Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - G. Saracco

Two examples of minimal Cheeger sets in the plane

created by saracco on 04 Sep 2017
modified on 15 Feb 2020


Published Paper

Inserted: 4 sep 2017
Last Updated: 15 feb 2020

Journal: Ann. Mat. Pura Appl. (4)
Volume: 197
Number: 5
Pages: 1511
Year: 2018
Doi: 10.1007/s10231-018-0735-y

ArXiv: 1709.00851 PDF


We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of Cheeger sets, as its perimeter does not coincide with the $1$-dimensional Hausdorff measure of its topological boundary. The second one is a kind of porous set, whose boundary is not locally a graph at many of its points, yet it is a weakly regular open set admitting a unique (up to vertical translations) non-parametric solution to the prescribed mean curvature equation, in the extremal case corresponding to the capillarity for perfectly wetting fluids in zero gravity.

Keywords: Cheeger problem, minimal Cheeger set, weak regularity, capillarity