Uniform Calderón-Zygmund estimates in multiscale elliptic homogenization

Technical Summary: Uniform Calderón-Zygmund Estimates in Multiscale Elliptic Homogenization

Problem Statement
This paper addresses the uniform regularity of solutions to linear elliptic equations in divergence form within a bounded $C^1$ domain $\Omega \subset \mathbb{R}^d$:
$$-\text{div}(A_\varepsilon \nabla u_\varepsilon) = \text{div} f,$$
where the coefficient matrix $A_\varepsilon(x)$ exhibits $n$ distinct oscillating scales:
$$A_\varepsilon(x) = A\left(\frac{x}{\varepsilon_1}, \frac{x}{\varepsilon_2}, \dots, \frac{x}{\varepsilon_n}\right).$$
Here, $A(y_1, \dots, y_n)$ is 1-periodic in each variable $y_i$, and the parameters $(\varepsilon_1, \dots, \varepsilon_n) \in (0, 1]^n$ are small. The primary challenge is to establish uniform $L^p$ bounds for the gradient $\nabla u_\varepsilon$ (the Calderón-Zygmund estimate) where the constant is independent of the small parameters, even when the scales are not well-separated (i.e., the ratios $\varepsilon_{i+1}/\varepsilon_i$ do not necessarily satisfy a power-law separation condition).

Previous work established uniform Hölder ($C^\alpha$) estimates for such systems without scale separation using compactness methods. However, gradient estimates ($L^p$ bounds) typically require quantitative convergence rates or fine properties of correctors, which were previously unavailable in the absence of scale separation.

Methodology
The authors develop a novel proof strategy that combines number theory with homogenization techniques to overcome the lack of scale separation. The core methodology involves three main components:

1. Dirichlet's Theorem and Reperiodization:
   The authors utilize Dirichlet's theorem on simultaneous Diophantine approximation to quantitatively separate the smallest scale from the others. Given a sequence of scales and a large parameter $Q$, they construct a new 1-periodic matrix $A^\sharp$ via a "reperiodization" technique. This transformation rewrites the original coefficient $A_\varepsilon$ as:
   $$A\left(\frac{x}{\varepsilon_1}, \dots, \frac{x}{\varepsilon_n}\right) = A^\sharp\left(\frac{x}{\varepsilon'_1}, \dots, \frac{x}{\varepsilon'_n}\right),$$
   where the new scales satisfy a separation condition: the smallest scale $\varepsilon'_n$ is $Q$-separated from the remaining $n-1$ scales (i.e., $\varepsilon'_i \ge Q \varepsilon'_n$). While this process may degrade the regularity of $A^\sharp$ in the last variable, it allows for the application of reiterated homogenization.

2. Quantitative Scale Reduction:
   Using the separation property established above, the authors perform a quantitative scale reduction. They prove that the solution to the $n$-scale problem can be uniformly approximated by the solution to a problem with at most $n-1$ scales. This approximation involves a corrector term derived from reiterated homogenization theory. The error in this approximation is controlled by powers of $Q^{-1}$ and the ratio of the smallest scale to the domain size.

3. Real-Variable Argument with Double Averaging:
   To establish the $L^p$ gradient estimates, the authors employ a refined real-variable argument (based on the framework of Caffarelli and Peral). A critical innovation is the use of a "double-averaging" technique.

   • The first averaging is taken at a large scale to handle the approximation error from the scale reduction.
   • The second averaging is taken at the scale of the periodic cell (dependent on $Q$) to exploit the periodic structure of the correctors.
     This dual approach allows the authors to bypass the need for uniform $L^p$ bounds on the gradients of the correctors in the reperiodized matrix (which would otherwise depend on $Q$ and blow up), relying instead on $L^2$ energy estimates and the periodicity of the correctors.

Key Results

• Theorem 1.1 (Main Result): For any $p \in (1, \infty)$, the gradient $\nabla u_\varepsilon$ satisfies the uniform estimate:
  $$\|\nabla u_\varepsilon\|_{L^p(\Omega)} \le C \|f\|_{L^p(\Omega)},$$
  where the constant $C$ depends only on the dimension $d$, the number of scales $n$, the ellipticity constant $\Lambda$, $p$, the Hölder continuity parameters $(\tau, L)$, and the domain $\Omega$. Crucially, $C$ is independent of the parameters $(\varepsilon_1, \dots, \varepsilon_n)$. This result holds for both Dirichlet and compatible Neumann boundary conditions.

• Corollary 1.2 (Quasiperiodic Homogenization): The main theorem implies uniform Calderón-Zygmund estimates for quasiperiodic coefficients $A(x/\varepsilon) = B(Mx/\varepsilon)$ without requiring the Diophantine condition on the matrix $M$. This resolves a gap in the literature where such estimates were previously known only under strict Diophantine conditions.

• Large-Scale Lipschitz Estimates: The paper also derives uniform large-scale or mesoscopic-scale Lipschitz estimates. For two scales ($n=2$), a Lipschitz estimate holds for radii $r \ge \varepsilon_2^{1-\alpha}$. For $n \ge 3$, similar estimates are obtained near any given scale, though a full-scale uniform Lipschitz estimate without scale separation remains an open problem.

Significance and Claims
The authors claim that this work provides the first proof of uniform $L^p$ gradient estimates for elliptic equations with arbitrary multiscale coefficients without any scale-separation assumptions.

• Novelty: The introduction of Dirichlet's theorem into homogenization theory is presented as a new tool that allows for the quantitative separation of scales where traditional geometric separation fails.
• Overcoming Limitations: The paper explicitly notes that soft compactness methods (sufficient for $C^\alpha$ estimates) are insufficient for $L^p$ gradient estimates. The proposed quantitative approach, which explicitly computes constants and utilizes correctors, is necessary to bridge this gap.
• Generality: The results significantly broaden the scope of homogenization theory to include quasiperiodic media with degenerate periodic cells and resonant frequencies (e.g., Liouville numbers) without imposing restrictive arithmetic conditions on the frequencies.
• Mechanism: The work suggests that the "averaging effect" in multiscale media occurs stably across all mesoscopic scales, even in the absence of a deterministic homogenized limit with a fixed rate, providing a deeper understanding of the regularity of solutions in complex media.