The Cheeger constant of a Jordan domain without necks

Gian Paolo Leonardi*, Robin Neumayer, Giorgio Saracco

*Corresponding author for this work

Research output: Contribution to journalArticle

8 Scopus citations

Abstract

We show that the maximal Cheeger set of a Jordan domain Ω without necks is the union of all balls of radius r= h(Ω) - 1 contained in Ω. Here, h(Ω) denotes the Cheeger constant of Ω , that is, the infimum of the ratio of perimeter over area among subsets of Ω , and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set Ω r is equal to πr2. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

Original languageEnglish (US)
Article number164
JournalCalculus of Variations and Partial Differential Equations
Volume56
Issue number6
DOIs
StatePublished - Dec 1 2017

Keywords

  • 35J93
  • Primary 49K20
  • Secondary 49Q20

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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