Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

Abstract

AbstractWe consider a convex solid cone $$\mathcal {C}\subset \mathbb {R}^{n+1}$$ C ⊂ R n + 1 with vertex at the origin and boundary $$\partial \mathcal {C}$$ ∂ C smooth away from 0. Our main result shows that a compact two-sided hypersurface $$\Sigma $$ Σ immersed in $$\mathcal {C}$$ C with free boundary in $$\partial \mathcal {C}\setminus \{0\}$$ ∂ C \ { 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 $$\mathcal {C}$$ C provided the boundary of $$\Sigma $$ Σ is away from the singular set of $$\partial \mathcal {C}$$ ∂ C .