TY - JOUR

T1 - The Cheeger constant of a Jordan domain without necks

AU - Leonardi, Gian Paolo

AU - Neumayer, Robin

AU - Saracco, Giorgio

N1 - Funding Information:
G.P. Leonardi and G. Saracco have been supported by GNAMPA projects: Problemi isoperimetrici e teoria
Funding Information:
geometrica della misura in spazi metrici (2015) and Variational problems and geometric measure theory in metric spaces (2016). R. Neumayer is supported by the NSF Graduate Research Fellowship under Grant DGE-1110007.
Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - 35J93

KW - Primary 49K20

KW - Secondary 49Q20

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U2 - 10.1007/s00526-017-1263-0

DO - 10.1007/s00526-017-1263-0

M3 - Article

AN - SCOPUS:85032983291

VL - 56

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 6

M1 - 164

ER -