Periodic rotating waves in an undulating annulus and their homogenization limit

We study a mean curvature flow equation in an annulus with periodically undulating boundaries and consider the homogenization limit problem as the period of the boundary undulation tends to zero. We first establish a necessary and sufficient condition for the existence of periodic rotating waves. Then we study how the average rotating speed of the periodic rotating wave depends on the geometry of the boundaries. Our results show that boundary undulation always lowers the speed of a rotating wave. We also determine the homogenization limit of the average rotating speed. Quite surprisingly, this homogenized speed depends only on the maximum opening angles of the domain boundaries.