Stable $(r+1)$-th capillary hypersurfaces

In this paper, we propose a new definition of stable $(r+1)$-th capillary hypersurfaces from variational perspective for any $1\leq r\leq n-1$. More precisely, we define stable $(r+1)$-th capillary hypersurfaces to be smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. Using the new concept of the stable $(r+1)$-th capillary hypersurfaces, we generalize the stability results of Souam \cite{Souam} in a Euclidean half-space and Guo-Wang-Xia \cite{GWX} in a horoball in hyperbolic space for capillary hypersurface to $(r+1)$-th capillary hypersurface case.