On stable capillary hypersurfaces with planar boundaries

We study stable immersed capillary hypersurfaces $\Sigma$ in domains B of R n+1 bounded by hyperplanes. When B is a half-space, we show $\Sigma$ is a spherical cap. When B is a domain bounded by k hyperplanes P 1 ,. .. , P k , 2 $\le$ k $\le$ n + 1, having independent normals, and $\Sigma$ has contact angle $\theta$ i with P i and does not touch the vertices of B, we prove there exists $\delta$ > 0, depending only on P 1 ,. .. , P k , so that if $\theta$ i $\in$ ($\pi$ 2 -- $\delta$, $\pi$ 2 + $\delta$) for each i, then $\Sigma$ has to be a piece of a sphere.