Large-scale harmonic measures and nontangential maximal functions in periodic homogenization

1. Problem Statement and Motivation

Context:
The paper investigates the boundary value problems for elliptic operators with rapidly oscillating periodic coefficients, defined as:
$$L_\varepsilon = -\nabla \cdot (A(x/\varepsilon)\nabla)$$
where $\varepsilon \ll \text{diam}(\Omega)$ is a microscopic scale, and $A$ is a real, symmetric, bounded, measurable matrix satisfying uniform ellipticity and periodicity conditions. The domain $\Omega$ is a bounded Lipschitz or $C^1$ domain.

The Challenge:
Classical regularity theory for elliptic operators (e.g., the solvability of Dirichlet, Neumann, and Regularity problems with $L^p$ estimates for nontangential maximal functions) typically requires smoothness assumptions on the coefficients (e.g., Hölder continuity). However, in periodic homogenization, the coefficients $A(x/\varepsilon)$ are merely bounded and measurable at the scale $\varepsilon$, lacking local smoothness.

Previous work by Kenig and Shen established uniform $L^p$ estimates for nontangential maximal functions ($N(u_\varepsilon)$ and $\widetilde{N}(\nabla u_\varepsilon)$) under the assumption that $A$ is smooth. The central question addressed in this paper is:

Can one establish uniform $L^p$ estimates for nontangential maximal functions for $L_\varepsilon$ without assuming any local smoothness on $A$, relying solely on the periodic structure?

The Approach:
The authors introduce the concept of large-scale estimates. They argue that while the coefficients are irregular at the $\varepsilon$-scale, the periodic structure induces regularity at scales $r \ge \varepsilon$ (large-scale Lipschitz estimates). The goal is to define appropriate "large-scale" maximal functions that measure solution behavior at a distance $\ge 10\varepsilon$ from the boundary and prove uniform estimates for these.

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2. Methodology and Key Tools

The proof strategy relies on a "two-scale" analysis, separating the problem into large-scale behavior (driven by periodicity) and small-scale behavior (driven by local smoothness, which is only invoked later for full-scale recovery).

A. Large-Scale Nontangential Maximal Functions

The authors define modified maximal functions that ignore the boundary layer of thickness $10\varepsilon$:

• $N_\varepsilon(F)(Q)$: The supremum of $|F(X)|$ for $X$ in the nontangential cone $\Gamma(Q)$ such that $\text{dist}(X, \partial\Omega) \ge 10\varepsilon$.
• $\widetilde{N}_\varepsilon(F)(Q)$: A modified version involving $L^2$ averages over balls $B(X, \delta(X)/2)$, also restricted to $\delta(X) \ge 10\varepsilon$.

B. Large-Scale $L_\varepsilon$-Harmonic Measures

A core innovation is the introduction of the large-scale harmonic measure $\omega^X_\varepsilon$. For a boundary point $Q$, it is defined by averaging the classical harmonic measure over a surface ball $\Delta_\varepsilon(Q)$ of radius $\varepsilon$:
$$\omega^X_\varepsilon(Q) := \frac{\omega^X_\varepsilon(\Delta_\varepsilon(Q))}{\sigma(\Delta_\varepsilon(Q))}$$
The authors prove that this measure satisfies a Reverse Hölder inequality (specifically, it is a $B_2$ weight) uniformly in $\varepsilon$. This is crucial for establishing $L^p$ estimates for the Dirichlet problem.

C. Difference Operators and Periodicity

For the Regularity and Neumann problems, the authors utilize a difference operator $Q_\varepsilon$:
$$Q_\varepsilon(u)(X) = \frac{u(X + \varepsilon e_d) - u(X)}{\varepsilon}$$
Due to the periodicity of $A$ in the transversal direction (after localization to graph domains), if $u_\varepsilon$ solves $L_\varepsilon u_\varepsilon = 0$, then $Q_\varepsilon(u_\varepsilon)$ also solves a similar equation. This allows the transfer of estimates from the solution to its difference quotient, effectively bypassing the lack of pointwise differentiability of the coefficients.

D. Large-Scale Rellich Estimates

The proofs rely heavily on large-scale Rellich estimates established in previous works ([16], [20]). These estimates bound the $L^2$ norm of the gradient in a boundary layer $\Omega_r$ (where $r \ge \varepsilon$) by the boundary data. The paper extends these to $L^p$ settings for $C^1$ domains using quantitative homogenization convergence rates and averaging operators ($M_\varepsilon$).

E. Real-Variable Arguments

The authors employ a large-scale version of the real-variable argument (originating from Caffarelli-Peral and refined by Shen). This technique allows the extension of estimates from $L^2$ to $L^p$ for $p > 2$ (via Reverse Hölder inequalities) and $p < 2$ (via interpolation with Hardy space atoms).

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3. Key Results

The paper establishes uniform $L^p$ estimates for the large-scale maximal functions for the Dirichlet, Neumann, and Regularity problems.

Theorem 1.1: Dirichlet Problem ($(D)_p$)

• Domain: Bounded Lipschitz domain.
• Result: For $2-\delta < p \le \infty$,
  $$\|N_\varepsilon(u_\varepsilon)\|_{L^p(\partial\Omega)} \le C \|S_\varepsilon(f)\|_{L^p(\partial\Omega)}$$
  where $S_\varepsilon(f)$ is a local maximal function of the boundary data.
• Refinement: If $\Omega$ is $C^1$, the range extends to $1 < p \le \infty$.

Theorem 1.2: Regularity Problem ($(R)_p$)

• Domain: Bounded Lipschitz domain.
• Result: For $1 < p < 2+\delta$,
  $$\|\widetilde{N}_\varepsilon(\nabla u_\varepsilon)\|_{L^p(\partial\Omega)} \le C \|f\|_{W^{1,p}(\partial\Omega)}$$
• Refinement: If $\Omega$ is $C^1$, the range extends to $1 < p < \infty$.

Theorem 1.3: Neumann Problem ($(N)_p$)

• Domain: Bounded Lipschitz domain.
• Result: For $1 < p < 2+\delta$,
  $$\|\widetilde{N}_\varepsilon(\nabla u_\varepsilon)\|_{L^p(\partial\Omega)} \le C \|g\|_{L^p(\partial\Omega)}$$
• Refinement: If $\Omega$ is $C^1$, the range extends to $1 < p < \infty$.

Corollary 1.4: Full-Scale Estimates

If the coefficients $A$ are Hölder continuous (providing small-scale regularity), the large-scale estimates can be combined with classical small-scale estimates to recover the full-scale nontangential maximal function estimates (i.e., without the $10\varepsilon$ cutoff).

• This recovers the results of Kenig and Shen for Lipschitz domains.
• Novelty: It provides the first full-scale $L^p$ estimates for $p \in (1, \infty)$ in $C^1$ domains for operators with merely bounded measurable periodic coefficients (once the small-scale smoothness is assumed).

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4. Significance and Impact

1. Removal of Smoothness Assumptions: The paper demonstrates that the periodic structure alone is sufficient to guarantee large-scale regularity (Lipschitz continuity and $L^p$ maximal function bounds) even when coefficients are discontinuous at the micro-scale. This is a fundamental step in understanding the regularity of homogenized solutions.
2. Optimal Ranges in $C^1$ Domains: The results establish the optimal range of $p$ ($1 < p < \infty$) for the Regularity and Neumann problems in $C^1$ domains for periodic operators, a result that was previously unknown for non-smooth coefficients.
3. Methodological Innovation: The introduction of large-scale harmonic measures and the systematic use of averaging operators ($M_\varepsilon$) to handle the lack of pointwise regularity provide a robust framework for future studies in homogenization and elliptic PDEs with rough coefficients.
4. Unified Framework: The paper unifies the treatment of Dirichlet, Neumann, and Regularity problems, showing that the large-scale behavior is governed by the same geometric and periodic principles, regardless of the specific boundary condition type.
5. Extension to Convex Domains: The authors note that their large-scale estimates hold in convex domains (and strongly quasiconvex domains), suggesting that the $C^1$ assumption is primarily needed for the small-scale recovery in Corollary 1.4, not the large-scale estimates themselves.

In summary, this paper bridges the gap between the rough coefficients of homogenization theory and the classical smooth theory of boundary value problems, proving that periodicity is a powerful substitute for local smoothness in establishing large-scale regularity.