Calculus of Variations and Geometric Measure Theory

V. Franceschi - A. Pinamonti - G. Saracco - G. Stefani

The Cheeger problem in abstract measure spaces

created by pinamonti on 01 Jul 2022
modified by saracco on 23 May 2024


Published Paper

Inserted: 1 jul 2022
Last Updated: 23 may 2024

Journal: J. London Math. Soc. (2)
Volume: 119
Number: 1
Pages: e12840
Year: 2024
Doi: 10.1112/jlms.12840

ArXiv: 2207.00482 PDF


We consider non-negative $\sigma$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.