Journal of Complex Analysis

Volume 2016 (2016), Article ID 8097095, 8 pages

http://dx.doi.org/10.1155/2016/8097095

## Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

Nazarbayev University, 53 Kabanbay Batyr Avenue, Astana 010000, Kazakhstan

Received 21 October 2015; Revised 15 December 2015; Accepted 10 January 2016

Academic Editor: Vladislav Kravchenko

Copyright © 2016 Bibinur Shupeyeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The parqueting-reflection method is applied to a nonregular domain and the harmonic Green function for the half hexagon is constructed. The related Dirichlet problem for the Poisson equation is solved explicitly.

#### 1. Introduction

The basic boundary value problems for the second-order complex partial differential equations are the harmonic Dirichlet and Neumann problems for the Laplace and Poisson equations. In order to find the solution in explicit or closed form diverse methods have been applied. In case a given domain is simply connected and has a piecewise smooth boundary the tools of complex analysis such as Schwarz reflection principle and conformal mapping serve perfectly. When a given domain is piecewise smooth polygonal and has corners the Schwarz-Christoffel formula can be used. Difficulties arise since the elliptic integrals appearing in the formula imply complicated computations and need to be solved numerically. As analogue to this formula, another method can be applied which gives the covering of the entire complex plane by reflection of the given domain at its boundary. The method is fully described in numerous papers of Begehr and other authors; see, for example [1–12]. Our aim is to find the solution of the Dirichlet boundary value problem for the Poisson equation through the Poisson integral formula. It is known that the Poisson kernel function is an analogue of the Cauchy kernel for the analytic functions and the Poisson integral formula solves the Dirichlet problem for the inhomogeneous Laplace equation. One way to obtain the Poisson kernel leads to the harmonic Green function which is to be constructed by use of the parqueting-reflection method.

In this paper we first consider the half hexagon domain and implement the parqueting-reflection method. The reflection points treated in a proper way help to construct the certain meromorphic functions needed to find the harmonic Green function and representation formula. The later one provides the solution to the harmonic Dirichlet problem which is shown in the last part.

#### 2. Half Hexagon Domain and Poisson Kernel

We consider a polygonal domain with corner points. The half hexagon denoted as with four corner points at 2, , , and lies in the upper half plane. A point will later serve as a pole of the Green function. Its complex conjugate does not lie in . is reflected at the real axis so that the entire hexagon (Figure 1) is obtained. The pole is reflected onto which will later become a zero of a certain meromorphic function related to the Green function. The points and from are reflected again through all the sides of the hexagon, starting with the right upper side and continuing in a positive direction. The successive reflections of give the points, which will later become zeros of the meromorphic function mentioned above. They areReflection of the point defines the poles of the meromorphic function in the hexagons . These points in turn are reflected through the sides of the new hexagons, except for reflecting to the original hexagon . Hence each hexagon includes now 3 poles and 3 zeros. Continuation of these operations reveals that all the points have the same coefficients of rotation: , , , and displacement , . Note that reflection includes rotation and shifting and the points from one hexagon can be expressed through the points of another one. In general the points from the hexagons differ by displacements in the direction of the real and in the direction of the imaginary axes. Thus the main period is .