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So, it is a famous fact that if $u:\mathbb{R}^n \to \mathbb{R}$ is an harmonic function, then its Kelvin transform $$ (Ku)(x) := \frac{1}{|x|^{n-2}} u\left(\frac{x}{|x|^2} \right) $$ is harmonic too. Apparently, all the proofs of this fact that I have seen are some variation of "do the computations and all the terms simplify". Moreover, I already did the computation once and so I am not interested in seeing the computations done in some other equivalent form.

My point is the following: why, really, is this function harmonic? I don't buy the fact that the first person that discovered this just randomly did the computations with the correct power of the norm in front. For $n=2$ this can be proved very easily by properties of holomorphic functions and indeed there's a geometric motivation, in this case $n=2$, on why it sends harmonic functions to harmonic functions. But what for general $n$? Does anyone have at least some geometric/physical intuition on why $Ku$ is harmonic if $u$ is?

I tried many characterizations of harmonic functions like the mean value property or integrating against test functions, and all seem not to bring the result in a clean way. Indeed, at some point, there's always a big computation (being that some tangential Jacobian or Laplacian of composition) that I find basically equivalent to doing the computation from the beginning.

I am searching for a proof or at least some motivation on why would someone think that this transformation is the natural candidate to send harmonic functions to harmonic functions.

echinodermata
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Michele Caselli
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    One guess as to why you'd want to look at (a variant) of this transformation: Conformal maps on $\mathbb{C}$ send harmonic functions to harmonic functions, and in higher dimensions, reflections across spheres are the prototype of conformal transformations (see Liouville's theorem. This means looking at $u(x/|x|^2)$ which is not harmonic, but the extra factor $|x|^{2-n}$ is the fundamental solution, so you might guess it based on the computations for $u(x/|x|^2)$ and formulas for the laplacian of a product. – Jose27 Oct 15 '22 at 07:56
  • I'd be really interested to see a geometric reason for the $|x|^{2-n}$ factor, without (or at least motivated) computations! – Jose27 Oct 15 '22 at 07:57
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    @Jose27 I am not sure what you said is really convincing. If the reason for this would be that $|x|^{2-n}$ is the fundamental solution in dimension $n\ge 3$ then for $n=2$ it would be $\log(|x|) u(x/|x|^2)$ but in this case it works without the $\log$ in front. – Michele Caselli Oct 15 '22 at 23:33
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    I could have phrased it a bit better: My comment is more along the lines of "why would we ever look at something like $u(x/|x|^2)$?" A possible answer is that this inversion is conformal. In two dimensions, you stop here, since composition with inversion does give a harmonic function. In higher dimensions this is no longer true, but you might guess the factor that's missing if you run the computations. I make no claim to justify why the factor $|x|^{2-n}$ is there other than it makes things work, hence my second comment. – Jose27 Oct 16 '22 at 03:53
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    Compute the expression for $\Delta$ in $y$ coordinates under the change of coordinates $x = y/|y|^2$. The Kelvin transform pops up. See Folland's PDE book. – Mason Oct 17 '22 at 05:30

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If the pull-back of a harmonic function $H$ is some radial multiple $f(r)$ of another harmonic function, the constant function 1 must be pulled back to $f(r)$, i.e. $f(r)$ is a radial harmonic function. This motivates trying $f(r)= r^{-(n-2)}$.

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