Effective motion of a curvature-sensitive interface through a heterogeneous medium

Abstract

This paper deals with the evolution of fronts or interfaces propagating with normal velocity $v_n=f-c\kappa$ where $f$ is a spatially periodic function, $c$ a constant and $\kappa$ the mean curvature. This study is motivated by the propagation of phase boundaries and dislocation loops through heterogeneous media. We establish a homogenization result when the scale of oscillation of $f$ is small compared to the macroscopic dimensions, and show that the overall front is governed by a geometric law $v_n=\bar{f}(n)$. We illustrate the results using examples. We also provide an explicit characterization of $\bar{f}$ in the limit $c\to \infty$.