Level set method for motion by mean curvature

TL;DR

This paper investigates the regularity of solutions to the level set method for motion by mean curvature, demonstrating that weak solutions are often twice differentiable and linking geometric insights with analytical proofs.

Contribution

It establishes the optimal regularity results for solutions to curvature-driven motion equations using a novel combination of analysis and geometry.

Findings

Weak solutions are always twice differentiable

Second derivatives are continuous only in rigid geometric cases

Analysis and geometry are intertwined in the proof

Abstract

Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second order differential equations on Euclidean space. One naturally wonders "what is the regularity of solutions?" A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.