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 $C$.

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关于旋转不变域的 Cheeger 问题

我们研究了旋转不变、有界域 $\Omega \subset \mathbb{R}^n$ 的 Cheeger 集的性质。对于旋转不变的 Cheeger 集 $C$，自由边界 $\partial C\cap\Omega$ 由多片 Delaunay 曲面组成，这些曲面是具有恒定平均曲率的旋转不变曲面。我们证明，如果 $\Omega$ 是凸的，那么 $C$ 的自由边界仅由球体和节点组成。当 $C$ 的生成曲线是闭合的、凸的并且属于 $\mathcal{C}^{1,1}$ 类时，该结果对于非凸域仍然有效。此外，我们提供了以下事实的数值证据，即对于一般的非凸域，在 $C$ 的自由边界中也可能出现不规则曲面或圆柱体的碎片。