Two dimensions are easier

Kawohl, Bernd

doi:10.1007/s00013-016-0953-8

Cited by 5 publications

(1 citation statement)

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“…On the other hand, in the higher dimensional case 𝑛 ≥ 3, there is a big variety of CMC surfaces, and a characterization of 𝐶 by "rolling" a ball inside Ω as in (1.3) can be violated, see [14,Remark 13]. Moreover, an explicit characterization of Cheeger sets seems to be known only for some particular domains such as ellipsoids of low eccentricity [2], spherical shells [8], and tubular neighbourhoods of complete curves [17], while even for three-dimensional cubes the problem is open [13].…”

Section: Introductionmentioning

confidence: 99%

On the Cheeger problem for rotationally invariant domains

Bobkov,

Parini

2019

Preprint

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We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains Ω ⊂ R 𝑛 . For a rotationally invariant Cheeger set 𝐶, the free boundary 𝜕𝐶 ∩ Ω consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if Ω 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 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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