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 vn = f - ck, where f is a spatially periodic function, c a constant and κ 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 vn = f̄(n). We illustrate the results using examples. We also provide an explicit characterization of f̄ in the limit c → ∞.