example of harmonic function on sphere

Question

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren hemisphere.

Answer 1

Let D be any connected, compact domain without boundary such as $S^2$. (1) By the maximum principle, a harmonic function having an interior extremum point on a connected set D must be constant. (2) Being continuous, a harmonic function on a compact set D must have an extremum point. (3) If D has no boundary points, then the extremum point must occur inside the domain. Thus on D, any harmonic function must be constant.

Answer 2

I didn't realize until just now how old a question this was. Oh well, to solve the problem we need $1$ integral: $$\begin{align}\int_0^1P_{\ell}(x)dx&=\frac1{2^{\ell}\ell!}\int_0^1\frac{d^{\ell}}{dx^{\ell}}(x^2-1)^{\ell}dx=\left.\frac1{2^{\ell}\ell!}\frac{d^{\ell-1}}{dx^{\ell-1}}(x^2-1)^{\ell}\right|_0^1\\ &=\left.-\frac1{2^{\ell}\ell!}\sum_{k=0}^{\ell}{\ell\choose k}(-1)^{\ell-k}\frac{d^{\ell-1}}{dx^{\ell-1}}x^{2k}\right|_{x=0}\\ &=-\frac1{2^{\ell}\ell!}\sum_{k=0}^{\ell}\left(\frac{\ell!}{k!(\ell-k)!}\right)(-1)^{\ell-k}(\ell-1)!\delta_{\ell-1,2k}\\ &=\begin{cases}0&\ell\text{ even}\\ \frac{(-1)^{\frac{\ell-1}2}(\ell-1)!}{2^{\ell}\left(\frac{\ell-1}2\right)!\left(\frac{\ell+1}2\right)!}&\ell\text{ odd}\end{cases}\end{align}$$ Then if we let $$f(r,\theta,\phi)=\sum_{\ell=0}^{\infty}\left(a_{\ell}r^{\ell}+\frac{b_{\ell}}{r^{\ell+1}}\right)P_{\ell}(\cos\theta)$$ And $$g(\theta)=\begin{cases}1&0\le\theta<\pi/2\\-1&\pi/2<\theta<\pi\end{cases}$$ So $f(1,\theta,\phi)=g(\theta)$. We get $$\begin{align}\frac{4\pi}{2\ell+1}(a_{\ell}+b_{\ell})&=\sum_{\ell^{\prime}=0}^{\infty}\left(a_{\ell^{\prime}}+b_{\ell^{\prime}}\right)\int_0^{2\pi}\int_0^{\pi}P_{\ell^{\prime}}(\cos\theta)P_{\ell}(\cos\theta)\sin\theta\,d\theta\,d\phi\\ &=\sum_{\ell^{\prime}=0}^{\infty}\left(a_{\ell^{\prime}}+b_{\ell^{\prime}}\right)\frac{4\pi}{2\ell+1}\delta_{\ell\ell^{\prime}}\\ &=\int_0^{2\pi}\int_0^{\pi}f(1,\theta,\phi)P_{\ell}(\cos\theta)\sin\theta\,d\theta\,d\phi\\ &=2\pi\int_0^{\pi}g(\theta)P_{\ell}(\cos\theta)\sin\theta\,d\theta\\ &=2\pi\int_0^{\pi/2}P_{\ell}(\cos\theta)\sin\theta\,d\theta-2\pi\int_{\pi/2}^{\pi}P_{\ell}(\cos\theta)\sin\theta\,d\theta\\ &=2\pi\int_0^{\pi/2}P_{\ell}(\cos\theta)\sin\theta\,d\theta-2\pi\int_{\pi/2}^0P_{\ell}(-\cos\theta)\sin\theta\,(-d\theta)\\ &=2\pi[1-(-1)^{\ell}]\int_0^{\pi/2}P_{\ell}(\cos\theta)\sin\theta\,d\theta\\ &=2\pi[1-(-1)^{\ell}]\int_0^1P_{\ell}(x)dx\end{align}$$ So $$a_{2\ell}+b_{2\ell}=0$$ and $$a_{2\ell+1}+b_{2\ell+1}=(4\ell+3)\frac{(-1)^{\ell}(2\ell)!}{2^{2\ell+1}\ell!(\ell+1)!}$$ We would need more boundary conditions to sort out the relative contributions of $a_{\ell}$ vs. $b_{\ell}$. Sort of reminds me of a problem from J. D. Jackson.

EDIT In fact this is problems 3.1 and 3.2 of the above reference.