Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains

Abstract

We consider two physically motivated problems: (1) Suppose the surface of a body in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ is charged with a constant density. If the induced single-layer potential is constant inside the body, does it have to be a ball? (2) Suppose a straight solid cylinder of unknown cross-section is dipped into a large plain liquid reservoir. If the liquid rises to the same height on the cylinder wall, does the cylinder necessarily have circular cross-section? Both questions are answered with yes, and both problems are shown to be of the type

where ${\partial }_{u}f\le 0$ and $\Omega ={\mathbb{R}}^N\bar{G}$ is the connected exterior of the smooth bounded domain $G$. The overdetermined nature of this possibly degenerate boundary value problem forces $\Omega$ to be radial. This is shown by a variant of the Alexandroff-Serrin method of moving hyperplanes, as recently developed for exterior domains by the author in [19]. The results extend to Monge-Ampere equations.