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Anisotropic geometric functionals and gradient flows

Tom 86 / 2009

Giovanni Bellettini, Luca Mugnai Banach Center Publications 86 (2009), 21-43 MSC: Primary 53C44; Secondary 74E10. DOI: 10.4064/bc86-0-2

Streszczenie

We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore, we discuss some relations between cylindrical anisotropies, the prescribed curvature problem and the capillarity problem. We conclude the paper by examining some higher order geometric functionals. In particular we discuss the anisotropic Willmore functional and compute its first variation in the smooth case.

Autorzy

  • Giovanni BellettiniDipartimento di Matematica
    Università di Roma “Tor Vergata”
    via della Ricerca Scientifica
    00133 Roma, Italy
    and
    INFN
    Laboratori Nazionali di Frascati
    via E. Fermi, 40, 00044 Frascati (Roma), Italy
    e-mail
  • Luca MugnaiMax Planck Institute for Mathematics in the Sciences
    Inselstr. 22
    D-04103 Leipzig, Germany
    e-mail

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