Internal and external one-sided homogeneous boundary value problems of conjugation for bicircular domains of the space \(\mathbb C^2\). (Russian) Zbl 1030.31002

Dzebisov, Kh. P.

Vladikavkaz. Mat. Zh. 2, No. 4, 3-10 (2000).

The article is devoted to the study of two-dimensional boundary-value problems of Riemann type for a bicircular domain of the complex plane \(\mathbb C^2\) in the class of functions \(M_{\sigma_1,\sigma_2}^{\alpha,\beta}\) which are defined by the formula \[ f(z) = \frac{1}{4\pi^{2} i}\int_{\alpha}^{\beta} d\tau \int_0^{2\pi} dt \int_{|\xi|= 1}\frac{F(t,\xi) d\xi}{\xi - u_1(\tau,\sigma)}, \] where \(z = (z_1,z_1)\in\mathbb C^2\), \(u_1(\tau,\sigma) = \tau^{\sigma_1}z_1 + \tau^{\sigma_1}z_2e^{it}\), \(0 < \alpha < \beta < 1\), \(\sigma_1 > 0\), \(\sigma_2 > 0\). Here \(F(t,\xi)\) satisfies the Lipschitz condition on the set \(\Delta = \{(t,\xi) : 0\leq t \leq 2\pi,\;|\xi|= 1\}\) with exponent \(0 < \alpha \leq 1\) over the variable \(\xi\) uniformly in \(t\).
The author examines various properties of this class of functions and applies them to solve the internal and external one-sided homogeneous boundary value problems of conjugation in the class \(M_{\sigma_1,\sigma_2}^{\alpha,\beta}\). As a result, a series of existence theorems is exposed in which the solutions to these problems are obtained by solving singular integral equations.

Reviewer: V.Grebenev (Novosibirsk)

MSC:

31A25	Boundary value and inverse problems for harmonic functions in two dimensions
31A20	Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31A10	Integral representations, integral operators, integral equations methods in two dimensions

Keywords:

boundary value problem; interior and exterior problem; singular integral equation