## Green function for annular region

Problem 2.5 Determine the Green’s function for the annular region bounded by two concenteric spheres in ${\mathbb{R}^n}$

Proof: Suppose the annular region is ${\Omega=B_R(0)/B_\rho(0)}$ and denote ${\lambda=\rho/R<1}$ and ${\displaystyle \bar{x}=\frac{R^2x}{|x|^2}}$, and ${\displaystyle \tilde{x}=\frac{r^2x}{|x|^2}}$.
Firstly

$\displaystyle h_1=\Gamma(x-y)-\Gamma(\frac{|x|}{R}|\bar{x}-y|)\quad x,y\in \Omega, x\neq y$

is a harmonic function in ${\Omega}$ and ${h_1|_{\partial B_R}=0}$ but not necessarily 0 on ${\partial B_\rho}$.

$\displaystyle h_2=\Gamma(\frac{|x|}{r}|\tilde{x}-y|)-\Gamma(\frac{|x|}{R}\frac{|\bar{x}|}{r}|\lambda^2x-y|)=\Gamma(\frac{|x|}{r}|\tilde{x}-y|)-\Gamma(\lambda^{-1}|\lambda^2x-y|)$

Then ${h_1-h_2}$ is harmonic in ${\Omega}$ and vanish on ${\partial B_\rho}$. Similarly construct

$\displaystyle h_3=\Gamma(\lambda|\lambda^{-2}x-y|)-\Gamma(\lambda\frac{|x|}{R}|\lambda^{-2}\bar{x}-y|)$

Then ${h_1-h_2+h_3}$ is harmonic in ${\Omega}$ and vanish on ${\partial B_R}$. We hope that

$\displaystyle h=h_1-h_2+h_3-h_4+h_5-\cdots$

is convergent, then ${h}$ is harmonic in ${\Omega}$ and vanish on both boundaries. Formally we get

$\displaystyle h=\Gamma(|x-y|)+\sum\limits_{k=1}^{\infty}[\Gamma(\lambda^{-k}|\lambda^{2k}x-y|)+\Gamma(\lambda^{k}|\lambda^{-2k}x-y|)]$

$\displaystyle -\sum\limits_{k=0}^{\infty}[\Gamma(\frac{|x|}{R}\lambda^{k}|\lambda^{-2k}\bar{x}-y|)+\Gamma(\frac{|x|}{r}\lambda^{-k}|\lambda^{2k}\tilde{x}-y|)]$

${h}$ is well defined, because the terms of infinite sum are geometric series, it is unifomly convergent when ${x\neq y}$. And very term is harmonic, thus ${h}$ is harmonic when ${x\neq y}$. One can verify that ${h}$ is 0 on the ${\partial \Omega}$.

$\Box$