  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.
