On the Cheeger problem for rotationally invariant 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 $C$.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.10474
 Bibcode:
 2019arXiv190710474B
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Differential Geometry;
 49Q15;
 49Q10;
 53A10;
 49Q20
 EPrint:
 18 pages, 8 figures. Minor improvements according to referee's suggestions. Ahead of print in Manuscripta Mathematica