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I am referring to the following post: Estimate the Green function for the Laplace equation in 2D, where there has been found an explicit expression for the integral $$\int\limits_{|y| \leq 1} G(x,y) \, dy$$ where $G$ is the Green function for the Dirichlet problem for the Laplace equation in the two dimensional unit disk.

Now I am trying to find out whether the following integral exists and if it does, how one could find an upper bound:

$$\int\limits_{|y| \leq 1} (G(x,y))^2 \, dy$$

Or even better, I am interested in the $L^2$ norm of $G$ over the unit disk (if it exists).

The culculaiton above will probably not work as we have to deal with a more complicated integral. So does anybody have an idea?

Mimimi
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    Sorry to ask another question, instead of answering, but how can you be sure that the integral is finite? – Giuseppe Negro Dec 26 '19 at 23:32
  • @GiuseppeNegro I am not sure. I should better ask whether it can be shown that it is finite and if yes, how it can be bounded. I will add this. Thanks! – Mimimi Dec 26 '19 at 23:39
  • Do I misunderstand things, or isn't $G(x,y)$ a constant multiple of $\log|x-y|$? – paul garrett Dec 27 '19 at 00:03
  • @paulgarrett Kind of...It is $G(x,y) = \frac{1}{2\pi}\ln \frac{|x-y|}{|x| \bigl|y - \frac{x}{|x|^2}\bigr|}$. Is the existence/non-existence obvious? – Mimimi Dec 27 '19 at 00:09
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    Ah, sorry, I did not read your question carefully... was thinking of the free-space Green's function... But, still, this is such a symmetrical situation that something should be clear. (Give me some hours... I'll come back to this if no one else has anything to say...) – paul garrett Dec 27 '19 at 00:13
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    I am too lazy/busy at the moment to carry this out, but I would treat the disc as being in $\mathbb C$, and use the fact that Moebius/linear-fractional transformations are transitive on the disk, and are holomorphic. Thus, with $\log |z|$ holomorphic except at $0$, use a Moebius transform to move $0$ to any other $w$ in the disk... – paul garrett Dec 27 '19 at 16:42
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    I have this vague idea of using the Cayley transform as a change of variable, so that we are reduced to $\iint_{y>0} G(x, y)^2 J(x,y), dxdy, $ where $G$ is now the Green function of the upper half-plane and $w$ is the Jacobian of the Cayley transform. This should be slightly easier but I am totally out of focus in these days and I cannot perform the computation right now. – Giuseppe Negro Dec 31 '19 at 13:51

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