Stable capillary hypersurfaces in a half-space or a slab

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