Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

We consider a convex solid cone $\mathcal{C}\subset {\mathbb{R}}^{n+1}$ with vertex at the origin and boundary $\mathrm{\partial }\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $\mathrm{\Sigma }$ immersed in $\mathcal{C}$ with free boundary in $\mathrm{\partial }\mathcal{C}\setminus \{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape.