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.

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

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.

We consider a polygonal domain with corner points. The half hexagon denoted as

Hexagons.

Obviously, the repeated reflections of the point

In general, all reflection points are either given by

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

The half hexagon can be viewed as the complement of the intersection of four half planes. We define them by

Let then

The Poisson kernels can be found from the respective Green functions

For the half plane

For the half plane

The boundary of the half plane

Finally, for the half plane

The method of reflections helps to find the harmonic Green function; see [

For the boundary part

The following lemmas will be needed to prove the Green representation formula below. The complete proofs of these lemmas are given in [

The infinite product

The equalities

The proof of this equality is based on the fact that the functions

The Green function must satisfy the following conditions; see [

The function

Any

We consider now the different forms of the Green function and take the derivatives

For the right-hand side, a boundary

For the boundary part

The representation formula in Theorem

At first the boundary behavior of the integral is to be studied. Let for

For

Let

For

Similarly, for the rest parts of the boundary

On

For

On

For

Obviously, similar calculations on the boundary parts imply the related sums to be convergent to zero, except for the boundary part

On

In the next lemma the boundary behavior of the function

If

The proof of this lemma is given in detail in [

The Dirichlet problem fo the Poisson equation

We need to prove that (

In order to construct the Pompeiu-type operator we consider the following term:

The author declares that there is no conflict of interests regarding the publication of this paper.