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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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  • [A] F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 54:8420
  • [BdC] João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339-353. MR 85k:58021c
  • [BM] Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513-541. MR 84h:58147
  • [BZ] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988, Translated from the Russian by A. B. Sosinski{\u{\i}}\kern.15em, Springer Series in Soviet Mathematics. MR 89b:52020
  • [Ch] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984. MR 86g:58140
  • [Gi] Enrico Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser Verlag, Basel, 1984. MR 87a:58041
  • [GMT] Eduardo Gonzalez, Umberto Massari, and Italo Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25-37. MR 84d:49043
  • [Gr] Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston Inc., Boston, MA, 1999, based on the 1981 French original MR 85e:53051, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates. MR 2000d:53065
  • [G1] Michael Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261-270. MR 87k:49050
  • [G2] -, Optimal regularity for codimension one minimal surfaces with a free boundary, Manuscripta Math. 58 (1987), no. 3, 295-343. MR 88m:49032
  • [GJ] Michael Grüter and Jürgen Jost, Allard type regularity results for varifolds with free boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129-169. MR 89d:49048
  • [LP] Pierre-Louis Lions and Filomena Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc. 109 (1990), no. 2, 477-485. MR 90i:52021
  • [M1] Frank Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. (to appear).
  • [M2] -, Riemannian geometry, second ed., A K Peters Ltd., Wellesley, MA, 1998, A beginner's guide. MR 98i:53001
  • [M3] -, Geometric measure theory, third ed., Academic Press Inc., San Diego, CA, 2000, A beginner's guide. MR 2001j:49001
  • [MJ] Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000), no. 3, 1017-1041. MR 2002e:53043
  • [MR] Frank Morgan and Manuel Ritoré, Isoperimetric regions in cones, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2327-2339 (electronic). MR 2003a:53089
  • [R1] Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom. 9 (2001), no. 5, 1093-1138. MR 2003a:53018
  • [R2] -, The isoperimetric problem in complete surfaces of nonnegative curvature, J. Geom. Anal. 11 (2001), no. 3, 509-517. MR 2002f:53109
  • [RV] Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19-33. MR 96h:53013
  • [R] César Rosales, Isoperimetric regions in rotationally symmetric convex bodies, Indiana U. Math. J. 52 (2003), no. 5, 1201-1214.
  • [S] Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983. MR 87a:49001
  • [SZ1] Peter Sternberg and Kevin Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63-85. MR 99g:58028
  • [SZ2] -, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199-220. MR 2000d:49062
  • [StZ] Edward Stredulinsky and William P. Ziemer, Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Anal. 7 (1997), no. 4, 653-677. MR 99k:49089
  • [W] Henry C. Wente, A note on the stability theorem of J. L. Barbosa and M. Do Carmo for closed surfaces of constant mean curvature, Pacific J. Math. 147 (1991), no. 2, 375-379. MR 92g:53010
  • [Wh] Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 19:309c
  • [Z] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989, Sobolev spaces and functions of bounded variation. MR 91e:46046

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Additional Information

Manuel Ritoré
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain

César Rosales
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, Spain

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

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