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On the Cheeger problem for rotationally invariant domains

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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.

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Notes

  1. Although the paper [25] deals with rotationally invariant, strictly convex domains, the convexity assumption is not used in the proof of [25, Lemma 3.1].

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Acknowledgements

The essential part of the present research was performed during a visit of E.P. at the University of West Bohemia and a visit of V.B. at Aix-Marseille University. The authors wish to thank the hosting institutions for the invitation and the kind hospitality. V.B. was supported by the Project LO1506 of the Czech Ministry of Education, Youth and Sports, and by the Grant 18-03253S of the Grant Agency of the Czech Republic. The authors also wish to thank the anonymous reviewer for valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00, Plzeň, Czech Republic

    Vladimir Bobkov

  2. Institute of Mathematics, Ufa Federal Research Centre, RAS, Chernyshevsky str. 112, Ufa, Russia, 450008

    Vladimir Bobkov

  3. Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, 39 Rue Frederic Joliot Curie, 13453, Marseille, France

    Enea Parini

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Correspondence to Vladimir Bobkov.

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Bobkov, V., Parini, E. On the Cheeger problem for rotationally invariant domains. manuscripta math. 166, 503–522 (2021). https://doi.org/10.1007/s00229-020-01260-9

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