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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Manuel Ritoré and César Rosales
Journal: Trans. Amer. Math. Soc. 356 (2004), 4601-4622
MSC (2000): Primary 53C20, 49Q20
Published electronically: April 27, 2004
MathSciNet review: 2067135
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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.


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Additional Information

Manuel Ritoré
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain
Email: ritore@ugr.es

César Rosales
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain
Email: crosales@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03537-8
PII: S 0002-9947(04)03537-8
Keywords: Isoperimetric regions, stability, hypersurfaces with constant mean curvature
Received by editor(s): March 6, 2003
Received by editor(s) in revised form: July 22, 2003
Published electronically: April 27, 2004
Additional Notes: Both authors were supported by MCyT-Feder research project BFM2001-3489
Article copyright: © Copyright 2004 American Mathematical Society