Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit

Abstract

We study a curvature-dependent motion of plane curves in a two-dimensional cylinder with periodically undulating boundary. The law of motion is given by $V=\kappa + A$, where $V$ is the normal velocity of the curve, $\kappa$ is the curvature, and $A$ is a positive constant. We first establish a necessary and sufficient condition for the existence of periodic traveling waves, then we study how the average speed of the periodic traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the period of the boundary undulation, denoted by $\epsilon$, tends to zero, and determine the homogenization limit of the average speed of periodic traveling waves. Quite surprisingly, this homogenized speed depends only on the maximum opening angle of the domain boundary and no other geometrical features are relevant. Our analysis also shows that, for any small $\epsilon>0$, the average speed of the traveling wave is smaller than $A$, the speed of the planar front. This implies that boundary undulation always lowers the speed of traveling waves, at least when the bumps are small enough.