Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

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.