  We study necessary conditions on the geometry and the topology of domains in
$\mathbb{R}^2$ that support a positive solution to a classical overdetermined
elliptic problem. The ideas and tools we use come from constant mean curvature
surface theory. In particular, we obtain a partial answer to a question posed
by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also
some boundedness properties of the solution $u$. Some of our results generalize
to higher dimensions.
