On the Cheeger problem for rotationally invariant domains

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Omega subset mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $partial C cap Omega$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $Omega$ is convex, then the free boundary of $C$ consists only of pieces of spheres and nodoids. This result remains valid for nonconvex domains when the generating curve of $C$ is closed, convex, and of class $mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the fact that, for general nonconvex domains, pieces of unduloids or cylinders can also appear in the free boundary of $C$.