Two examples of minimal Cheeger sets in the plane

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.

19 pages, 6 figures

References in corpus (1)

• The Cheeger constant of a Jordan domain without necks

Cited by in corpus (5)

• Rigidity and trace properties of divergence-measure vector fields
• Minimizers of the prescribed curvature functional in a Jordan domain with no necks
• The Cheeger constant of curved tubes
• Geometric criteria for the existence of capillary surfaces in tubes
• The isoperimetric problem in $2$d domains without necks