On Solvability of the Hilbert Homogeneous Boundary Value Problem for Quasiharmonic Functions in Circular Domains

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