Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 5 / i / Solution

Let $B_r(x_0)$ have closure contained in the domain of a harmonic function $u$. Put ${\omega }_n=|B_1(0)|$, so $|B_r|={\omega }_n{r}^n$ and $|\partial B_r|=n{\omega }_n{r}^{n-1}$. The sphere and ball mean value properties for harmonic functions areTo prove the sphere formula, define its average using a fixed unit sphere:Differentiating under the integral and applying the divergence theorem givesContinuity gives $M(r)\to u(x_0)$ as $r\downarrow 0$, proving the spherical average identity. Integrating that identity over radii with the polar-coordinate weight yieldswhich proves the ball identity. For $n=1$, the sphere average is the average of the two endpoints, and the same formulas hold with the counting surface measure.