The theorem state:
If $u \in C^2(U)$ satisfies
$$ u(x) = \frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)} u(y) dS(y)$$
for each ball $B(x,r) \in U$, then u is harmonic.
The issue that I have is with the proof of the theorem. He asserts to show it by contradiction, to assume that $\Delta u > 0$. Then for
$$\phi(r) = \frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)} u(y) dS(y)$$
$$ 0 = \phi ' (r) = \frac{r}{n}\frac{1}{|B(x,r)|} \int_{B(x,r)} \Delta u(y) dy >0$$
a contradiction.
My issue is that I'm pretty sure that we show $\phi ' (r) = 0$ in the opposite direction by using the fact that $u$ is harmonic in the first place, so I don't see how we can use that fact here, especially when we're assuming the opposite. I feel like there is something very sly going on here. Can someone explain the proof?
Best Answer
There is nothing sly going on here: the reason Evans claims $\phi'(r) = 0$ is for an entirely different reason this time around. Namely, the mean-value property in the hypothesis is precisely the statement that $\phi(r)$ is constant in $r$, and in particular equal to $u(x)$.