How can I find the Green's function for the subspace: $K=\{(x,y)\in R^2,\quad y>0\}$?
To find the Green's function of the Laplacian for the freespace, I solved the problem: $\bigtriangledown^2G(r,0)=\delta(x)$, with $r\neq0$. Thus: $$\bigtriangledown^2G(r,0)=0\RightarrowG_{rr}\frac1rG_r=0\Rightarrow G(r,0)=c\ln{r}$$
to find $c$:
$$\int_{R^2}\bigtriangledown{G(r,0)}\bigtriangledown{\phi(r)}drd\theta=\phi(0)\Rightarrow c\int_{0}^{2\pi}d\theta \int_{0}^{\infty}\frac{d\phi}{dr}dr=\phi(0) \Rightarrow c=\frac{1}{2\pi}$$ So: $$ G(r,0)=\frac{1}{2\pi}\ln{r}=\frac{1}{2\pi}\ln{\frac1r}$$
For the subspace $K$, I'm thinking of repeating the same process, with setting the upper limit of the polar angle integral from $\theta=2\pi$ to $\pi$, since $y>0$. Is this right?
