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arXiv · 1709.00851

Two examples of minimal Cheeger sets in the plane

Gian Paolo Leonardi·Giorgio Saracco
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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.

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Gian Paolo Leonardi, Giorgio Saracco. 2017-09-04. Two examples of minimal Cheeger sets in the plane. https://doi.org/10.1007/s10231-018-0735-y

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