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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[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**0420406****[BdC]**João Lucas Barbosa and Manfredo do Carmo,*Stability of hypersurfaces with constant mean curvature*, Math. Z.**185**(1984), no. 3, 339–353. MR**731682**, 10.1007/BF01215045**[BM]**Pierre Bérard and Daniel Meyer,*Inégalités isopérimétriques et applications*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 3, 513–541 (French). MR**690651****[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. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419****[Ch]**Isaac Chavel,*Eigenvalues in Riemannian geometry*, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR**768584****[Gi]**Enrico Giusti,*Minimal surfaces and functions of bounded variation*, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR**775682****[GMT]**E. Gonzalez, U. Massari, and I. 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**684753**, 10.1512/iumj.1983.32.32003**[Gr]**Mikhael Gromov,*Structures métriques pour les variétés riemanniennes*, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR**682063**

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 [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR**1699320****[G1]**Michael Grüter,*Boundary regularity for solutions of a partitioning problem*, Arch. Rational Mech. Anal.**97**(1987), no. 3, 261–270. MR**862549**, 10.1007/BF00250810**[G2]**Michael Grüter,*Optimal regularity for codimension one minimal surfaces with a free boundary*, Manuscripta Math.**58**(1987), no. 3, 295–343. MR**893158**, 10.1007/BF01165891**[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**863638****[LP]**Pierre-Louis Lions and Filomena Pacella,*Isoperimetric inequalities for convex cones*, Proc. Amer. Math. Soc.**109**(1990), no. 2, 477–485. MR**1000160**, 10.1090/S0002-9939-1990-1000160-1**[M1]**Frank Morgan,*Regularity of isoperimetric hypersurfaces in Riemannian manifolds*, Trans. Amer. Math. Soc. (to appear).**[M2]**Frank Morgan,*Riemannian geometry*, 2nd ed., A K Peters, Ltd., Wellesley, MA, 1998. A beginner’s guide. MR**1600519****[M3]**Frank Morgan,*Geometric measure theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR**1775760****[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**1803220**, 10.1512/iumj.2000.49.1929**[MR]**Frank Morgan and Manuel Ritoré,*Isoperimetric regions in cones*, Trans. Amer. Math. Soc.**354**(2002), no. 6, 2327–2339. MR**1885654**, 10.1090/S0002-9947-02-02983-5**[R1]**Manuel Ritoré,*Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces*, Comm. Anal. Geom.**9**(2001), no. 5, 1093–1138. MR**1883725**, 10.4310/CAG.2001.v9.n5.a5**[R2]**Manuel Ritoré,*The isoperimetric problem in complete surfaces of nonnegative curvature*, J. Geom. Anal.**11**(2001), no. 3, 509–517. MR**1857855**, 10.1007/BF02922017**[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**1338315**, 10.1007/BF01263611**[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**756417****[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**1650327****[SZ2]**Peter Sternberg and Kevin Zumbrun,*On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint*, Comm. Anal. Geom.**7**(1999), no. 1, 199–220. MR**1674097**, 10.4310/CAG.1999.v7.n1.a7**[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**1669207**, 10.1007/BF02921639**[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**1084716****[Wh]**Hassler Whitney,*Geometric integration theory*, Princeton University Press, Princeton, N. J., 1957. MR**0087148****[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**1014685**

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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

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