content_image/308002.jpg content_image/308005.jpg content_image/308003.jpg content_image/308004.jpg content_image/308006.jpg
By Remark 2 and taking into account Lemma 2, we see that the functions ̂ u , $^{∂$_{η}$}$̂ u are continuously differentiable in the region D × G ′ for both cases η > 0 and η < 0. Then using (4.3), we obtain that for both cases η > 0 and η < 0 the function ̂ u satisfies the equation 

in the classical sense. 

<img src='content_image/308002.jpg'>

Putting ̂ u = p + √ − 1 q and separating the real and the imaginary parts of the left-hand side of equation (4.4) for both η > 0 and η < 0, we have the following equations with respect to the functions ∂$_{η}$p and ∂$_{η}$ q respectively: 

where 

<img src='content_image/308005.jpg'>

<img src='content_image/308003.jpg'>

<img src='content_image/308004.jpg'>

In this work, the uniqueness of the solution of the problem is investigated under the assumption of the existence of the solution . Hence it is assumed that there exists a solution u ( x, ξ ) to equation (3.5), i.e., there exists a solution ̂ u = p + √ − 1 q to equation (4.3) which satisfies the properties indicated in Remark 2 and the condition 

<img src='content_image/308006.jpg'>

Lemma 3. Let the conditions of Lemma 1 and condition (3.6) be satisfied. 

