A level-set method for a mean curvature flow with a prescribed boundary

TL;DR

This paper introduces a level-set method for mean curvature flow with a prescribed boundary, interpreted as an obstacle, ensuring a global-in-time solution that aligns with classical and previous level-set flows under Dirichlet conditions.

Contribution

The authors develop a globally solvable level-set approach for mean curvature flow with prescribed boundaries, extending applicability without initial hypersurface restrictions.

Findings

Method provides global-in-time solutions with prescribed boundaries.

Solution agrees with classical mean curvature flow under Dirichlet conditions.

Matches previous level-set flow constructions in specific geometric settings.

Abstract

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hypersurface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition. We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P. Sternberg and W. P. Ziemer (1994), where the initial hypersurface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.