I learned the following theorem in Folland's Introduction to Partial Differential Equations(p.69 Chapter 2):
Suppose $u$ is harmonic on an open set $\Omega\subset{\mathbb R}^n$. If $x\in\Omega$ and $r>0$ is small enough so that $\overline{B_r(x)}\subset\Omega$, then $u(x)=\frac{1}{r^{n-1}\omega_n}\int_{S_r(x)}u(y)d\sigma(y)=\frac{1}{\omega_n}\int_{S_1(0)}u(x+ry)d\sigma(y),$ where $\omega_n=\frac{2\pi^{n/2}}{\Gamma(n/2)}.$
I found that I could not immediately reconstruct a proof for the theorem. A key point is that one needs to use the Green's identity, which is a basic property of harmonic functions. But I don't see any "clue" that how people actually come up with this theorem and such proof. (Maybe this is the common problem, at least for me, for most of the textbooks.) A curious search in Google returns nothing satisfactory to me. Since this is a basic property of harmonic functions, I am wondering that if one needs to know this history of harmonic functions in order to know this theorem well.
Here is my question:
- Can any one here come up with a motivation of this theorem in PDE?
My second question may be more vague:
- How can I approach the proof of this theorem more "naturally" instead of just remembering bunch of facts? (In the language of Polya, any heuristics here?)