Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

Abstract

Abstract. We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

“…Many kinds of relative isoperimetric inequalities have been studied for manifolds-with-boundary (see e.g. a survey [43]), including singular boundaries of sectorial type [5][6][7][8][9]15,32] (or more generally of conical type; see [4,29,37,41], and also [14, Sect. 5] and references therein).…”

Differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with $$C^0$$-locally asymptotic bounded geometry

“…If M is compact, classical compactness arguments of geometric measure theory combined with the direct method of the calculus of variations provide a short proof of the continuity of I M in any dimension n, [1] Proposition 1. Finally, if M is complete, non-compact, and V(M) < +∞ , an easy consequence of Theorem 2.1 in [19] yields the possibility of extending the same compactness argument valid in the compact case and to prove the continuity of the isoperimetric profile, see for instance Corollary 2.4 of [16]. Recently Manuel Ritoré (see for instance [18]) showed that a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function has continuous and nondecreasing isoperimetric profile Ĩ M .…”

Area-minimizing regions with small volume in Riemannian manifolds with boundary

“…Recall that the partitioning problem in consists in finding, for a given v < | | g , a critical point of the perimeter functional ᏼ g ( • , ) in the class of Borel sets in that enclose a volume v. A set that minimizes the perimeter will be called an isoperimetric region. It is clear that the boundary of a smooth solution to the partitioning problem in have constant mean curvature and, if it touches ∂ , it will intersect it orthogonally; see for example [Ros and Vergasta 1995]. In light of standard results in geometric measure theory, minimizers do exist for any given volume and may have various topologies; see the survey [Ros 2005].…”

“…Up to now the complete description of minimizers has been achieved only in special cases; see for example [Bürger and Kuwert 2008;Ros and Vergasta 1995;Ritoré and Rosales 2004;Sternberg and Zumbrun 1998]. However, the study of existence and geometric and topological properties of stationary surfaces (not necessarily minimizers) is far from complete.…”

“…Assume that the boundary of ∂ is nonempty and is contained in ∂ . From the first variation of area (see for instance [Ros and Vergasta 1995]), E is a critical point for the perimeter functional under variations that keep the volume invariant if and only if n H ≡ const in and N ∂ ,…”