equation under constraints that led to an infinite index of logarithmic order of the accompanying problem for analytical functions. We have obtained a formula for the general solution of the Riemann problem and conducted a complete study of the existence and the number of solutions of a boundary value problem for generalized analytic functions with a singular line.

Bibliographie

[1] Vekua I. N., Generalized Analytic Functions, Nauka, M., 1988, 507 pp. (in Russian)

[2] Mikhailov L. G., New Classes of Singular Integral Equations and Their Application to Differential Equations with Singular Coefficients, TadjikNIINTI, Dushanbe, 1963, 183 pp. (in Russian) | MR

[3] Radzhabov N. R., Integral Representations and Boundary Value Problems for Some Differential Equations with Singular Line or Singular Surfaces, v. 1, Tadzhikskiy gosuniversitet Publ., Dushanbe, 1980, 147 pp. (in Russian)

[4] Radzhabov N. R., Integral Representations and Boundary Value Problems for Some Differential Equations with Singular Line or Singular Surfaces, v. 2, Tadzhikskiy gosuniversitet Publ., Dushanbe, 1981, 170 pp. (in Russian)

[5] Radzhabov N. R., “Integral representations and boundary value problems for the generalized Cauchy – Riemann system with a singular line”, Doklady Akademii Nauk SSSR, 267:2 (1982), 300–305 (in Russian) | MR

[6] Radzhabov N. R., Rasulov A. B., “Integral representations and boundary value problems for a class of systems of differential equations of elliptic type with singular manifolds”, Differentsial'nye Uravneniya, 25:7 (1989), 1279–1981 (in Russian) | MR

[7] Usmanov Z. D., Generalized Cauchy – Riemann Systems with a Singular Point, Routledge, New York, 1997, 232 pp. | DOI | MR

[8] Begehr H., Dao-Qing Dai, “On continuous solutions of a generalized Cauchi – Riemann system with more than one singularity”, Journal of Differential Equations, 196:1 (2004), 67–90 | DOI | MR

[9] Meziani A., “Representation of solutions of a singular CR equation in the plane”, Complex Variables and Elliptic Equations, 53:12 (2008), 1111–1130 (accessed 2 August 2022) https://www.tandfonline.com/doi/abs/10.1080/17476930802509239 | DOI | MR

[10] Rasulov A. B., “Representation of the variety of solutions and investigation of the boundary value problems for some generalized Cauchy – Riemann systems with one and two singular lines”, Izvestiya AN Tadzh. SSR. Seriia fiziko-matematicheskikh, khimicheskikh i geologicheskikh nauk, 1982, no. 2 (84), 23–32 (in Russian)

[11] Rasulov A. B., Soldatov A. P., “Boundary value problem for a generalized Cauchy – Riemann equation with singular coefficients”, Differential Equations, 52:5 (2016), 616–629 | DOI | MR

[12] Fedorov Iu. S., Rasulov A. B., “Hilbert type problem for a Cauchy – Riemann equation with singularities on a circle and at a point in the lower-order coefficients”, Differential Equations, 57:1 (2021), 127–131 | DOI | DOI | MR

[13] Rasulov A. B., “The Riemann problem on a semicircle for a generalized Cauchy – Riemann system with a singular line”, Differential Equations, 40:9 (2004), 1364–1366 | DOI | MR

[14] Rasulov A. B., “Integral representations and the linear conjugation problem for a generalized cauchy-riemann system with a singular manifold”, Differential Equations, 36:2 (2000), 306–312 | DOI | MR

[15] Govorov N. V., Riemann's Boundary Problem with Infinite Index, Operator Theory: Advances and Applications, 67, Birkhauser Basel, Berlin, 1994, 263 pp. | DOI | MR

[16] Monakhov V. N., Semenko E. V., Boundary Value Problems and Pseudodifferential Operators on Riemann Surfaces, Fizmatlit, M., 2003, 416 pp. (in Russian)

[17] Ostrovskii I. V., “Homogeneous Riemann boundary value problem with infinite index on a curved contour”, Theory of Functions, Functional Analysis and Their Applications, 1991, no. 56, 95–105 (in Russian)

[18] Yurov P. G., “Inhomogeneous Riemann boundary value problem with infinite index of logarithmic order $\alpha\geq1$”, Materials of the All-Union Conference on Boundary Value Problems, Kazan State University Publ, Kazan, 1970, 279–284 (in Russian)

[19] Yurov P. G., “The homogeneous Riemann boundary value problem with an infinite index of logarithmic type”, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1966, no. 2, 158–163

[20] Salimov R. B., Khasanova E. N., “The solution of the homogeneous boundary value problem of Riemann with infinite index of logarithmic order on the beam by a new method”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 17:2 (2017), 160–171 (in Russian) | DOI