*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

**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.

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

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