On solvability of the Hilbert homogeneous boundary value problem for quasiharmonic functions in circular domains

K. M. Rasulov ; T. I. Timofeeva

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 4, pp. 33-40

Résumé

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.

Détail

Keywords: boundary value problem, Hilbert-type boundary value problem, quasiharmonic function, differential equation, unit circumference, non-unit circumference.
Mots-clés : cyclic domain