Multiscale analysis and homogenization of nonlocal thin films

Technical Summary: Multiscale Analysis and Homogenization of Nonlocal Thin Films

Problem Statement
This paper addresses the asymptotic analysis of nonlocal variational models for thin films. The authors consider a thin domain $\Omega_\gamma := \omega \times (-\gamma, \gamma)$, where $\omega \subset \mathbb{R}^{d-1}$ is the cross-section and $\gamma > 0$ represents the thickness. The energy functionals are of convolution type, defined on this thin domain, where the interaction range between material points is governed by a parameter $\varepsilon > 0$.

The central challenge lies in the interplay between two vanishing scales: the thickness of the domain ($\gamma \to 0$) and the nonlocal interaction horizon ($\varepsilon \to 0$). Unlike local models, where dimension reduction is well-understood via $\Gamma$-convergence, nonlocal energies do not a priori scale proportionally to the volume of the domain. The paper seeks to identify the correct scaling factor, establish a general $\Gamma$-convergence result to a local integral functional on the lower-dimensional cross-section, and characterize the limit energy density under various regimes of the ratio $\varepsilon/\gamma$.

Methodology
The analysis relies on the calculus of variations, specifically the theory of $\Gamma$-convergence. The methodology proceeds through the following steps:

1. Scaling Identification: Through heuristic and rigorous computations, the authors determine that the appropriate scaling factor for the nonlocal energy is $\max\{\varepsilon/\gamma^2, 1/\gamma\}$. This scaling accounts for the geometric constraints of the thin domain on the interaction neighborhood, distinguishing between regimes where the thickness is larger or smaller than the interaction horizon.
2. Rescaling and Compactness: The problem is reformulated on a fixed domain $\Omega = \omega \times (-1, 1)$ via a change of variables. The authors establish a compactness result (Theorem 3.6) for sequences with uniformly bounded energy. This involves separate controls of planar and vertical interactions, proving that limit functions are independent of the vertical variable and belong to a Sobolev space $W^{1,p}$.
3. Integral Representation: Using the localization method of $\Gamma$-convergence, the authors prove that the $\Gamma$-limit of the rescaled functionals is a local integral functional of the form $\int_\omega f_0(x_\alpha, \nabla_\alpha v) dx_\alpha$. The effective density $f_0$ is shown to be quasiconvex.
4. Periodic Homogenization: The study extends to the case where the energy density is periodic in the planar variables. The authors analyze three distinct regimes based on the limit of the ratio $\delta = \lim_{\varepsilon \to 0} \varepsilon/\gamma$:
   • Proportional Scales ($\delta \in (0, \infty)$): The thickness and interaction range vanish at the same rate.
   • Nonlocal Dominance ($\delta = 0$, i.e., $\varepsilon \ll \gamma$): The interaction range vanishes faster than the thickness.
   • Dimension Reduction Dominance ($\delta = \infty$, i.e., $\gamma \ll \varepsilon$): The thickness vanishes faster than the interaction range.

Key Contributions and Results

• General $\Gamma$-Compactness: Theorem 4.1 establishes that the family of nonlocal thin-film functionals $\Gamma$-converges to a local integral functional defined on the cross-section, regardless of the specific relationship between $\varepsilon$ and $\gamma$, provided suitable growth and decay conditions on the interaction kernels are met.
• Homogenization Formulas:
  • In the proportional regime ($\varepsilon = \delta \gamma$), the limit energy density $f^\delta_{\text{hom}}$ is characterized by an asymptotic formula involving a minimization over periodic cells (Theorem 5.1). In the convex case, this reduces to a cell formula (Theorem 5.2).
  • In the separation of scales regimes, the paper investigates whether the limit can be obtained by performing two successive $\Gamma$-limits (first nonlocal-to-local, then dimension reduction, or vice versa).
• Separation of Scales:
  • Case $\varepsilon \ll \gamma$: Under additional symmetry assumptions on the kernel (specifically independence from the vertical coordinate and evenness in the vertical interaction variable), Proposition 6.2 proves that the limit density coincides with the result of performing a nonlocal-to-local homogenization followed by dimension reduction. Specifically, $f^0_{\text{hom}}(M) = 2Q f_{\text{bulk}}(M)$, where $Q$ denotes quasiconvexification and $f_{\text{bulk}}$ is the full-dimensional homogenized density.
  • Case $\gamma \ll \varepsilon$: Under convexity and independence from vertical variables, Proposition 6.4 shows that the limit density reduces to a $(d-1)$-dimensional cell formula, effectively decoupling the vertical dimension entirely.
• Non-Commutativity of Limits: A crucial finding is presented in Example 7.3, which demonstrates that, in general, the nonlocal-to-local limit and the dimension reduction procedure do not commute. The order in which the limits are taken can yield different effective energy densities, highlighting the complexity of the multiscale interaction in nonlocal thin films.

Significance and Claims
The paper claims to provide the first systematic variational framework for the dimension reduction of nonlocal thin films. Its significance lies in:

1. Flexibility: Unlike discrete models, the nonlocal approach allows for a continuum of interaction lengths without a predetermined minimal scale, making it suitable for numerical simulations and modeling materials with long-range interactions.
2. Rigorous Scaling: It resolves the ambiguity regarding the correct scaling of nonlocal energies in thin domains, showing that the scaling depends critically on the relative magnitude of the thickness and the interaction horizon.
3. Multiscale Interplay: The work clarifies the conditions under which "separation of scales" occurs. It shows that while such separation is possible under specific structural assumptions (e.g., convexity, symmetry), it is not a universal property of nonlocal thin films. The example of non-commuting limits serves as a cautionary result for modeling strategies that assume the order of limits is irrelevant.

The authors conclude that their results bridge the gap between the well-established local theory of thin films and the emerging theory of nonlocal functionals, offering a robust mathematical foundation for analyzing materials where nonlocal effects and geometric thinness coexist.