Homogenization in fractional phase transitions at the critical scale

Technical Summary: Homogenization in Fractional Phase Transitions at the Critical Scale

Problem Statement
This paper investigates the asymptotic behavior of fractional phase-transition energies in the presence of periodic heterogeneities, specifically focusing on the critical scaling regime where the fractional order is $s = 1/2$. In the absence of heterogeneities, it is known that the energy scaling for $s=1/2$ involves a logarithmic factor ($1/(\varepsilon |\log \varepsilon|)$), leading to a sharp-interface limit independent of the specific double-well potential $W$. The central problem addressed is how this critical regime interacts with oscillatory coefficients $a(x/\delta, y/\delta)$ representing material heterogeneities, where $\delta = \delta(\varepsilon) \to 0$ as $\varepsilon \to 0$. The authors aim to determine whether standard homogenization effects survive at this scale or if the interaction between the critical scaling and the oscillations produces a novel effective behavior.

Methodology
The analysis is conducted within the framework of $\Gamma$-convergence with respect to $L^2$ convergence. The primary functional under study is:
$$F_\varepsilon(u) = \frac{1}{\varepsilon |\log \varepsilon|} \int_0^1 W(u(t)) dt + \frac{1}{|\log \varepsilon|} \int_0^1 \int_0^1 a\left(\frac{s}{\delta}, \frac{t}{\delta}\right) \frac{|u(s) - u(t)|^2}{|s-t|^2} ds dt$$
where $W$ is a double-well potential and $a$ is a continuous, strictly positive, 1-periodic coefficient.

The methodology relies on a discretization procedure and a multi-scale analysis of the energy contributions near jump points of the limit function. Key technical tools include:

1. Discretization and Index Sets: The domain is partitioned into intervals of size $\theta(\varepsilon)$. The authors define index sets ($P_1, P_2, P_3$) based on the measure of the sets where the function $u_\varepsilon$ is close to the potential wells ($-1$ or $1$). This allows for the separation of regions where the function is "stable" versus regions where transitions occur.
2. Scale Separation: The proof distinguishes between interactions occurring at scales much smaller than the oscillation period $\delta$ and those at scales much larger than $\delta$.
   • Small scales ($<\delta$): These interactions effectively sample the local values of the coefficient $a$.
   • Large scales ($>\delta$): These interactions average out the oscillations of $a$.
3. Discrete Rearrangement: To estimate the non-local interactions between disjoint sets of indices, the authors employ a discrete rearrangement lemma for non-increasing kernels, showing that minimal interaction is achieved when sets are arranged in opposite extremal positions.
4. Recovery Sequence Construction: For the $\Gamma$-limsup inequality, explicit recovery sequences are constructed. These sequences are designed to align the transition layers of $u_\varepsilon$ with specific points in the periodic cell of $a$ to minimize the energy, depending on the relative rate of convergence of $\delta$ and $\varepsilon$.

Key Results
Let $\lambda_0 = \lim_{\varepsilon \to 0} \frac{\log \delta}{\log \varepsilon}$ and $\lambda = \min\{\lambda_0, 1\}$. The main theorem establishes the $\Gamma$-limit of $F_\varepsilon$ as a sharp-interface functional defined on $BV((0,1); \{-1, 1\})$:
$$F(u) = 8 \left( (1-\lambda) a_{\min} + \lambda \bar{a} \right) \#S(u)$$
where:

• $\#S(u)$ is the number of jump points of $u$.
• $a_{\min} = \min_{t \in [0,1]} a(t,t)$ is the minimum value of the coefficient on the diagonal.
• $\bar{a} = \int_0^1 \int_0^1 a(s,t) ds dt$ is the mean value of the coefficient over the unit cell.

The result demonstrates a coexistence of homogenization and averaging effects. The effective surface energy density is a weighted combination of the minimum value of the oscillatory parameter and its mean value. The weights $(1-\lambda)$ and $\lambda$ correspond to the relative contributions of the small-scale (local/minimal) and large-scale (averaged) interactions, respectively.

Significance and Claims
The paper claims that this specific behavior—where the effective coefficient is a linear combination of the minimum and the mean—is a novel phenomenon specific to the critical logarithmic scaling ($s=1/2$).

• Contrast with Supercritical Regimes: In the supercritical case ($s > 1/2$), the limiting energy depends exclusively on either the minimum value or the mean value depending on the regime, but not a weighted coexistence of both.
• Contrast with Other Heterogeneous Problems: The authors note that in heterogeneous vortex problems, oscillations lead to homogenized coefficients rather than averaged ones. Similarly, in capacitary problems, the effective coefficient arises from a harmonic mean. The linear combination found here is distinct to the fractional phase-transition context at the critical scale.
• Independence from Potential: Consistent with the homogeneous critical case, the limit functional is independent of the specific shape of the double-well potential $W$, depending only on the geometry of the wells and the coefficient $a$.

The work provides the first extension of heterogeneous phase-transition results to the critical fractional regime, highlighting that the interaction between critical scaling and oscillations preserves a memory of both local minima and global averages in the effective energy.