Diffuse Interface Energies with Microscopic Heterogeneities I: Homogenization

Technical Summary: Diffuse Interface Energies with Microscopic Heterogeneities I: Homogenization

Problem Statement
This paper investigates the macroscopic behavior of diffuse interfaces in materials characterized by microscopic heterogeneities. Specifically, the authors analyze the $\Gamma$-limit of Allen-Cahn-type energy functionals in the regime where the length scale of the heterogeneities, $\delta$, is much smaller than the interface width, $\epsilon$ (i.e., $\delta \ll \epsilon$). The energy functional is given by:
$$F_{\epsilon,\delta}(u; U) = \int_U \left( \frac{\epsilon}{2} a(\delta^{-1}x)\nabla u \cdot \nabla u + \frac{1}{\epsilon}\theta(\delta^{-1}x)W(u) \right) dx$$
where $a$ is a uniformly elliptic matrix field and $\theta$ is a positive function, both modeled as samples of stationary, ergodic random fields. The potential $W$ is a double-well potential with minima at $\pm 1$.

The central question is whether the limit as $\epsilon \to 0$ (sharp interface limit) commutes with the homogenization limit ($\delta \to 0$). While it is known that if $\delta \sim \epsilon$, the limit is an anisotropic surface energy determined by the joint statistics of $a$ and $\theta$, the behavior when $\delta \ll \epsilon$ is more subtle. The authors aim to determine conditions under which the effective energy is simply the homogenized surface energy (as if coefficients were replaced by their averages) versus cases where "rare events" (atypical local configurations) dominate the limit.

Methodology
The proof strategy relies on a reduction to planar cell problems, a standard technique in the $\Gamma$-convergence of Allen-Cahn functionals. The authors establish that $\Gamma$-convergence of the global functional is equivalent to the convergence of the energy of minimizers in large cubes with planar boundary conditions to a specific surface tension $\bar{\sigma}$.

The core of the analysis involves a two-step approach:

1. Deterministic Relative Error Estimates: The authors first derive deterministic estimates for the difference between the heterogeneous energy $F_{\gamma}$ (where $\gamma = \delta/\epsilon$) and the homogenized energy $F$. This is achieved by decomposing the macroscopic domain into mesoscopic subcubes of fixed size $r_c$. Within each subcube, they compare the heterogeneous minimizer to a homogenized minimizer. This comparison utilizes:

   • Correctors: Gradient correctors $\phi$ and flux correctors $\sigma$ associated with the elliptic operator $-\nabla \cdot (a \nabla)$.
   • Regularity: De Giorgi-Nash-Moser estimates and Meyer's estimates to control the $L^p$ norms of gradients and the Hölder continuity of minimizers.
   • Clearing-out Property: A lemma ensuring that in low-energy regimes, minimizers are close to the pure phases ($\pm 1$), allowing for linearization of the potential $W$.
   • Regularization: A modified potential $W_{\text{reg}}$ is used to ensure strict convexity at small scales, facilitating convexity arguments.

2. Probabilistic Averaging: The deterministic error bounds depend on the "sublinearity" of the correctors and the oscillations of $\theta$ measured in the $H^{-1}$ norm. The authors define stationary fields $Sub_x(r)$ and $Osc_x(r)$ to quantify these quantities. They prove that if the ratio $\delta/\epsilon$ decays sufficiently fast, the probability that these fields exceed a threshold vanishes rapidly enough to ensure that the homogenization error is negligible in the limit.

Key Contributions and Results

• Main Theorem (Homogenization Regime): The primary result (Theorem 1) establishes that if the ratio $\delta(\epsilon)/\epsilon$ decays fast enough relative to the probabilistic decay rates of the sublinearity of correctors and the oscillations of $\theta$, the energy $\Gamma$-converges in probability to the homogenized surface energy $\bar{E}$. The effective surface tension is given by $\bar{\sigma}(\nu)^2 = \sigma_W^2 \bar{\theta} \nu \cdot \bar{a} \nu$, where $\bar{a}$ is the homogenized matrix and $\bar{\theta}$ is the expected value of $\theta$.

  • The sufficient condition is expressed via the decay of probabilities:
    $$\lim_{\epsilon \to 0} \epsilon^{-d} P\left\{ Sub_0\left(\frac{\epsilon}{\delta(\epsilon)}\right) > \nu \right\} = 0$$
    and a similar condition for $Osc_0$.

• Sharpness and Rare Events: The paper notes that if $\delta/\epsilon$ decays too slowly, the $\Gamma$-limit may be strictly smaller than the homogenized prediction. This "rare events regime," where improbable local configurations of the medium lower the energy, is the subject of a companion paper [49]. The current work proves that the derived conditions are sharp for the random checkerboard model in dimension $d=1$.

• Periodic and Almost Periodic Media: The authors prove unconditional homogenization results for periodic and uniformly almost periodic (Besicovitch space $B_\infty$) media (Corollary 2). In these settings, the convergence of correctors and oscillations is uniform in space, meaning homogenization holds for any sequence $\delta(\epsilon)$ such that $\delta(\epsilon)/\epsilon \to 0$, regardless of the rate of decay.

• Almost-Sure Convergence: Under slightly stronger summability conditions on the probabilities of large deviations (Corollary 1), the convergence is upgraded from convergence in probability to almost-sure convergence for sequences.

• Technical Tools: The paper introduces a metric $d_\Gamma$ for $\Gamma$-convergence in probability and utilizes recent quantitative stochastic homogenization techniques (specifically the minimal radius and sublinearity estimates from Gloria, Neukamm, and Otto [38]) to quantify the necessary decay rates.

Significance
The paper provides a rigorous characterization of the interplay between microscopic heterogeneity scales and mesoscopic interface widths in phase transition models. It clarifies that homogenization is not automatic in the $\delta \ll \epsilon$ regime; rather, it requires the heterogeneity scale to be sufficiently small relative to the interface width to "average out" the medium before the interface sharpens.

The significance lies in:

1. Generalizing Previous Results: It extends known periodic homogenization results to the general stationary ergodic setting without requiring quantitative mixing conditions, provided the scale separation is sufficient.
2. Identifying the Threshold: It explicitly identifies the probabilistic threshold (via $Sub$ and $Osc$ fields) that separates the homogenization regime from the rare events regime.
3. Unconditional Results for Structured Media: It demonstrates that for periodic and uniformly almost periodic media, the "rare events" phenomenon does not occur, and homogenization is robust regardless of the specific rate of scale separation (as long as $\delta \ll \epsilon$).

The authors emphasize that their results are specific to the scalar case and rely on techniques (like De Giorgi-Nash-Moser estimates) that may not be available in the vector-valued setting. The work serves as the first part of a two-paper series, with the companion paper [49] detailing the counterexamples and the mechanics of the rare events regime.