Homogenization in one-dimensional higher-order non-local models of phase transitions

Technical Summary: Homogenization in One-Dimensional Higher-Order Non-Local Models of Phase Transitions

Problem Statement
This paper investigates the asymptotic behavior (specifically $\Gamma$-convergence) of a class of one-dimensional functionals modeling phase transitions. These functionals generalize the classical Cahn–Hilliard energy by replacing the local gradient term with a higher-order non-local term involving fractional derivatives, while simultaneously introducing a rapidly oscillating coefficient (homogenization kernel).

The functionals are defined on the fractional Sobolev space $H^{k+s}((0, 1))$ with $k \in \mathbb{N}$, $s \in (0, 1)$, and $k+s > 1/2$. The energy functional $F_{\varepsilon, \delta}$ takes the form:
$$F_{\varepsilon, \delta}(u) = \frac{1}{\varepsilon} \int_0^1 W(u) \, dx + \varepsilon^{2(k+s)-1} \int_0^1 \int_0^1 a\left(\frac{x}{\delta}, \frac{y}{\delta}\right) \frac{|u^{(k)}(x) - u^{(k)}(y)|^2}{|x-y|^{1+2s}} \, dx \, dy$$
where $W$ is a double-well potential (minima at $\pm 1$) and $a$ is a 1-periodic homogenization kernel. The study focuses on the limit as $\varepsilon, \delta \to 0$, analyzing the interplay between the transition layer scale ($\varepsilon$) and the oscillation scale ($\delta$).

Methodology
The authors employ the calculus of variations, specifically $\Gamma$-convergence theory, to derive sharp-interface limits. The methodology relies on several key technical components:

1. Scaling and Rescaling: The analysis involves rescaling the functionals to identify the effective surface tension in different regimes. The authors define rescaled functionals $\Phi_a$ to study optimal profile problems on the real line.
2. Compactness and Coercivity: A significant portion of the work establishes equicoercivity and improved coercivity properties for sequences with bounded energy. Unlike standard local problems, the non-local nature requires careful handling of the "tails" of the functions. The authors prove that bounded energy sequences are bounded in $H^{k+s}_{loc}$ and that their $k$-th derivatives are bounded in $L^2$.
3. Approximation Lemmas: To handle the non-local interactions, the authors develop specific modification lemmas:
   • Flattening Lemma: Allows modifying a function near the minima of the potential $W$ to be exactly constant without significantly increasing the energy. This is crucial for extending functions to infinity and isolating transition layers.
   • Vanishing Tails Lemma: Controls the energy contribution of the function's "tails" (far from the transition layer), ensuring that truncating the domain does not alter the asymptotic energy significantly.
4. Regime Analysis: The proof distinguishes three distinct regimes based on the ratio $\lambda = \lim_{\varepsilon \to 0} \delta/\varepsilon$:
   • Subcritical ($\delta/\varepsilon \to +\infty$): The oscillation scale is much larger than the transition layer.
   • Supercritical ($\delta/\varepsilon \to 0$): The oscillation scale is much smaller than the transition layer.
   • Critical ($\delta/\varepsilon \to \lambda \in (0, \infty)$): The scales are comparable.

Key Contributions and Results
The paper identifies three distinct regimes for the $\Gamma$-limit, each converging to a sharp-interface functional of the form:
$$F(u) = \sum_{\omega \in \{+1, -1\}} m^\omega_\lambda \# S_\omega(u)$$
where $S_\omega(u)$ denotes the set of jumps of direction $\omega$, and $m^\omega_\lambda$ is the effective surface tension (transition energy).

1. Critical Regime ($\delta \sim \varepsilon$):
   When $\delta/\varepsilon \to \lambda \in (0, \infty)$, the surface tension $m^\omega_\lambda$ is given by a variational problem where the oscillating kernel $a$ is evaluated at the scale $\lambda$:
   $$m^\omega_\lambda = \inf \left\{ \int_{\mathbb{R}} W(u) dx + \iint_{\mathbb{R}^2} a\left(\frac{x}{\lambda}, \frac{y}{\lambda}\right) \frac{|u^{(k)}(x) - u^{(k)}(y)|^2}{|x-y|^{1+2s}} dx dy \right\}$$
   This represents a competition between the oscillations and the transition layer, with no separation of scales.

2. Supercritical Regime ($\delta \ll \varepsilon$):
   When $\delta/\varepsilon \to 0$, the oscillations are averaged out before the transition layer forms. The surface tension involves the arithmetic mean of the kernel, $\bar{a} = \iint_{(0,1)^2} a(x,y) dx dy$:
   $$m^\omega_0 = (\bar{a})^{\frac{1}{2(k+s)}} m^\omega$$
   where $m^\omega$ is the transition energy for the homogeneous case ($a \equiv 1$). The authors highlight a sharp contrast with the local gradient case (where the harmonic mean appears); here, the arithmetic mean arises due to the non-local nature of the fractional derivative.

3. Subcritical Regime ($\delta \gg \varepsilon$):
   When $\delta/\varepsilon \to +\infty$, assuming $a$ is continuous, the transition layer is confined to a scale $\varepsilon$ where the kernel $a$ appears constant. The surface tension is determined by the minimum of the kernel on the diagonal:
   $$m^\omega_\infty = (\inf_{t \in \mathbb{R}} a(t,t))^{\frac{1}{2(k+s)}} m^\omega$$
   This result demonstrates a "separation-of-scales" effect and an additional concentration on the diagonal $x=y$, distinct from the local case.

Significance and Claims
The paper claims to extend the existing theory of phase transitions from local gradient models to higher-order non-local models with oscillating coefficients.

• Generalization: It generalizes previous results by Savin, Valdinoci, and Solci (who treated the homogeneous case) and Ansini et al. (who treated local inhomogeneities) to the fractional, higher-order, and inhomogeneous setting.
• Non-Local vs. Local Distinction: The authors emphasize that the homogenization behavior in the supercritical regime differs fundamentally from the local gradient case. While local models yield a harmonic mean for the effective coefficient, the non-local fractional models yield an arithmetic mean. This discontinuity is attributed to the singular nature of minimum problems as $s \to 0$ or $s \to 1$.
• Technical Rigor: The work provides a rigorous proof of $\Gamma$-convergence in the one-dimensional setting, explicitly handling the technical difficulties arising from the non-local double integrals and the lack of compactness in the fractional setting. The authors note that the one-dimensional restriction is necessary to avoid superposing the difficulties of fractional settings with the technicalities of homogenization on interfaces in higher dimensions.

The paper does not propose new applications or experimental setups but serves as a theoretical foundation for understanding how non-local interactions and micro-oscillations jointly influence phase transition energies.