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In Evans' book on PDE it is stated (p.50 in my edition) that a Green's function for solving Laplace's equation on $B_1(0)$ is given by $$G(x,y)=\Phi(y-x)-\Phi(|x|\cdot (y-\tilde x)),$$ where $x,y\in B_1(0)$, $x\not= y$, $\tilde x:=\frac{x}{|x|^2}, x\not=0$ and $\Phi$ denotes the fundamental solution of Laplace's equation on $\mathbb{R}^n\setminus\{0\}$. But why can one define $G$ for $x=0$ like this? $\Phi$ has a singularity at $x=0$, it is not even defined there.

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  • $\begingroup$ I think that when the author says that $G(x,y) = \Phi(y - x) - \Phi(|x|(y - \tilde{x}))$ for $x \neq y$ he is implicitly assuming that $x \neq 0$ since otherwise $\tilde{x}$ would not be defined. $\endgroup$ – Giovanni Oct 4 '15 at 23:50
  • $\begingroup$ Ok - so we found a Green's function on $B_1(0)\setminus\{0\}$ instead of the entire ball. $\endgroup$ – JohnSmith Oct 4 '15 at 23:58
  • $\begingroup$ @Giovanni Not defined literally, but the limit as $x\to 0$ exists. $\endgroup$ – user147263 Oct 6 '15 at 5:31
  • $\begingroup$ No, this is Green's function of $B_1(0)$. $\endgroup$ – user147263 Oct 6 '15 at 5:35
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The singularity at $x=0$ is removable, provided $y\ne 0$. Indeed, as $x\to 0$, we have $|x|y\to 0$. The vector $|x|\tilde x$ does not have a limit as $x\to 0$, but its magnitude stays at $1$, and $\Phi$ is radially symmetric. So, $\Phi(|x|(y-\tilde x)) $ has a limit as $x\to 0$ (it's whatever value $\Phi$ has on the unit sphere), and this is used to extend the definition of $G$ to the case $x=0$.

The only singularity $G$ has is when $y=x$, but this is to be expected from Green's function.

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  • $\begingroup$ that's a very good point! $\endgroup$ – Giovanni Oct 6 '15 at 5:40

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