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Crystalline Mean Curvature Flow of Convex Sets

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Abstract

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133, Roma, Italy

    Giovanni Bellettini

  2. Dept. de Tecnologia, Universitat Pompeu Fabra, Passeig de Circumvalacio 8, 08003, Barcelona, Spain

    Vicent Caselles

  3. CMAP, CNRS UMR 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Antonin Chambolle

  4. Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127, Pisa, Italy

    Matteo Novaga

Authors
  1. Giovanni Bellettini
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  2. Vicent Caselles
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  3. Antonin Chambolle
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  4. Matteo Novaga
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Correspondence to Giovanni Bellettini.

Additional information

Communicated by L.C. Evans

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Bellettini, G., Caselles, V., Chambolle, A. et al. Crystalline Mean Curvature Flow of Convex Sets. Arch. Rational Mech. Anal. 179, 109–152 (2006). https://doi.org/10.1007/s00205-005-0387-0

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  • DOI: https://doi.org/10.1007/s00205-005-0387-0

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