Regularity theorems for random elliptic operators on domains

1. Problem Statement

The paper addresses a critical gap in the quantitative theory of stochastic homogenization: the lack of large-scale boundary regularity for random elliptic operators on general smooth domains and convex polytopes.

• Context: In stochastic homogenization, one studies the behavior of solutions $u_\varepsilon$ to the equation $-\nabla \cdot (a(x/\varepsilon)\nabla u_\varepsilon) = 0$ as $\varepsilon \to 0$, where $a$ is a random, stationary, and ergodic coefficient field. While large-scale interior regularity (e.g., $C^{1,\alpha}$ estimates) is well-established for the whole space $\mathbb{R}^d$, extending these results to bounded domains with Dirichlet boundary conditions has been an open problem.
• The Challenge: On general domains, the standard two-scale expansion $u_\varepsilon \approx \bar{u} + \varepsilon \phi_i(x/\varepsilon)\partial_i \bar{u}$ fails to satisfy the boundary conditions because the corrector $\phi_i$ does not vanish on the boundary. This necessitates the introduction of a boundary layer corrector $\theta^\varepsilon_i$, defined as the solution to a Dirichlet problem with oscillating boundary data.
• The Gap: Prior to this work, the theory was limited to Lipschitz estimates ($C^{0,1}$) on general smooth domains. A full $C^{1,\alpha}$ theory (which implies pointwise gradient bounds and excess decay) was only known for the half-space $\mathbb{H}^d_+$. The behavior near corners and edges (in polytopes) was also not fully understood in the stochastic setting.

2. Methodology

The authors employ a combination of quantitative stochastic homogenization techniques and deterministic PDE analysis. The core methodology involves:

1. Boundary Layer Analysis: The central technical hurdle is controlling the boundary layer corrector $\theta^\varepsilon_i$. The authors derive a suboptimal decay estimate for $\theta^\varepsilon_i$ away from the boundary. This is achieved via a "homogenization-infused Campanato iteration," comparing the heterogeneous boundary layer to a homogenized counterpart at large scales.
2. Flattening the Boundary: For general $C^{1,1}$ domains, the authors use a coordinate transformation to locally straighten the boundary, reducing the problem to the half-space case, while carefully tracking the errors introduced by the curvature.
3. Weighted Energy Estimates: They utilize a weighted Hardy inequality (Lemma 10) and a weighted Meyers estimate (Proposition 7). These tools allow for the control of gradients near the boundary and the derivation of fluctuation bounds.
4. Transfer of Regularity: The strategy follows the Avellaneda-Lin paradigm:
   • Establish regularity for the homogenized (constant-coefficient) operator.
   • Use the decay of the boundary layer corrector and the corrector estimates to "transfer" this regularity to the heterogeneous operator at scales $r \gg \varepsilon$.
5. Handling Polytopes: For convex polytopes, the authors analyze the singular behavior of harmonic functions near edges and vertices. They show that in the convex case, the singular contributions scale in a way that allows for a full $C^{1,\alpha}$ estimate (unlike non-convex cases where singularities dominate).

3. Key Contributions and Results

A. Suboptimal Decay of the Boundary Layer (Proposition 3)

The authors prove that the boundary layer corrector $\theta^\varepsilon_i$ decays away from the boundary. Specifically, for $x_0 \in O$:
$$|\varepsilon \nabla \theta^\varepsilon_i(x_0)| \lesssim C(a_\varepsilon, x_0) \left(1 + \frac{\text{dist}(x_0, \partial O)}{\varepsilon}\right)^{-1/3}$$
This result is crucial as it provides the necessary input for the regularity proofs, even though the decay rate ($1/3$) is suboptimal compared to the optimal rate ($1$) derived in their companion paper for the half-space.

B. $C^{1,\alpha}$ Regularity on Smooth Domains (Theorem 4)

The main result establishes a large-scale $C^{1,\alpha}$ regularity theory for random elliptic operators on bounded $C^{1,1}$ domains (or the half-space) with homogeneous Dirichlet boundary conditions.

• Excess Decay: For any solution $u$, the gradient can be approximated by a linear combination of the homogenized correctors and the boundary layer corrector, with an error that decays algebraically as the scale $r$ decreases relative to $R$.
• Mean Value Property: A large-scale mean value property for the gradient is established, uniform in $\varepsilon$.
• Pointwise Estimates: Combining large-scale regularity with small-scale Schauder estimates (requiring Hölder continuity of coefficients, Assumption A4), they derive pointwise gradient bounds:
  $$|\nabla u(x_0)| \leq C(a_\varepsilon, x_0) \left( \fint_{B_r(x_0) \cap O} |\nabla u|^2 \right)^{1/2}$$
  where the constant $C$ has stretched exponential moments and is independent of $\varepsilon$.

C. Regularity on Convex Polytopes (Theorem 8)

The authors extend the theory to convex polytopes.

• They prove that despite the presence of corners and edges, the gradient of $a_\varepsilon$-harmonic functions decays as one approaches the $(d-2)$-dimensional hyperedges.
• The estimate takes the form:
  $$|\nabla u(x_0)| \leq C \left(\frac{\text{dist}(x_0, E)}{\rho}\right)^\delta \left( \fint_{O \cap B_\rho(x_0)} |\nabla u|^2 \right)^{1/2}$$
  where $E$ is the set of hyperedges and $\delta > 0$ depends on the geometry. This result is significant for numerical homogenization methods (like the Local Orthogonal Decomposition method) which often use cubic domains.

D. Weighted Meyers Estimate (Proposition 7)

A new large-scale weighted Meyers estimate is proven for random elliptic operators on regular domains. This provides higher integrability for the gradient of solutions to non-homogeneous equations with weighted norms. This is a key auxiliary result used to derive fluctuation bounds in their companion paper [17].

4. Assumptions

The results rely on standard assumptions in quantitative stochastic homogenization:

• (A1) Uniform ellipticity and boundedness of the coefficient field $a$.
• (A2) Stationarity of the ensemble.
• (A3) Quantitative ergodicity via a spectral gap inequality (ensuring sufficient decorrelation).
• (A4) Hölder continuity of $a$ with stretched exponential moments (required for small-scale Schauder estimates and pointwise bounds).
• Domain: $C^{1,1}$ smoothness for the main theorems; convexity for the polytope results.
• Dimension: $d \geq 3$ (results for $d=2$ hold with logarithmic corrections).

5. Significance and Impact

• Completing the Theory: This work completes the large-scale regularity theory for stochastic homogenization by extending $C^{1,\alpha}$ results from the whole space and half-space to general smooth domains and convex polytopes.
• Numerical Homogenization: The results on convex polytopes are directly applicable to numerical methods (e.g., LOD, RVEs) where computational domains are typically cubes or polytopes. It justifies the use of these domains for capturing effective properties with optimal error rates.
• Fluctuation Theory: The derived estimates, particularly the weighted Meyers estimate and the boundary layer decay, are foundational for the companion paper [17], which establishes optimal fluctuation bounds for the boundary layer corrector. This is essential for understanding the statistical variance in homogenization errors.
• Methodological Advance: The proof techniques, specifically the handling of boundary layers in general domains via Campanato iteration and weighted Hardy inequalities, provide a robust framework for future analysis of boundary value problems in random media.

In summary, the paper bridges the gap between interior and boundary regularity in stochastic homogenization, providing the rigorous analytical tools necessary for high-precision numerical simulations and fluctuation analysis in complex geometries.