|The talk will discuss some mathematical questions motivated by the
motion of materials phase boundaries in heterogeneous medium.
The ultimate goal is to derive effective, homogenized equation and
study the property of the solution in large space-time regime. Motion
by mean curvature is used as an illustrative example. It already
involves interesting mathematical analysis due to the nonlinear
interaction between the curvature and the background heterogeneity.
In this talk, we will concentrate on periodic background environment.
For some linearized version of motion by mean curvature flow, we derive
the scaling behavior for the averaged velocity in some pinning and
de-pinning regime. For the fully nonlinear version, we prove the
existence, uniqueness and stability of pulsating waves (above the
pinning threshold) for any normal direction. Furthermore, the
effective speed of propagation is a Lipschitz continuous function of
the normal. Connection with homogenization will be discussed.
(This talk is based on joint works with Nicolas Dirr and Georgia