О КАРТИНЕ РАЗРЕШИМОСТИ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ ГИЛЬБЕРТА ДЛЯ КВАЗИГАРМОНИЧЕСКИХ ФУНКЦИЙ В КРУГОВЫХ ОБЛАСТЯХ

Abstract

A Hilbert-type boundary value problem in the classes of quasi-harmonic functions is considered. Quasi-harmonic functions are regular solutions of an elliptic differential equation form $\frac{\partial ^2 W}{\partial z\partial \overline z} + \frac{n(n + 1)}{(1 + z \overline z)^2}W = 0$, where $\frac{\partial }{\partial z} = \frac{1}{2} ( \frac{\partial }{\partial x} - i\frac{\partial }{\partial y})$ , $\frac{\partial }{\partial \overline z} = \frac{1}{2} (\frac{\partial }{\partial x} + i\frac{\partial }{\partial y})$ , and n is a given positive integer. Using the fact that a circle is an analytic curve, we have developed an explicit method for finding solutions of the Hilbert homogeneous boundary value problem for quasi-harmonic functions in circular domains. The principal logic of this method consists of two stages. At stage one we are using a representation of quasi-harmonic function via analytic function and its derivatives to reduce the problem to the classical Hilbert problem for some auxiliary analytic function in the circular domain. A solution Φ( z ) for this problem will be used at stage two, when we solve the linear differential Euler equation of order n with the right-hand side Φ( z ). General solution for the problem can be explicitly expressed in terms of the solution of the Euler equation. Moreover, we have established that the solvability for the considered boundary-value problem depends essentially on whether a unit circumference is the carrier of boundary conditions or a non-unit circle