I was looking for the Green Function of the following problem, on the upper half plane:
$ u : \mathbb{R}\times \mathbb{R}^+\rightarrow \mathbb{R}^2$, $\mathbb{R}^+$ corresponds to the non-negative positive real axis.
$\omega: \mathbb{R}\times \mathbb{R}^+\rightarrow \mathbb{R}$ prescribed
$\partial_xu_x+\partial_y u_y = 0$
$\partial_x u_y - \partial_y u_x = \omega $
$u_x(x,y=0) = u_y(x,y=0) = 0 $
In this case, the Green Function would correspond to the case $\omega = \delta(y-y_0) \delta(x-x_0)$
Alternatively:
$\psi : \mathbb{R}\times \mathbb{R}^+\rightarrow \mathbb{R}$
$ u_x = \partial_y \psi$
$ u_y = -\partial_x \psi$
$- (\partial_x^2+\partial_y^2)\psi = -\nabla^2 \psi = \omega$
$\partial_x \psi(x,y=0) = \partial_y\psi(x,y=0) = 0$
Do anyone have any good sugestion where to look for green functions of this problem? Any references for books or papers are more than welcome.
I know how to work with Green-Functions without boundary constraints, but I don't think that techniques with Fourier transform would work in this particular case.