Stable capillary hypersurfaces in a half-space or a slab

Abstract

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $Σ$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $Σ$ has genus zero, (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $π/2$ and each component of $\partialΣ$ is embedded, (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<π/2$ and each component of $\partialΣ$ is embedded. Moreover, in case (2), if not a right circular cylinder, the hypersurface has to be graphical over a domain in $\partial\mathcal B.$ In case (3), the hypersurface is a spherical cap.