Generic mean curvature flow with obstacles

TL;DR

This paper introduces a novel approach to mean curvature flow with obstacles by incorporating a penalization term, leading to a unique level set formulation that extends previous work and ensures solutions satisfy obstacle constraints.

Contribution

It develops a new singular perturbation method for mean curvature flow with obstacles, establishing existence, uniqueness, and distributional solutions in the level set framework.

Findings

Unique solutions up to fattening are obtained.

The method extends previous work by Evans, Spruck, and Ullrich.

Level sets are shown to be distributional solutions of the obstacle problem.

Abstract

We study the obstacle problem associated to mean curvature flow. We add to the geometric vanishing-viscosity approximation of Evans and Spruck a singular perturbation that penalizes the violation of the constraint, and pass to the limit. The resulting level set formulation has unique solutions - up to fattening. Extending the work of Evans and Spruck and a work by Ullrich and one of the authors, we show that generic level sets of this flow are distributional solutions of the obstacle problem.