Harmonic boundary value problems in half disc and half ring
type: Item
instance of: scholarly article
title: Harmonic boundary value problems in half disc and half ring
zbMATH Open document ID: 1183.30039
author: Heinrich Begehr
author: Tatyana Vaitekhovich
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-10-26T00:00:00Z
review text: The authors consider a Poisson kernel which is developed on the basis of the reflection principle leading to the solution of the Schwarz problem for the inhomogeneous Cauchy-Riemann equation for the half unit disc \[ \mathbb{D}^{+}:=\{ z : |z| < 1, 0 < \text{Im}\,\, z \}. \] With the help of the harmonic Green and Neumann functions, the Dirichlet and Neumann boundary value problems are solved explicitly for the Poisson equation in the upper half disc. Particular attention is paid to the two corner points of the boundary for the Dirichlet problem. Also, these boundary value problems are treated in similar manners in the upper half circular ring \[ R^{+} :=\{ z : r < |z| < 1, 0 < \text{Im}\,\,z \}. \] The fundamental tool for complex boundary value problems is the Cauchy-Pompeiu representation, which just has to be properly modified. Theorem. Any \(w \in C^{1} (D;\mathbb{C})\cap C(\overline{D};\mathbb{C})\), for a regular complex domain \(D\subset \mathbb{C}\), can be represented as \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{\partial D}^{ }} w (\zeta) \frac{d \zeta}{\zeta - z} - \frac{1}{\pi} \displaystyle{\int_{D}^{ }} w_{\overline{\zeta}}(\zeta) \frac{d \xi d \eta}{\zeta - z}} \] or \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{\partial D}} w (\zeta) \frac{d \overline{\zeta}}{\overline{\zeta - z}} - \frac{1}{\pi} \int_{D} w_{\zeta} (\zeta) \frac{d\xi d \eta}{\overline{\zeta - z}}}. \] As one knows, the right-hand sides vanish if \(z\) it is not in \(\overline{D}\), the closure of \(D\). With regard to the Schwarz problem, the Cauchy-Pompeiu formula is modified by combining it in a proper way with the Cauchy-Pompeiu formula evaluated at the reflections of the point \(z\) at the boundary \(\partial D, 1/z\) and \(\overline{z}\), and for symmetry also at the reflection \(1/z\) of \(1/\overline{z}\) at the real axis, or, what is the same, the reflection \(1/z\) of \(\overline{z}\) at the unit circle. Theorem. Any \(w \in C^{1} (\math…
reviewed by: Mugur Acu
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 31A25
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5620898
zbMATH Keywords: Schwarz problem
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Neumann problem
zbMATH Keywords: Cauchy-Riemann equation
zbMATH Keywords: Poisson equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.7169/facm/1246454030
OpenAlex ID: W2086766501
DOI: 10.7169/FACM/1246454030
description: scientific article; zbMATH DE number 5620898