On the Cheeger problem for rotationally invariant domains. (English) Zbl 1477.49063

Bobkov, Vladimir; Parini, Enea

Manuscr. Math. 166, No. 3-4, 503-522 (2021).

Summary: 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 \(C\).

MSC:

49Q15	Geometric measure and integration theory, integral and normal currents in optimization
49Q10	Optimization of shapes other than minimal surfaces
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20	Variational problems in a geometric measure-theoretic setting

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