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

2004; American Mathematical Society; Volume: 356; Issue: 11 Linguagem: Inglês

10.1090/s0002-9947-04-03537-8

1088-6850

Manuel Ritoré, César Rosales,

Advanced Mathematical Modeling in Engineering

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.