Overdetermined Neumann boundary value problem in unbounded domains

The studying of overdetermined boundary value problems for elliptic partial differential equations was initiated by J. Serrin in 1971. In his work, he established a property of radial symmetry for solutions of some overdetermined Poisson problem. Apart of a significant independent interest, the problems of such kind have important applications in the potential theory, integral geometry, hydrodynamics and capillarity theory. Usually, the resolving of these problems is based on Hopf lemma on an angular boundary point and the method of hyperplanes motion introduced by A.A. Alexandrov for studying some geometric problems related with characterizing the spheres. Among other more modern methods not involving the maximum principle for the considered problems we mention the duality method, the method of volume derivative as well as an integral method.
In the present paper we consider an overdetermined Neumann problem for the Laplace equation $\Delta f=0$ in planar unbounded domains. We show that under some conditions, see Theorem 1 in Section 1, such problem is solvable only for the exterior of a ball. A specific feature of Theorem 1 is that in this theorem, for the first time, we obtain an exact condition for the growth of $f$ at infinity. Moreover, as Theorem 2 in Section 2 shows, other conditions in Theorem 1 are also necessary. In contrast to the earlier works, the proof of Theorem 1 employs some boundary properties of conformal mappings, Smirnov theorem on functions in a class $H_p$ and Fejer-Riesz theorem on non-negative trigonometrical polynomials.