I have been struggling for the past couple of days to demonstrate that the Kelvin transform: $$ w(x) = |x|^{2-n}u\left(\frac{x}{|x|^2}\right) $$ is harmonic given that u(x) is harmonic. I am interested in showing this for only n=3, and I have tried to utilize spherical coordinates (a brute force computational approach is desired). My difficulty is in considering the second order partial derivatives of $u(x/|x|^2)$. I understand that this is a composite function, but I am unsure of how to consider the partial derivatives here. I would appreciate if somebody could guide me through this in spherical coordinates, or at least point me towards the necessary formula/theorem or literature for attempting this.
Asked
Active
Viewed 118 times
1
-
Hi, I have seen this question before. I am under the assumption that the proof would be convenient to compute in spherical coordinates for n=3. The linked question works in cartesian coordinates - it is a bit too heavy for my understanding. – mathWizzzz Oct 30 '22 at 13:34
-
This doesn't yet have an answer, but looks like the same question. (Just noting for posterity in the hope of avoiding duplicated effort.) – Andrew D. Hwang Oct 30 '22 at 15:13
-
See pages 111-114 of Folland's PDE book. If you consider the change of coordinates from $x \in \mathbb{R}^n \setminus 0$ to $y \in \mathbb{R}^n \setminus 0$ given by reflection across the boundary of the unit sphere, $x = \tilde{y} = \frac{1}{|y|}\frac{y}{|y|}$, then $$(\Delta u)^y(y) = |y|^{n + 2}\sum_{j = 1}^{n}\frac{\partial^2}{\partial y_j^2}(|y|^{2 - n}u^y(y)) = |y|^{n + 2}\sum_{j = 1}^{n}\frac{\partial^2}{\partial y_j^2}((Ku)(y)).$$ Here I use $f^{y}(y) = f(\tilde{y})$ for the coordinate representation of a function $f$ in $y$-coordinates. – Mason Oct 30 '22 at 22:24