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Abstract

We study a curvature flow equation in a band domain with (almost) periodic undulating boundaries.  First we give the definition and prove the existence of (almost) periodic traveling waves, then we study how the average speed of the (almost) periodic traveling wave depends on the geometry of the domain boundaries.  More specifically, we consider the homogenization problem as the period of the boundary undulation tends to zero, and determine the homogenization limit of the average speed of 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 the boundary undulation always lowers the speed of traveling waves, at least when the boundary bumps are small enough.