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 $\Bbb R^{n+1}.$ We prove that such a hypersurface $\Sigma$ is rotationally symmetric in the following cases:   (1) $n=2$, $\mathcal B$ is a slab and $\Sigma$ has genus zero,   (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $\pi/2$ and each component of $\partial\Sigma$ is embedded,   (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<\pi/2$ and each component of $\partial\Sigma$ 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. 
Abdelhamid Ainouz and Rabah Souam
math.DG