Quantitative homogenization of forced geometric motions through random fields of obstacles

TL;DR

This paper proves a quantitative homogenization result for interfaces moving through sparse random obstacles, showing large-scale effective motion with error estimates, even allowing for local pinning effects, advancing understanding of interface dynamics in disordered media.

Contribution

It introduces a homogenization framework for forced mean curvature flow with obstacles, accommodating local pinning and negative forcing, unlike previous models requiring positive forcing everywhere.

Findings

Effective large-scale interface motion governed by constant speed.

Error estimate for front arrival times of order ε^{1/9-}.

Applicable to models with local pinning and obstacles.

Abstract

We establish a quantitative homogenization result for an interface moving through a field of sufficiently sparse but possibly impenetrable random obstacles. From a physical viewpoint, such problems arise e.g. in the context of the motion of dislocations or magnetic domain walls in a material with impurities. More precisely, given an interface moving by forced mean curvature flow -- with a positive global driving force plus a spatially fluctuating (negative) driving force modeling the obstacles -- , we prove that the effective large-scale behavior of the forward front is governed by a constant-speed effective motion. For typical values of the global forcing, on large scales of the order $\varepsilon^{-1}$ we obtain a (relative) error estimate for arrival times of the front of the order $\varepsilon^{1/9-}$. Previous stochastic homogenization results for forced mean curvature motion in the literature have required a positive pointwise lower bound on the combined forcing, which implies the absence of any actual obstacles capable of locally blocking the interface motion. In contrast, our results are valid even in the presence of islands with locally negative forcing, potentially allowing for locally pinned interfaces and eventually enclosures left behind the main front. Thus, our homogenization result applies to settings closer to (but still strictly away from) the pinning-depinning transition.