
Abstract
We study a curvaturedependent motion of plane curves in a
twodimensional 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.
Mathematics Subject Classification: Primary: 35B27, 53C44; Secondary: 35B40, 35K55.
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