A discussion about the homogenization of moving interfaces

There has been considerable interest lately in the homogenization theory for first- and second-order partial differential equations in periodic/almost periodic and random, stationary, ergodic environments. Of special interest is the study of the averaged behavior of moving interfaces. In this note we revisit the last issue. We present several new results concerning interfaces moving by either oscillatory first-order or curvature dependent coupled with oscillatory forcing normal velocity in periodic environments and analyze in detail their behavior. Under sharp assumptions we show that such fronts may homogenize, get trapped or oscillate.