An overdetermined problem in potential theory (Q374429)

It is trivial to observe that, if \(\Omega \) is a half-space in \(\mathbb{R}^{N}\), or the exterior of a closed ball, then there is a positive harmonic function on \(\Omega \) that vanishes on \(\partial \Omega \) and has constant normal derivative there. It has been shown by \textit{L. Hauswirth, F. Hélein} and \textit{F. Pacard} [Pac. J. Math. 250, No. 2, 319--334 (2011; Zbl 1211.35207)] that this phenomemon also occurs in a much less obvious case, namely where \(\Omega \) is the plane domain \(\Omega _{0}=\{\left| y\right| <\pi /2+\cosh x\}\). They suggested that these were essentially the only possibilities in the plane, and also that there might be higher dimensional axially symmetric analogues of \(\Omega _{0}\). The present paper addresses both these questions. In the case of the plane, it shows that these are indeed the only possibilities provided that (i) \(\Omega \) is a Smirnov domain, and (ii) either \(\Omega \) is simply connected or \(\mathbb{R}^{2}\backslash \Omega \) is a continuum. It also shows that, in \(\mathbb{R}^{4}\), there is no axially symmetric analogue of \(\Omega _{0}\) that contains its axis of symmetry.NEWLINENEWLINEThe authors note that \textit{M. Traizet} [``Classification of the solutions to an overdetermined elleptic problem in the plane'', Preprint, \url{arXiv 1301.6927}] has independently, and by different methods, shown that the above three examples of plane domains with the desired properties are the only ones having finitely many boundary components.