Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

Abstract

We consider a convex solid cone C in R^{n+1} with vertex at the origin and boundary smooth away from 0. Our main result shows that a compact two-sided hypersurface Sigma immersed in C with free boundary away from 0 and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape. The technique of proof also works for a non-smooth convex cone C provided the boundary of Sigma is away from the singular set of the boundary of C.