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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.

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migrated from physics.stackexchange.com Mar 10 '16 at 18:00

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  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Mar 10 '16 at 17:35
  • $\begingroup$ Maybe, but I wasn't sure, since it's a more practical question I thought that someone here would have already done something similar. $\endgroup$ – Hydro Guy Mar 10 '16 at 18:00

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