Geometric Minimizing Movements

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.

Appendix

The variational approach for motion by mean curvature is due to Almgren et al. [2]. The variational approach for crystalline curvature flow is contained in a paper by Almgren and Taylor [1].

The homogenization of the flat flow essentially follows the discrete analog contained in the paper by Braides et al. [3]. In that paper more effects of the microscopic geometry are described for more general initial sets. The homogenization with forcing term is part of ongoing work with A. Malusa and M. Novaga.

Geometric motions with a non-trivial homogenized velocity are described in the paper by Braides and Scilla [4], where example are shown of geometries which do not influence the crystalline perimeter obtained as Γ-limit, but do influence various features of the evolution.