# Question on Laplace eq. in a ball

1. Nov 15, 2013

### yungman

For $\nabla^2 u(r,\theta, \phi)=0$, $u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi)$.

But I have issue with this, for spherical coordinates:

$$\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)$$

Let $u=R(r)Y(\theta,\phi)$ where $Y(\theta,\phi)$ is the spherical harmonics.

$$\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0$$
and
$$\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partial {\phi}^2}+\mu Y=0$$

For Euler equation: $r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-\mu R=0$ where $\mu=n^2$. and the solution is $R=r^n$.

Here, because of the condition, only $\mu=n(n+1)$ is used for bounded solution.
$$r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R$$

Which gives

$$R=r^{(n+1/2)}$$

What have I done wrong?

Last edited: Nov 15, 2013