On stable capillary hypersurfaces with planar boundaries

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

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