Alexandrov's soap bubble theorem

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Alexandrov's soap bubble theorem is a mathematical theorem from geometric analysis that characterizes a sphere through the mean curvature. The theorem was proven in 1958 by Alexander Danilovich Alexandrov.^[1]^[2] In his proof he introduced the method of moving planes, which was used after by many mathematicians successfully in geometric analysis.

Soap bubble theorem

Let $\Omega \subset \mathbb {R} ^{n}$ be a bounded connected domain with a boundary $\Gamma =\partial \Omega$ that is of class $C^{2}$ with a constant mean curvature, then $\Gamma$ is a sphere.^[3]^[4]

Literature

Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". p. 1. arXiv:1811.05202.
Smirnov, Yurii Mikhailovich; Aleksandrov, Alexander Danilovich (1962). "Nine Papers on Topology, Lie Groups, and Differential Equations". American Mathematical Society Translations. 2. Vol. 21. American Mathematical Soc. ISBN 0821817213. {{cite encyclopedia}}: ISBN / Date incompatibility (help)

References

^ Alexandrov, Alexander Danilovich (1962). "Uniqueness theorem for surfaces in the large". American Mathematical Society Translations. 2. Vol. 21. American Mathematical Soc. pp. 412–416.
^ Alexandrov, Alexander Danilovich (1962). "A characteristic property of spheres". Annali di Matematica. 58: 303–315. doi:10.1007/BF02413056.
^ Magnanini, Rolando; Poggesi, Giorgio (2017). "Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities". Indiana University Mathematics Journal. 69. arXiv:1708.07392. doi:10.1512/iumj.2020.69.7925.
^ Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". p. 1. arXiv:1811.05202.

[1] Alexandrov, Alexander Danilovich (1962). "Uniqueness theorem for surfaces in the large". American Mathematical Society Translations. 2. Vol. 21. American Mathematical Soc. pp. 412–416.

[2] Alexandrov, Alexander Danilovich (1962). "A characteristic property of spheres". Annali di Matematica. 58: 303–315. doi:10.1007/BF02413056.

[3] Magnanini, Rolando; Poggesi, Giorgio (2017). "Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities". Indiana University Mathematics Journal. 69. arXiv:1708.07392. doi:10.1512/iumj.2020.69.7925.

[4] Ciraolo, Giulio; Roncoroni, Alberto (2018). "The method of moving planes: a quantitative approach". p. 1. arXiv:1811.05202.

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