Determine Green's function for the annular region bounded by two concentric spheres in $\mathbb R^n$.

Question

Determine Green's function for the annular region bounded by two concentric spheres in $\mathbb R^n$.

Don't know where to start. It's actually a problem in Trudinger's PDE book. Any hints would be appreciated.

Answer 1

I think this is very similar to the case of determining the green function in an infinite strip, say $ |x_3| < L$ . The key word is method of images. Consider first how you would go about determining the green function of upper half plane ( or for that matter the green function in a sphere) ; This is done by looking at the fundamental Green function $G(x;x_0)$ and then introducing a reflection point $ x_0^{*}$ to the x-axis and then simply looking at $ G(x;x_0) - k G(x;x_0^{*})$ For the case of upper half plane $k=1$ and $x_0^{*}$ is the reflection of $x_0$ to x-axis. In the case of the sphere, $x_0^{*}$ is the inverse of the point $x_0$.

Now when one has an infinite strip or an annular region this method can be utilized via an infinite reflection procedure much like how you get infinite reflections of an image sitting in between two mirrors. Thus without much difficulty you should be able to get the green function in the annular region from the fundamental green function via an infinite alternating series.