Two examples of minimal Cheeger sets in the plane

TL;DR

This paper constructs two minimal Cheeger sets in the plane, illustrating counterexamples to regularity properties and exploring their geometric and analytical features in relation to prescribed mean curvature equations.

Contribution

It provides explicit examples of minimal Cheeger sets that challenge existing regularity assumptions and connects geometric properties with solutions to mean curvature problems.

Findings

Counterexample to weak regularity of Cheeger sets

Porous Cheeger set with non-graph boundary segments

Existence of a weakly regular Cheeger set solving prescribed mean curvature equation

Abstract

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.