Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

TL;DR

This paper proves short-time existence of classical solutions for a system of three mean curvature flows of graphs in two dimensions that intersect along a moving curve with fixed angles, addressing a free boundary problem.

Contribution

It establishes the short-time existence of classical solutions for a mean curvature flow system with triple junctions in two dimensions, a novel result for this geometric free boundary problem.

Findings

Short-time existence of classical solutions proven.

Solutions maintain fixed angles at the triple junction.

Results apply to sufficiently regular initial data.

Abstract

We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to $2\pi/3$.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic H\"{o}lder spaces), for sufficiently regular initial data satisfying a compatibility condition.