Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

The basic boundary value problems for the second-order complex partial differential equations are the harmonic Dirichlet andNeumannproblems for the Laplace andPoisson equations. In order to find the solution in explicit or closed form diverse methods have been applied. In case a given domain D is simply connected and has a piecewise smooth boundary ∂D the tools of complex analysis such as Schwarz reflection principle and conformal mapping serve perfectly. When a given domain D 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 C by reflection of the given domain D 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 theDirichlet 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-reflectionmethod.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.


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 C 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][2][3][4][5][6][7][8][9][10][11][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.

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, 1 +  √ 3, −1 +  √ 3, and −2 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 are Reflection of the point  ∈  defines the poles of the meromorphic function in the hexagons  1 , . . .,  6 .These points in turn are reflected through the sides of the new hexagons, except for reflecting to the original hexagon .
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 6 in the direction of the real and 2 √ 3 in the direction of the imaginary axes.Thus the main period is Obviously, the repeated reflections of the point  ∈  + are representable in different ways, using either of the points which are connected by the relations ž 2 =  1 − 6 − 2 √ 3 and In general, all reflection points are either given by or by where   = 3 +  √ 3 such that  +  ∈ 2Z.
We choose zeros as direct reflection of poles and poles as direct reflection of zeros.Then having a set of zeros and a set of poles, one can construct the Schwarz kernel for  + and treat the related Schwarz problem [9] and Riemann-Hilbert-type boundary value problem.
The half hexagon can be viewed as the complement of the intersection of four half planes.We define them by  − 1 being the right-hand half plane which has the boundary line passing through the points 2 and 1 +  √ 3,  − 2 being the upper half plane with the border line through the points ±1 +  √ 3,  − 3 being the left-hand half plane with the border line passing through the points −1 +  √ 3 and −2, and  − 4 being the half plane which is below the real axis.
Let then  + 1 ,  + 2 ,  + 3 ,  + 4 be the complementary half planes of those listed above.The Green functions of these half planes are, in fact, the Green functions for the complementary half planes  − 1 , . . .,  − 4 .The outward normal derivatives of the Green function on the boundary is the Poisson kernel.The kernel provides the boundary condition  =  in the Dirichlet problem.
The Poisson kernels can be found from the respective Green functions  1 (, ),  =  + ,  =  +  as described below.
For the half plane  + 1 with the boundary described by the relation where For the half plane  + 2 the relation on the boundary is given as  =  + 2 √ 3; then here where Finally, for the half plane  + 4 with the boundary described by  = , we have

Green Representation Formula
The method of reflections helps to find the harmonic Green function; see [3][4][5].The reflection points given in (3) or ( 4) are used to construct a meromorphic function: where where Here  is considered as a parameter and  ∈ C is the variable.
The following lemmas will be needed to prove the Green representation formula below.The complete proofs of these lemmas are given in [9].
We consider now the different forms of the Green function and take the derivatives    1 (, ),    1 (, ).
Proof.Let  0 be defined on different boundary parts and consider the boundary behavior when  →  0 .