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 Ωr is equal to π r^2. 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.

The Cheeger constant of a Jordan domain without necks / Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 2016, 56:6(2017), pp. 164.1-164.29. [10.1007/s00526-017-1263-0]

The Cheeger constant of a Jordan domain without necks

Leonardi, Gian Paolo;Saracco, Giorgio
2017-01-01

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 Ωr is equal to π r^2. 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.
2017
6
Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio
The Cheeger constant of a Jordan domain without necks / Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 2016, 56:6(2017), pp. 164.1-164.29. [10.1007/s00526-017-1263-0]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/228716
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