Stable anisotropic capillary hypersurfaces in a half-space

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape. On the other hand, we prove a Bernstein-type theorem for stable anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area growth assumption.