On level-set approach to motion of manifolds of arbitrary codimension

Abstract

Ambrosio and Soner have shown that level-set methods can be used to characterize classical, and define generalized, evolution by mean curvature of manifolds of arbitrary codimension in ${\mathbb{R}}^n$. We investigate for what (other) normal velocities, and how, can the level-set methods be used to treat motion of manifolds of arbitrary codimension by the given velocity. Two variants of the level-set approach are studied. One uses the properties of the distance function to describe the motion. In the other one, the moving manifolds are represented as a zero-level set of a solution to a parabolic differential equation. Necessary conditions and sufficient conditions for these approaches to be applicable are given. The motion of curves in ${\mathbb{R}}^n$, by a velocity that is parallel to the normal vector, is studied in greater detail and the velocities to which the level-set methods apply are partially classified. We also compare the level-set approach with the use of minimal barriers of De Giorgi, and show that for many velocities the two approaches are essentially equivalent. That in turn provides new information on applicability of barriers of De Giorgi.