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Periodic traveling waves of a mean curvature flow in
heterogeneous media
We consider a curvature flow in heterogeneous media in the plane: $
V= a(x,y) \kappa + b$, where for a plane curve, $V$ denotes its
normal velocity, $\kappa$ denotes its curvature, $b$ is a constant
and $a(x,y)$ is a positive function, periodic in $y$. We study
periodic traveling waves which travel in $y$-direction with given
average speed $c \geq 0$. Four different types of traveling waves
are given, whose profiles are straight lines, ''V"-like curves,
cup-like curves and cap-like curves, respectively. We also show
that, as $(b,c)\rightarrow (0,0)$, the profiles of the traveling
waves converge to straight lines. These results are connected with
spatially heterogeneous version of Bernshteĭn's Problem and De
Giorgi's Conjecture, which are proposed at last.