The problem is to prove that Kelvin transformation preserves harmonic functions. That is
If $u$ is a harmonic function, then $v(x)=|x|^{2-n}u(\frac{x}{|x|^2})$ is harmonic whenever it is defined.
I know there is a way to calculate $\Delta v$ explicitly ($\Delta v=|x|^{-n-2}\Delta u(x/|x|^2)$), which will give me the conclusion. But I was told that this fact can be proven easily by using the fact that if $\int_{\Omega}\langle \nabla v,\nabla \phi\rangle dx=0$, for any test function $\phi$ with $\phi|_{\partial \Omega}=0$, then $v$ is harmonic.
So I calculated $\nabla v$ and yielded that it is equal to $$(2-n)|x|^{-n}u(\frac{x}{|x|^2})x+|x|^{-n}\nabla u(\frac{x}{|x|^2})-2|x|^{-2-n}(\nabla u(\frac{x}{|x|^2})\cdot x)x $$ Then I don't know what to do. Can somebody give any idea how this should work?
