Annals of Global Analysis and Geometry

, Volume 43, Issue 4, pp 331–365 | Cite as

Existence of isoperimetric regions in \({\mathbb{R}^{n}}\) with density

  • Frank Morgan
  • Aldo PratelliEmail author


We prove the existence of isoperimetric regions in \({\mathbb{R}^{n}}\) with density under various hypotheses on the growth of the density. Along the way, we prove results on the boundedness of isoperimetric regions.


Isoperimetric sets \({\mathbb{R}^{n}}\) with density Existence of optimal sets Boundedness of optimal sets 

Mathematics Subject Classification

Primary 49Q05 Secondary 53A10 49Q20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  2. 2.
    Burago Y.D., Zalgaller V.A.: Geometric Inequalities, Grund. math. Wiss., vol. 285. Springer, Berlin (1988)Google Scholar
  3. 3.
    Cañete A., Miranda M. Jr, Vittone D.: Some isoperimetric problems in planes with density. J. Geom. Anal. 20(2), 243–290 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dí az, A., Harman, N., Howe, S., Thompson, D.: Isoperimetric problems in sectors with density. Adv. Geom. (available at (2012, to appear)
  5. 5.
    Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)zbMATHCrossRefGoogle Scholar
  7. 7.
    Kawohl B.: On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems. Arch. Ration. Mech. Anal. 94, 227–243 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Morgan F.: Riemannian Geometry: A Beginner’s Guide, 2nd edn. A.K. Peters, Wellesley (1998)zbMATHGoogle Scholar
  9. 9.
    Morgan F.: Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. AMS 355, 5041–5052 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Morgan F.: Manifolds with density. Notices Am. Math. Soc. 52, 853–858 (2005)zbMATHGoogle Scholar
  11. 11.
    Morgan F.: Geometric Measure Theory: A Beginner’s Guide, 4th edn. Academic Press, San Diego (2009)zbMATHGoogle Scholar
  12. 12.
  13. 13.
    Morgan F., Johnson D.L.: Some sharp isoperimetric theorems for Riemannian manifolds. Indiana Univ. Math. J. 49(3), 1017–1041 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Morgan F., Howe S., Harman N.: Steiner and Schwarz symmetrization in warped products and fiber bundles with density. Revista Mat. Iberoamericana 27, 909–918 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rosales C., Cañete A., Bayle V., Morgan F.: On the isoperimetric problem in Euclidean space with density. Calc. Var. Partial Differ. Equ. 31(1), 27–46 (2008)zbMATHCrossRefGoogle Scholar
  16. 16.
    Vol’pert A.I.: Spaces BV and quasilinear equations. Math. USSR Sb. 17, 225–267 (1967)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  2. 2.Dipartimento di MatematicaUniversità di Pavia “F. Casorati”PaviaItaly

Personalised recommendations