On the Cheeger problem for rotationally invariant domains
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Vladimir Bobkov
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
· Enea Parini
On the Cheeger problem for rotationally invariant domains Received: 24 July 2019 / Accepted: 30 October 2020 Abstract. We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains ⊂ Rn . For a rotationally invariant Cheeger set C, the free boundary ∂C ∩ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if 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.
1. Introduction Let be a bounded domain in Rn , n ≥ 2. The Cheeger problem consists in finding subsets C of which solve the minimization problem h() = inf
E⊂
P(E) , |E|
(1.1)
where P(E) = P(E; Rn ) is the distributional perimeter of a subset E of measured with respect to Rn and |E| stands for the Lebesgue measure of E. The value of h() is called Cheeger constant of , and any minimizer C of (1.1) is called Cheeger set of . An overview of general properties of the Cheeger problem, such as the existence of the Cheeger set and its regularity, can be found in surveys [19,24], see also Sect. 2.2. Let us particularly note that the Cheeger set always exists and if ∂C ∩ is nonempty, then it is a constant mean curvature surface (CMC surface) with mean curvature h() . (1.2) H= n−1 V. Bobkov (B): Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzeˇn, Czech Republic. e-mail: [email protected] V. Bobkov: Institute of Mathematics, Ufa Federal Research Centre, RAS, Chernyshevsky str. 112, Ufa, Russia 450008. E. Parini: Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, 39 Rue Frederic Joliot Curie, 13453 Marseille, France. e-mail: [email protected] Mathematics Subject Classification: 49Q15 · 49Q10 · 53A10 · 49Q20
https://doi.org/10.1007/s00229-020-01260-9
V. Bobkov, E. Parini
Hereinafter, ∂C ∩ will be called free boundary of C. Despite the geometric nature of the problem (1.1), an explicit analytical description of Cheeger sets is, in general, a difficult task. Such description is relatively well-established in the planar case, thanks to the fact that the only planar CMC surface is a circular arc, see [15,18,21] and references therein. In particular, if ⊂ R2 is convex, then its Cheeger set C is unique and can be characterized by “rolling” a disk Br (x) inside : C= Br (x), (1.3) x∈r 1 where r = h() and r = {x ∈ : dist(x, ∂) ≥ r }, see [15]. 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 as in (1.3) can be violated, see [14, Remark 13]. Moreover, an exp
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