## Global harmonic miracle

That harmonic functions would satisfy the mean value property for infinitesimally small circles is intuitively somewhat clear. But that this property should persist on circles of arbitrary radius is quite miraculous. To recall, if $h$ is harmonic on a domain $\Omega\subset{\mathbb C}$ and one considers a disk $D(z_0,r)\subset\Omega$, then

$\displaystyle h(z_0)=\frac{1}{\pi r^2}\int_{D(z_0,r)}h(w)dA(w)$.

One might wonder whether such a global property occurs for other shapes, say ellipses. The answer is no.

Fact: Suppose $U\subset {\mathbb C}$ is open and has finite area $A$, $z_0\in U$, and

$\displaystyle h(z_0)=\frac{1}{A}\int_U h(w)dA(w)$

for every integrable harmonic function $h$ on $U$. Then $U$ is a disk centered at $z_0$.

To see why, pick a point $z_1\not\in U$ that is closest to $z_0$. Then play with the function

$\displaystyle h(z)=2{\mathcal Re}\left(\frac{z-z_0}{z-z_1}\right).$