Mean curvature flow with obstacles: existence, uniqueness and regularity of solutions

TL;DR

This paper proves short-term existence and uniqueness of solutions to the mean curvature flow with obstacles of class C^{1,1}, and demonstrates long-term existence and convergence to minimal hypersurfaces for periodic graphs.

Contribution

It establishes the existence, uniqueness, and regularity of solutions to the mean curvature flow with obstacles, including long-term behavior and convergence results.

Findings

Short time existence and uniqueness of C^{1,1} solutions.

Long time existence for periodic graph initial interfaces.

Convergence to minimal constrained hypersurfaces.

Abstract

We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.