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

By Manuel Ritor'e, César Rosales. 2003

93 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0f3ff26246e0f3634e3e00978abb551b57292b18

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Overdetermined problems and constant mean curvature surfaces in cones (2018, background): “10 of [25], providing all details for the convenience of the reader.”
A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones (2012, background): “The equality cases were considered in [27] for the special case when C \ {0} is smooth (see also [34]).”
The relative isoperimetric inequality outside convex domains in Rn (2007, unknown): “If C is bounded then, from the results in [13], it can be proved that any perimeter minimizing sequence of sets in RC of given volume has a subsequence converging to an isoperimetric region.”
Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures (2013, methodology): “The uniqueness of general isoperimetric minimizers in the Lions–Pacella scenario in the case that Σ\{0} is smooth was obtained by Lions and Pacella themselves [21], and using a different approach by Ritoré and Rosales [26]; the smoothness assumption has been recently removed by Figalli and Indrei [1”
Sharp isoperimetric inequalities via the ABP method (2013, unknown): “gin are the unique minimizers in (1.4) up to translations that leave invariant the cone (if they exist). This had been established in [37] in the particular case when ∂Σ \ {0} is smooth (and later in [49], which also classiﬁed stable hypersurfaces in smooth cones). The following is the notion of ani”
On the isoperimetric problem in Euclidean space with density (2006, background): “In general, for a Riemannian manifold with density, standard compactness arguments of Geometric Measure Theory (see [Si, 27.3 and 31.2] or [M2, 5.5 and 9.1], and [M1, 4.1]or [ RiR , Thm.”