# On the Cheeger problem for rotationally invariant domains

## TL;DR

This paper studies the shape and properties of Cheeger sets in rotationally invariant domains, revealing geometric structures of their free boundaries and providing numerical evidence of complex boundary pieces.

## Contribution

It characterizes the free boundary of Cheeger sets in convex and certain nonconvex domains, showing they consist of Delaunay surfaces, including spheres, nodoids, unduloids, or cylinders.

## Key findings

- Free boundary consists of Delaunay surfaces in rotationally invariant domains.
- In convex domains, free boundary pieces are spheres and nodoids.
- Numerical evidence shows unduloids or cylinders can appear in nonconvex domains.

## Abstract

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 $C$.