Green's function in elliptic box

[Einj]

Hello everyone! Does anyone know if there is a know expression for the Green's function for Poisson's equation that vanishes on an ellipse in 2 dimensions?
I'm essentially looking for a solution to:
$$
\nabla^2G(\vec x-\vec x_0)=\delta^2(\vec x-\vec x_0)
$$
in 2 dimensions where
$$G(\vec x-\vec x_0)=0$$
when $\vec x$ lies on an ellipse.
The solution for a circle is well know but I wanted to know if there any kind of generalization.

Thanks a lot!

 

[the_wolfman]

If I remember correctly Laplace's equation on an ellipse is separable if you use elliptical coordinates, and the solution can be expressed in terms of Mathieu functions. From this one should be able to construct the Green's function.

 

[Einj]

Well, I don't really want to solve Laplace equation since the Laplacian of the Green's function is not equal to zero but to the Dirac delta. Do you think the same technique still holds? Do you have any reference?

 

[the_wolfman]

If you think about it the Green's function equation is equal to Laplace's equation everywhere except for at one point. You can use the solution to Laplace's equation to construct the Green's function. It takes a little bit of algebra, but it is tractable. Of course there are other ways to solve for the Green's function. I bet most of them will probably use elliptical coordinates and Mathieu funtions.

I don't have references on hand, but Googling elliptical greens function produces some promising hits.

 

[Einj]

Ok I will definitely do that! Thanks a lot.