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Green's function for a subspace of $R^2$

Multivariate Calculus
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Green's function for a subspace of $R^2$


Post by andrew.tzeva » Wed May 23, 2018 4:47 pm

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 free-space, I solved the problem: $-\bigtriangledown^2G(r,0)=\delta(x)$, with $r\neq0$. Thus:
$$-\bigtriangledown^2G(r,0)=0\Rightarrow-G_{rr}-\frac1rG_r=0\Rightarrow G(r,0)=c\ln{r}$$

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?
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