1

Let $u(x)$ be a harmonic function in $\mathbb{R^n}$ and $A$ an orthogonal matrix then I wish to show that $u(Ax)$ is also harmonic. I understand how to do this by showing that $$\triangle{u(Ax)}=0$$ But would like to show this using the Mean value property and believe that I need to show that

$$u(Ax)=1/\alpha(n)r^n\int_{B(x,r)}u(Ay)dy=1/n\alpha(n)r^{n-1}\int_{\partial{B(x,r)}}u(Ay)dS(y)$$ However, I cannot seem to get started on this at all.

OEB
  • 21
  • Look at this: https://math.stackexchange.com/questions/50274/intuitive-interpretation-of-the-laplacian/50285#50285 – Christian Blatter Nov 20 '19 at 18:50
  • Thank you Christian unfortunately due to my lack of expertise I am unable to progress. Further direction would be appreciated. – OEB Nov 21 '19 at 15:29

0 Answers0