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.