  We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard,
namely, to characterize all the domains in the plane that admit a "roof
function", i.e., a positive harmonic function which solves simultaneously a
Dirichlet problem with null boundary data, and a Neumann problem with constant
boundary data. Under some a priori assumptions, we show that the only three
examples are the exterior of a disk, a halfplane, and a nontrivial example. We
show that in four dimensions the nontrivial simply connected example does not
have any axially symmetric analog containing its own axis of symmetry.
