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
$\frac{\partial^2W}{\partial z\partial\overline{z}}+\frac{n(n+1)}{(1+z\overline{z})^2}W=0$,
where $\frac{\partial}{\partial z}=\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)$,
$\frac{\partial}{\partial \overline{z}}=\frac12\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)$, 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 $\Phi(z)$ 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 $\Phi(z)$. 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.
Keywords:boundary value problem, Hilbert-type boundary value problem, quasiharmonic function, differential equation, cyclic domain, unit circumference, non-unit circumference.
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