The Journal of Geometric Analysis

, Volume 27, Issue 2, pp 1086–1105 | Cite as

Existence of Isoperimetric Sets with Densities “Converging from Below” on \({\mathbb {R}}^N\)

  • Guido De Philippis
  • Giovanni Franzina
  • Aldo PratelliEmail author


In this paper, we consider the isoperimetric problem in the space \({\mathbb {R}}^N\) with a density. Our result states that, if the density f is lower semi-continuous and converges to a limit \(a>0\) at infinity, with \(f\le a\) far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331–365, 2013.


Isoperimetric problem Perimeter with density Existence of optimal sets 

Mathematics Subject Classification

49Q05 49Q20 53A10 



The work of the three authors was supported through the ERC St.G. 258685. We also wish to thank Michele Marini and Frank Morgan for useful discussions and their comments.


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Copyright information

© Mathematica Josephina, Inc. 2016

Authors and Affiliations

  • Guido De Philippis
    • 1
  • Giovanni Franzina
    • 2
  • Aldo Pratelli
    • 2
    Email author
  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Department MathematikUniversity of ErlangenErlangenGermany

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