Integral Representations for the Generalized Cauchy-Riemann Equation with a Supersingular Point in a Half Plane
DOI:
https://doi.org/10.24160/1993-6982-2020-1-105-108Keywords:
Cauchy-Riemann operator, Vekua operator, singular point, half-planeAbstract
In the class of first-order elliptic systems, the generalized Cauchy-Riemann system (GCRS) occupies a special place. If the coefficients at the lowest terms and the right-hand side of the GCRS belong to the space of summable functions with degree p > 2 the representation of the general solution of the equation of this system from the Helder class was obtained by I.N. Vekua.
The problems for GCRS with coefficients having first-order singularities at an isolated singular point were studied in works by L.G. Mikhailov, Z.D. Usmanov, A.P. Soldatov, N.R. Rajabov, A. Tungatarov, R. Saks, and others. The concept of a super-singular singularity was suggested by N.R. Rajabov. Great interest in GCRS is also stemming from its numerous applications. For example, GCRS with a singular point are used in the theory of infinitesimal flexures of positively curved surfaces with flattening points, and GCRS with a singular line reduces to the Ernst version of the Maxwell-Einstein equation. In all of published papers, GCRS with singularities in the lower coefficients was mainly studied in a finite domain.
In this article, an integral representation of the solution in the class of bounded functions has been found for an equation with a Cauchy- Riemann operator with an inner supersingular point in lower coefficients on a half-plane.
References
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Для цитирования: Расулов А.Б., Дорофеева И.Н. Интегральные представления для обобщенного уравнения Коши-Римана со сверхсингулярной точкой на полуплоскости // Вестник МЭИ. 2020. № 1. С. 105—108. DOI: 10.24160/1993-6982-2020-1-105-108.
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7. Meziani A. Representation of Solutions of a Singular CR Equation in the Plane. Complex Var. and Elliptic Eq. 2008;53:1111—1130.
8. Reissig M., Timofeev A. Dirichlet Problems for Generalized Cauchy-Riemann Systems with Singular Coefficients. Complex Variables. 2005;50;7 (11): 653—672.
9. Begehr H, Dao-Qing Dai. On Continuous Solutions of a Generalized Cauchy-Riemann System with More Than One Singularity. J. Differential Equations. 2004;196: 67—90.
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For citation: Rasulov A.B., Dorofeeva I.N. Integral Representations for the Generalized Cauchy-Riemann Equation with a Supersingular Point in a Half Plane. Bulletin of MPEI. 2020;1:105—108. (in Russian). DOI: 10.24160/1993-6982-2020-1-105-108.
