Spherical rigidity for an exterior overdetermined problem with Neumann data prescribed by mean curvature

Abstract

We study an overdetermined elliptic free boundary problem for exterior domains in $\mathbb{R}^N$, $N \ge 2$, introduced by F. Morabito [Comm. PDE 46 (2021), 1137-1161]. The overdetermining condition prescribes the Neumann data as a multiple of the boundary mean curvature, with parameter $\Gamma$, together with a spherical compatibility condition. For $N \ge 3$, we prove rigidity of the spherical solution among star-shaped domains when $\Gamma \ge N-2$; in the borderline case $\Gamma = N-2$, the star-shapedness assumption can be removed, and rigidity holds among all bounded domains. The proof combines the Pohozaev identity, geometric identities, and the sharp boundary inequality of Agostiniani and Mazzieri for capacitary potentials. We also obtain rigidity among bounded domains for $\Gamma \le 0$ via Serrin's moving plane method. In dimension two, the unit disc is the only admissible domain for every $\Gamma$.