Curvature-Driven Flows: A Variational Approach

Abstract

This paper introduces a new mathematical approach to the study of time evolutions of solids $K(t)$ in n-space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy $\Phi (\partial K(t))$. These “flat $\Phi $ curvature flows” are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized. The construction works equally well for smooth elliptic $\Phi $’s, for nondifferentiable crystalline $\Phi $’s, and for anything in between. The flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane.