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Geometric Minimizing Movements

  • Andrea Braides
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2094)

Abstract

Geometric motions; i.e., motions of sets governed by geometric quantities, such as mean curvature flow, can sometime be cast in the framework of minimizing movements following the approach of Almgren, Taylor and Wang. In this case the energy is a perimeter functional, and the distance term must be suitably rewritten as an anisotropic integral. With this approach, it is possible to prove existence for motions by curvature; e.g., mean curvature or crystalline curvature. For oscillating perimeter energies, we apply the approach of the minimizing movements along a sequence, computing an effective motion showing pinning for large sets and a discontinuous dependence of the velocity on the curvature. Additional effects are shown to appear for minimizing movements related to the homogenization of perimeter functionals with oscillating forcing terms.

References

  1. 1.
    Almgren, F., Taylor, J.E.: Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differ. Geom. 42, 1–22 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature driven flows: a variational approach. SIAM J. Control Optim. 50, 387–438 (1983)MathSciNetGoogle Scholar
  3. 3.
    Braides, A, Gelli, M.S., Novaga, M.: Motion and pinning of discrete interfaces. Arch. Ration. Mech. Anal. 95, 469–498 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Braides, A., Scilla, G.: Motion of discrete interfaces in periodic media. Interfaces Free Bound. 15 (2013), to appear.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrea Braides
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly

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