Existence of Isoperimetric Sets with Densities “Converging from Below” on RN

Guido De Philippis; Giovanni Franzina; Aldo Pratelli

doi:10.1007/s12220-016-9711-1

Existence of Isoperimetric Sets with Densities “Converging from Below” on R^N

Guido De Philippis, Giovanni Franzina, Aldo Pratelli

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we consider the isoperimetric problem in the space 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≤ 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.

Original language	English (US)
Pages (from-to)	1086-1105
Number of pages	20
Journal	Journal of Geometric Analysis
Volume	27
Issue number	2
DOIs	https://doi.org/10.1007/s12220-016-9711-1
State	Published - Apr 1 2017

Keywords

Existence of optimal sets
Isoperimetric problem
Perimeter with density

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.1007/s12220-016-9711-1

Cite this

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abstract = "In this paper, we consider the isoperimetric problem in the space RN 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≤ 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.",

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