Quantitative homogenization of forced geometric motions through random fields of 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.

Figures (21)

• Figure 1.1: A comparison of the evolution by forced mean curvature flow in a field of random obstacles with the evolution according to a homogeneous first-order motion. The initial set is depicted in grey, the evolution by forced mean curvature flow in green, and the evolution by the first-order motion in yellow. The target area for which the arrival times are compared is depicted in blue. Left: The length scale of the initial set is of the same order as the range of dependence. Right: The length scale of the initial set is much larger than the range of dependence, and homogenization is observed.
• Figure 3.1: An illustration of our notion of h-envelopment $S_2 \subset_{_} S_1$. A set $S_1 \subset {\mathbb{R}^{d}}$ (depicted in green) which $h$-envelops a set $S_2\subset {\mathbb{R}^{d}}$ (depicted in grey).
• Figure 3.2: An illustration of our notion of effective minimum speed. Left: An initial set S (depicted in grey) and its evolution ${{\mathscr{R}}_{t}^{}(*){\raisebox{0pt}{${\newline\mathopen{}S}$}}}$ after some time $t$ (depicted green). The dotted line corresponds to a further expansion of ${{\mathscr{R}}_{t}^{}(*){\raisebox{0pt}{${\newline\mathopen{}S}$}}}$ by $\overline{B}_h$. Right: If an effective minimum speed $v$ holds for $S$, then after an additional time step $v^{-1} h$ the evolved set ${{\mathscr{R}}_{t+v^{-1} h}^{}(*){\raisebox{0pt}{${\newline\mathopen{}S}$}}}$ (depicted in green) must $h$-envelop the dotted area.
• Figure 3.3: An illustration of our notion of a stable stable $(h,c)$-approximation $S_h$ of a set $S$. Left: The set $S\subset {\mathbb{R}^{d}}$ to be approximated is depicted in grey. The subset of $S$ that must be included in $S_h$ is depicted with a dotted boundary. Center: In addition to the aforementioned sets, the approximation $S_h$ is depicted in green. Right: After a time step $ch$, the evolved set ${{\mathscr{R}}_{ch}^{}(*){\raisebox{0pt}{${\newline\mathopen{}S_h}$}}}$ (depicted in green) must $h$-envelop the set $S$.
• Figure 3.4: An illustration of our box criterion (Example \ref{['ex_ex_box']}) sufficient for Assumption \ref{['veff_aP_star']} in dimension $d=2$. Left: A rectangular box with a sample of the obstacles in the interior (depicted in brown) and the artificial obstacle from the restriction surrounding the boundary (also depicted in brown) with an initial stable set $S$ in the bottom of the box (depicted in green). Right: After some time $t$, we require the evolution ${{\mathscr{R}}_{t}^{}(*){\raisebox{0pt}{${\newline\mathopen{}S}$}}}$ (depicted in green) of the set $S$ to have reached the top of the box.

Theorems & Definitions (127)

• Theorem 1.1: Homogenization for Poisson-distributed obstacles
• Definition 1.2: The set evolution in the heterogeneous environment
• Remark 1.3: Hyperbolic scaling regime
• Remark 1.4: Selection criterion for fattening
• Definition 1.5: The homogenized set evolution
• Definition 1.6: Arrival times
• Theorem 1.7: Informal main result
• Remark 1.8
• Remark 3.1
• Remark 3.2: On Assumption \ref{['sett_aP_gen']}