Mean curvature flow with obstacles

TL;DR

This paper develops a variational method to analyze the evolution of fronts by mean curvature in the presence of obstacles, establishing existence and uniqueness of solutions in 2D before singularities occur.

Contribution

It introduces a new variational approach for mean curvature flow with obstacles and proves existence and uniqueness of solutions in two dimensions under regularity assumptions.

Findings

Existence of weak solutions via variational methods.

Uniqueness and regularity of solutions in 2D before singularities.

Application to positive mean curvature flow.

Abstract

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.